# Amalgamation in classes of involutive commutative residuated lattices

Sándor Jenei Institute of Mathematics and Informatics, University of Pécs,
Ifjúság u. 6, H-7624 Pécs, Hungary

###### Abstract.

The amalgamation property and its variants are in strong relationship with
various syntactic interpolation properties of substructural logics, hence its
investigation in varieties of residuated lattices is of particular interest.
The amalgamation property is investigated in some classes of non-divisible,
non-integral, and non-idempotent involutive commutative residuated lattices in
this paper. It is proved that the classes of odd and even totally ordered,
involutive, commutative residuated lattices fail the amalgamation property. It
is also proved that their subclasses formed by their idempotent-symmetric
algebras have the amalgamation property but fail the strong amalgamation
property. Finally, it is shown that the variety of semilinear, idempotent-
symmetric, odd, involutive, commutative residuated lattices has the
amalgamation property, and hence also the transferable injections property.

###### Key words and phrases:

Involutive residuated lattices, representation, ordered abelian groups,
amalgamation

###### 2010 Mathematics Subject Classification:

Primary 97H50, 20M30; Secondary 06F05, 06F20, 03B47.

The present scientific contribution was supported by the GINOP
2.3.2-15-2016-00022 grant and the Higher Education Institutional Excellence
Programme 20765-3/2018/FEKUTSTRAT of the Ministry of Human Capacities in
Hungary.

##  1\. Introduction

Varieties of (pointed) residuated lattices are algebraic counterparts of
substructural logics [5]. In this paper the amalgamation property is
investigated in classes of involutive commutative residuated lattices.
Historically, amalgamations were first considered for groups in the form of
amalgamated free products [17]. In residuated lattices the amalgamation
property has been analyzed mostly in varieties in which the algebras are
linear or semilinear, i.e., subdirect products of linearly ordered ones, or
conic or semiconic. In addition, the investigated classes in the literature
have mostly been either divisible and integral [14] or idempotent [1, 6]. The
scope of the present paper is investigating the amalgamation property in some
varieties of residuated lattices which are neither divisible nor integral nor
idempotent.
We just mention here a few of the most recent related developments (see [5,
14] and the references therein for a more complete picture). As shown in [7],
the varieties of semilinear residuated lattices and semilinear cancellative
residuated lattices fail the amalgamation property. There is a comprehensive
account of the interrelationships between algebraic properties of varieties
and properties of their free algebras and equational consequence relations,
including equivalences between different kinds of amalgamation properties and
the Robinson property, the congruence extension property, and the deductive
interpolation property in [14]. Also, the presence or failure of amalgamation
is established for some subvarieties of residuated lattices, notably for all
subvarieties of commutative GMV-algebras. As shown in [6], the variety
generated by totally ordered commutative idempotent residuated lattices has
the amalgamation property, and even an example of a non-commutative variety of
idempotent residuated lattices that has the amalgamation property has been
presented.

The amalgamation property is quite rare for general varieties. In the present
paper for two classes of involutive (pointed) residuated lattices, namely, the
classes of odd and even involutive totally ordered pointed commutative
residuated lattices, we show the failure of the amalgamation property. An
insight is gained, too: in these classes amalgamation fails for the same
reason as the reason of the failure of the amalgamation property for discrete
linearly ordered groups with positive normal homomorphisms [9]. On that basis,
we consider two subclasses which contain only algebras which are idempotent-
symmetric (to be defined here), and show that those two classes have the
amalgamation property. The proofs make use of a representation theorem, in
which odd or even totally ordered involutive commutative pointed residuated
lattices are represented as direct systems of abelian $o$-groups equipped with
some further structure [12] (see Theorem 2.5 below). After some preliminaries
on residuated lattices, amalgamation, and direct systems in Section 2, the
main construction and results are presented in Section 5. Combining our main
result in Theorem 5.3 with previous results, a strengthening of the
amalgamation property, the transferable injection property can be deduced for
the variety generated by the class of odd, involutive, idempotent-symmetric,
commutative residuated chains (Corollary 5.5). Our main result yields, as a
by-product, a new algebraic proof for amalgamation in the classes of odd and
even totally ordered Sugihara monoids (Corollary 5.4).

##  2\. Preliminaries

Algebras will be denoted by bold capital letters, their underlying sets by the
same regular letter unless otherwise stated. Let $\mathbf{X}=(X,\leq)$ be a
poset. For $x\in X$ define the upper neighbor $x_{\uparrow}$ of $x$ to be the
unique cover of $x$ if such exists, and $x$ otherwise. Define $x_{\downarrow}$
dually. A partially ordered algebra with a poset reduct will be called
discretely ordered if for any element $x$, $x_{\downarrow}<x<x_{\uparrow}$
holds. An FLe-algebra111Other terminologies for FLe-algebras are: pointed
commutative residuated lattices or pointed commutative residuated lattice-
ordered monoids. is a structure
$\mathbf{X}=(X,\wedge,\vee,{\mathbin{*\mkern-9.0mu\circ}},\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}},t,f)$
such that $(X,\wedge,\vee)$ is a lattice,
$(X,\leq,{\mathbin{*\mkern-9.0mu\circ}},t)$ is a commutative residuated
monoid, and $f$ is an arbitrary constant, called the falsum constant.
Commutative residuated lattices are the $f$-free reducts of FLe-algebras.
Sometimes the lattice operators will be replaced by their induced ordering
$\leq$ in the signature, in particular, if an FLe-chain is considered (that
is, when the order is total). Being residuated means that there exists a
binary operation $\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}$,
called the residual operation of ${\mathbin{*\mkern-9.0mu\circ}}$, such that
${x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{y}\leq z$ if and only if
${x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{z}\geq
y$. This equivalence is called adjointness condition,
(${\mathbin{*\mkern-9.0mu\circ}},\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}$)
is called an adjoint pair. Equivalently, for any $x,z$, the set $\\{v\ |\
{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{v}\leq z\\}$ has its greatest
element, and
${x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{z}$, the
residuum of $x$ and $z$, is defined as this element:
${x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{z}:=\max\\{v\
|\ {x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{v}\leq z\\}$; this is called
the residuation condition. Being residuated implies that
${\mathbin{*\mkern-9.0mu\circ}}$ is lattice ordered. One defines the residual
complement operation by
${x}{{}^{\prime}}={x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{f}$
and calls an FLe-algebra involutive if ${({x}{{}^{\prime}})}{{}^{\prime}}=x$
holds. In this case
${x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{y}={({x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{y}{{}^{\prime}}})}{{}^{\prime}}$
holds, and

(2.1) for $x\geq t$,
${x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}\leq x$

holds if the order is total [12, Lemma 2.1]. Call the elements $x\geq t$
positive. An involutive FLe-algebra is odd (or rank $0$) if the residual
complement operation leaves the unit element fixed, that is,
${t}{{}^{\prime}}=t$, and even (or rank $1$) if the unit element is the unique
cover of its residual complement. In our subsequent discussion there are a few
classes of algebras which will play a significant role. These are given
distinguished notation as listed below.

$\begin{array}[]{ll}\mathfrak{C}&\mbox{the class of totally ordered sets
(chains)}\\\ \mathfrak{A}^{\mathfrak{c}}&\mbox{the class of abelian
$o$-groups}\\\ \mathfrak{A}^{\mathfrak{l}}&\mbox{the class of abelian
$\ell$-groups}\\\ \mathfrak{A}^{\mathfrak{c}\mathfrak{d}}&\mbox{the class of
discrete abelian $o$-groups}\\\ \mathfrak{I}&\mbox{the class of involutive
FL${}_{e}$-algebras}\\\ \end{array}$

Adjunct to $\mathfrak{I}$,

  * •

the superscript $\mathfrak{c}$ means restriction to totally ordered algebras,

  * •

the superscript $\mathfrak{sl}$ (semilinear) means restriction to algebras
which are subdirect products of totally ordered ones,

  * •

the subscript $\mathfrak{1}$ means restriction to even (that is, rank $1$)
algebras,

  * •

the subscript $\mathfrak{0}$ means restriction to odd (that is, rank $0$)
algebras,

  * •

the subscript $\mathfrak{symm}$ means restriction to those algebras in which
${x}{{}^{\prime}}$ is idempotent whenever $x$ is idempotent. Call these
algebras idempotent-symmetric222It takes a half-line proof to show that in
such an algebra the residual complement of every negative idempotent element
is a positive idempotent element. Therefore, such an algebra is idempotent-
symmetric if and only the following holds true: $x$ is idempotent if and only
if ${x}{{}^{\prime}}$ is idempotent. So this property captures that the set of
idempotent elements is symmetric with respect to $t$..

For instance $\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$
refers to the class of odd involutive, idempotent-symmetric, FLe-chains.

###  2.1. Amalgamation

Let $\mathfrak{U}$ be a class of algebras. Call
$\langle\mathbf{A},\mathbf{B}_{1},\mathbf{B}_{2},\iota_{1},\iota_{2}\rangle$ a
V-formation in $\mathfrak{U}$, if
$\mathbf{A},\mathbf{B}_{1},\mathbf{B}_{2}\in\mathfrak{U}$ and $\iota_{k}$ is
an embedding of $\mathbf{A}$ into $\mathbf{B}_{k}$ ($k=1,2$). The V-formation
can be amalgamated in $\mathfrak{U}$ if there exists a triple
$\langle\mathbf{C},\iota_{3},\iota_{4}\rangle$, called an amalgam of the
V-formation, such that $\mathbf{C}\in\mathfrak{U}$, $\iota_{k}$ is an
embedding of $\mathbf{B}_{k}$ into $\mathbf{C}$ ($k=3,4$) such that
$\iota_{3}\circ\iota_{1}=\iota_{4}\circ\iota_{2}$, see Fig. 17.

Figure 1. Amalgamation

If every V-formation in $\mathfrak{U}$ can be amalgamated in $\mathfrak{U}$,
then $\mathfrak{U}$ is said to have the amalgamation property. The triple is
called a strong amalgamation of the V-formation if it amalgamates in such a
way that $\iota_{3}(b_{1})=\iota_{4}(b_{2})$ implies $b_{1}\in\iota_{1}(A)$
and $b_{2}\in\iota_{2}(A)$. If all V-formations in $\mathfrak{U}$ can be
strongly amalgamated in $\mathfrak{U}$, then $\mathfrak{U}$ is said to have
the strong amalgamation property. In order to keep the complexity of notation
on a reasonable level, without loss of generality, we will always assume in
the sequel that $\iota_{1}$ and $\iota_{2}$ are inclusion maps.
It is known that the classes $\mathfrak{A}^{\mathfrak{c}}$ and
$\mathfrak{A}^{\mathfrak{l}}$ have the amalgamation property [15, 16], and the
class $\mathfrak{C}$ has the strong amalgamation property [16, Lemma 2.2]. One
way of constructing an amalgam, in general, can be to consider the amalgamated
free product (if it exists). For instance, for $\mathfrak{A}^{\mathfrak{l}}$
it was shown in [20, Theorem 12.2.1], that for
$\textbf{{G}},\textbf{{K}},\textbf{{M}}\in\mathfrak{A}^{\mathfrak{l}}$ with
embeddings $\iota_{1}:\textit{G}\to\textit{K}$ and
$\iota_{2}:\textit{G}\to\textit{M}$, there exists the free product in
$\mathfrak{A}^{\mathfrak{l}}$ of K and M with G amalgamated, denoted by
$\textbf{{K}}\ast_{\textbf{{G}}}\textbf{{M}}$, which is an algebra in
$\mathfrak{A}^{\mathfrak{l}}$ such that

  1. (P1)

there exists embeddings $\iota_{3}$ and $\iota_{4}$ which make the upper part
of the diagram in Fig. 2 commute, and

  2. (P2)

for every $\textbf{{P}}\in\mathfrak{A}^{\mathfrak{c}}$ and homomorphisms
$f_{1}:\textbf{{K}}\to\textbf{{P}}$ and $f_{2}:\textbf{{M}}\to\textbf{{P}}$
which make the outer square commute, there is a unique homomorphism
$f:\textbf{{K}}\ast_{\textbf{{G}}}\textbf{{M}}\to\textbf{{P}}$ which makes the
two triangles in the lower part of the diagram in Fig. 2 commute.

Figure 2. Amalgamated free product in $\mathfrak{A}^{\mathfrak{l}}$

As with all universal constructions, the amalgamated free product, if it
exists, is unique up to a unique isomorphism. As shown in [20, Theorem
12.2.2], one way of constructing an amalgam in $\mathfrak{A}^{\mathfrak{c}}$
of a V-formation in $\mathfrak{A}^{\mathfrak{c}}$ is to consider its
amalgamated free product in $\mathfrak{A}^{\mathfrak{l}}$ first, and then to
extend its lattice order to a total one by the Szpilrajn extension theorem
[18]. Since the extension can, in general, be done in more than one ways, the
obtained algebra is not unique, and thus there cannot exists the free product
in $\mathfrak{A}^{\mathfrak{c}}$ of K and M with G amalgamated. This will make
our construction somewhat more involved, see the part between (5.3) and (5.5).

###  2.2. Direct systems and direct limits in $\mathfrak{A}^{\mathfrak{c}}$

A directed partially ordered set is a nonempty set together with a reflexive,
antisymmetric, and transitive binary relation, with the additional property
that every pair of elements has an upper bound. Let
$\langle\alpha,\leq\rangle$ be a directed partially ordered set. Let
$\\{\mathbf{A}_{i}\in\mathfrak{U}:i\in\alpha\\}$ be a family of algebras of
the same type and $f_{i\to j}$ be a homomorphism333Homomorphisms are
understood in the corresponding setting. for every $i,j\in\alpha$, $i\leq j$
with the following properties:

  1. (D1)

$f_{i\to i}$ is the identity of $\mathbf{A}_{i}$, and

  2. (D2)

$f_{i\to k}=f_{j\to k}\circ f_{i\to j}$ for all $i\leq j\leq k$.

Then $\langle\mathbf{A}_{i},f_{i\to j}\rangle$ is called a direct system of
algebras in $\mathfrak{U}$ over $\alpha$.

Sometimes we also write $\langle\mathbf{A}_{i},f_{i\to j}\rangle_{\alpha}$.
The direct limit of $\langle\mathbf{A}_{i},f_{i\to j}\rangle$ is an algebra in
$\mathfrak{U}$, denoted by $\underset{\longrightarrow}{\lim}\,\mathbf{A}_{i}$,
along with canonical homomorphisms
$\pi_{i}:\mathbf{A}_{i}\to\underset{\longrightarrow}{\lim}\,\mathbf{A}_{i}$
such that for all $i\leq j$, $\pi_{j}\circ f_{i\to j}=\pi_{i}$, and such that
the following universal property holds: if $\mathbf{A}\in\mathfrak{U}$ and for
every $i\in\alpha$, $\sigma_{i}:\mathbf{A}_{i}\to\mathbf{A}$ is a
homomorphisms satisfying $\sigma_{j}\circ f_{i\to j}=\sigma_{i}$ (for $i\leq
j$) then there exists a unique homomorphism
$\phi:\underset{\longrightarrow}{\lim}\,\mathbf{A}_{i}\to\mathbf{A}$ such that
for every $i\in\alpha$, $\phi\circ\pi_{i}=\sigma_{i}$ holds. The direct limit
can be represented as follows (also in case of general algebras, see [19,
Exercise 1.4.23]). Its universe is the disjoint union of the $A_{i}$’s modulo
the following equivalence relation: if $x_{i}\in A_{i}$ and $x_{j}\in A_{j}$
then $x_{i}\sim x_{j}$ if there exists $k\in\alpha$, $i,j\leq k$ such that
$f_{i\to k}(x_{i})=f_{j\to k}(x_{j})$, the canonical homomorphism $\pi_{i}$
sends each element to its equivalence class, and the algebraic operations on
$\underset{\longrightarrow}{\lim}\,\mathbf{A}_{i}$ are defined such that these
maps become homomorphisms. It is easy to see that the direct limit, if exists,
is unique up to isomorphism, and that the direct limit of
$\langle\mathbf{A}_{i},f_{i\to j}\rangle_{\alpha}$ coincides with the direct
limit of $\langle\mathbf{A}_{i},f_{i\to j}\rangle_{\beta}$ for any cofinal
subset $\beta$ of $\alpha$.
Direct limits exist in $\mathfrak{U}=\mathfrak{A}$ [3, Section 11]. Direct
limits exist in the class of partially ordered abelian groups with positive
homomorphisms444Positive homomorphisms are abelian group homomorphisms that
map positive elements to positive elements. Equivalently, they are order
preserving: $x\leq y$ implies $f(x)\leq f(y)$., too: Let
$\langle\mathbf{A}_{i},f_{i\to j}\rangle_{\alpha}$ be a direct system of
partially ordered abelian groups and positive homomorphisms. Let $\mathbf{A}$
be its abelian group direct limit and $\pi_{i}$’s be the canonical
homomorphisms. Then, as shown in [10, Proposition 1.15], $\mathbf{A}$ can be
made into a partially ordered abelian group with positive cone
$A^{+}=\cup_{i\in\alpha}\pi_{i}(A_{i}^{+})$, and $\mathbf{A}$ together with
the maps $\pi_{i}$ is a direct limit for $\langle\mathbf{A}_{i},f_{i\to
j}\rangle_{\alpha}$ in the class of partially ordered abelian groups. The same
construction works in $\mathfrak{A}^{\mathfrak{c}}$. Indeed, when applied to
abelian $o$-groups, it results is an abelian $o$-group: Consider $x\notin
A^{+}$. Then there is some $i\in\alpha$ and $y\in A_{i}$ such that
$x=\pi_{i}(y)$. Since $x\notin A^{+}$, $y\notin A_{i}^{+}$ follows. Hence
$y^{-1}\in A_{i}^{+}$ since $\mathbf{A}_{i}$ is totally ordered. Therefore,
$x^{-1}=\pi_{i}(y)^{-1}=\pi_{i}(y^{-1})\in A^{+}$ and we are done. These lead
to

######  Proposition 2.1.

Direct limits exist in $\mathfrak{A}^{\mathfrak{c}}$. ∎

We are going to refer to homomorphisms between direct limits which are induced
by homomorphisms between the corresponding direct systems. If
$\mathcal{A}=\langle\mathbf{A}_{i},f_{i\to j}\rangle$ and
$\mathcal{B}=\langle\mathbf{B}_{i},g_{i\to j}\rangle$ are two direct systems
over the same index $\alpha$ set then by a homomorphism
$\Phi:\mathcal{A}\to\mathcal{B}$ is meant a system of homomorphisms
$\Phi=\\{\Phi_{i}:A_{i}\to B_{i}:i\in\alpha\\}$ such that for every $i<j$ the
diagram in Fig. 3

Figure 3.

is commutative.

######  Lemma 2.2.

[3, Theorem 11.2] If $\mathcal{A}$ and $\mathcal{B}$ are direct systems and
$\Phi:\mathcal{A}\to\mathcal{B}$ is a homomorphism then there exists a unique
homomorphism
$\Phi^{\star}:\mathbf{A}^{\star}=\underset{\longrightarrow}{\lim}\,\mathbf{A}_{i}\to\mathbf{B}^{\star}=\underset{\longrightarrow}{\lim}\,\mathbf{B}_{i}$
such that for every $u\in\alpha$ the diagram in Fig. 4

Figure 4.

is commutative. Moreover, $\Phi^{\star}$ is embedding if all the $\Phi_{i}$’s
are embeddings.

###  2.3. A link between $\mathfrak{I}^{\mathfrak{c}}_{0}$,
$\mathfrak{I}^{\mathfrak{c}}_{1}$ and some direct systems in
$\mathfrak{A}^{\mathfrak{c}}$

Theorem 2.5 is a representation theorem of odd and even involutive FLe-chains
by means of some direct systems of abelian $o$-groups. It establishes a one-
to-one correspondence between the class
$\mathfrak{I}^{\mathfrak{c}}_{0}\cup\mathfrak{I}^{\mathfrak{c}}_{1}$
containing all odd and even involutive FLe-chains and the class of bunches of
layer groups. Therefore, a bunch of layer groups will also be referred to as
the group representation of the algebra in question. In a bunch of layer
groups $\langle\textbf{{G${}_{u}$}},\textbf{{H${}_{u}$}},\varsigma_{u\to
v}\rangle_{\langle\kappa_{o},\kappa_{J},\kappa_{I},\leq_{\kappa}\rangle}$ of
an algebra $\mathbf{X}$ there are three pairwise disjoint sets $\kappa_{o}$,
$\kappa_{J}$, and $\kappa_{I}$, their union $\kappa$ is totally ordered by
$\leq_{\kappa}$ such that $\kappa$ has a least element $t$. It can be seen
whether the corresponding $\mathbf{X}$ is odd, or even with a non-idempotent
falsum constant, or even with an idempotent falsum constant, by looking at
whether $t$ is in $\kappa_{o}$, in $\kappa_{J}$, or in $\kappa_{I}$,
respectively. This explains the role of Table 2.3 in

######  Definition 2.3.

[12, Definition 6.1] Call
${\mathcal{X}}=\langle\textbf{{G${}_{u}$}},\textbf{{H${}_{u}$}},\varsigma_{u\to
v}\rangle_{\langle\kappa_{o},\kappa_{J},\kappa_{I},\leq_{\kappa}\rangle}$ a
bunch of layer groups, where $(\kappa,\leq_{\kappa})$ is a totally ordered set
with least element $t$, the ordered triple
$\langle\\{t\\},\bar{\kappa}_{J},\bar{\kappa}_{I}\rangle$ is a partition of
$\kappa$, where $\bar{\kappa}_{I}$ and $\bar{\kappa}_{J}$ can also be empty,
$\kappa_{o}$, $\kappa_{J}$, and $\kappa_{I}$ are defined by one of the rows of

$\kappa_{o}$ | $\kappa_{J}$ | $\kappa_{I}$
---|---|---
{t} | $\bar{\kappa}_{J}$ | $\bar{\kappa}_{I}$
$\emptyset$ | $\bar{\kappa}_{J}\cup\\{t\\}$ | $\bar{\kappa}_{I}$
$\emptyset$ | $\bar{\kappa}_{J}$ | $\bar{\kappa}_{I}\cup\\{t\\}$

$\textbf{{G${}_{u}$}}=(G_{u},\preceq_{u},\cdot_{u},\ {}^{-1_{u}},u)$ is a
family of abelian $o$-groups indexed by elements of $\kappa$, and
$\textbf{{H${}_{u}$}}=(H_{u},\preceq_{u},\cdot_{u},\ {}^{-1_{u}},u)$ is a
family of abelian $o$-groups indexed by elements of $\kappa_{I}$, such that

for $u\in\kappa_{J}$, Gu is discrete,  for $u\in\kappa_{I}$,
$\textbf{{H${}_{u}$}}\leq\textbf{{G${}_{u}$}}$,

and such that for every $u,v\in\kappa$, $u\leq_{\kappa}v$, there exists a
homomorphism $\varsigma_{u\to v}:G_{u}\to G_{v}$ satisfying

  1. (G1)

$\varsigma_{u\to u}=id_{G_{u}}$ and $\varsigma_{v\to w}\circ\varsigma_{u\to
v}=\varsigma_{u\to w}$ (direct system property),

  2. (G2)

for $v\in\kappa_{I}$, $\varsigma_{u\to v}$ maps into $H_{v}$.

  3. (G3)

for $u\in\kappa_{J}$, $\varsigma_{u\to v}(u)=\varsigma_{u\to
v}(u_{\downarrow_{u}})$,

Call the Gu’s and the Hu’s the layer groups and layer subgroups of
$\mathcal{X}$, respectively, call $\langle\kappa,\leq_{\kappa}\rangle$ the
skeleton of $\mathcal{X}$, call
$\langle\kappa_{o},\kappa_{J},\kappa_{I}\rangle$ the partition of the
skeleton, and call $\langle\textbf{{G${}_{u}$}},\varsigma_{u\to
v}\rangle_{\kappa}$ the direct system of $\mathcal{X}$. Note that $\kappa$ can
be recovered from its partition, (and ultimately, from $\mathcal{X}$) via
$\kappa=\kappa_{o}\cup\kappa_{J}\cup\kappa_{I}$.

######  Remark 2.4.

Bunches of layer groups are direct systems of abelian $o$-groups over a
totally ordered set, equipped with a few extra properties.

The following theorem demonstrates a one-to-one correspondence in a
constructive manner between the class
$\mathfrak{I}^{\mathfrak{c}}_{0}\cup\mathfrak{I}^{\mathfrak{c}}_{1}$ and the
class of bunches of layer groups. Because of this, if $\mathbf{X}$ denotes the
algebra corresponding to the bunch $\mathcal{X}$, then we also say that the
Gu’s and the Hu’s are the layer groups and layer subgroups of $\mathbf{X}$, we
call $\langle\kappa,\leq_{\kappa}\rangle$ the skeleton of $\mathbf{X}$,
$\langle\kappa_{o},\kappa_{J},\kappa_{I}\rangle$ the partition of the
skeleton, and call $\langle\textbf{{G${}_{u}$}},\varsigma_{u\to
v}\rangle_{\kappa}$ the direct system of $\mathbf{X}$.

######  Theorem 2.5.

[12, Theorem 8.1]

  1. (A)

Given an odd or an even involutive FLe-chain
$\mathbf{X}=(X,\leq,{\mathbin{*\mkern-9.0mu\circ}},\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}},t,f)$
with residual complement operation ′,

$\mathcal{X}_{\mathbf{X}}=\langle\textbf{{G${}_{u}$}},\textbf{{H${}_{u}$}},\varsigma_{u\to
v}\rangle_{\langle\kappa_{o},\kappa_{J},\kappa_{I},\leq_{\kappa}\rangle}$

is bunch of layer groups, called the bunch of layer groups of $\mathbf{X}$,
where

$\kappa=\\{{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}:x\in
X\\}=\\{u\geq t:u\mbox{ is idempotent}\\}\mbox{ is ordered by $\leq$,}$
$\bar{\kappa}_{I}=\\{u\in\kappa\setminus\\{t\\}:{u}{{}^{\prime}}\mbox{ is
idempotent}\\},$
$\bar{\kappa}_{J}=\\{u\in\kappa\setminus\\{t\\}:{u}{{}^{\prime}}\mbox{ is not
idempotent}\\},$

$\kappa_{o}$, $\kappa_{J}$, $\kappa_{I}$ are defined by

$\kappa_{o}$ | $\kappa_{J}$ | $\kappa_{I}$ |
---|---|---|---
{t} | $\bar{\kappa}_{J}$ | $\bar{\kappa}_{I}$ |  if $\mathbf{X}$ is odd
$\emptyset$ | $\bar{\kappa}_{J}\cup\\{t\\}$ | $\bar{\kappa}_{I}$ |  if $\mathbf{X}$ is even and $f$ is not idempotent
$\emptyset$ | $\bar{\kappa}_{J}$ | $\bar{\kappa}_{I}\cup\\{t\\}$ |  if $\mathbf{X}$ is even and $f$ is idempotent

for $u\in\kappa$,

$\begin{array}[]{llll}\textbf{{G${}_{u}$}}&=&(G_{u},\leq,{\mathbin{*\mkern-9.0mu\circ}},\
{}^{-1},u)&\mbox{if $u\in\kappa$,}\\\
\textbf{{H${}_{u}$}}&=&(H_{u},\leq,{\mathbin{*\mkern-9.0mu\circ}},\
{}^{-1},u)&\mbox{if $u\in\kappa_{I}$,}\\\ \end{array}$

where $X_{u}=\\{x\in
X:{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}=u\\}$,
$H_{u}=\\{x\in
X_{u}:{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}{{}^{\prime}}}<x\\}$,
$\accentset{\bullet}{H}_{u}=\\{{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}{{}^{\prime}}}:x\in
H_{u}\\}$,

(2.2) $G_{u}=\left\\{\begin{array}[]{ll}X_{u}&\mbox{if $u\notin\kappa_{I}$}\\\
X_{u}\setminus\accentset{\bullet}{H}_{u}&\mbox{if $u\in\kappa_{I}$}\\\
\end{array}\right.$ (2.3)
$x^{-1}={x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{u},$

and for $u,v\in\kappa$ such that $u<v$, $\varsigma_{u\to v}:G_{u}\to G_{v}$ is
defined by

$\varsigma_{u\to v}(x)={v}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{x}.$

  1. (B)

Given a bunch of layer groups
$\mathcal{X}=\langle\textbf{{G${}_{u}$}},\textbf{{H${}_{u}$}},\varsigma_{u\to
v}\rangle_{{\langle\kappa_{o},\kappa_{J},\kappa_{I},\leq_{\kappa}\rangle}}$
with $\textbf{{G${}_{u}$}}=(G_{u},\preceq_{u},\cdot_{u},\ {}^{-1_{u}},u)$

$\mathbf{X}_{\mathcal{X}}=(X,\leq,{\mathbin{*\mkern-9.0mu\circ}},\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}},t,{t}{{}^{\prime}})$

is an involutive FLe-chain with residual complement ′, called the involutive
FLe-chain of $\mathcal{X}$, where
$\kappa=\kappa_{o}\cup\kappa_{J}\cup\kappa_{I}$, for $u\in\kappa$,

(2.4) $X_{u}=\left\\{\begin{array}[]{ll}G_{u}&\mbox{ if
$u\not\in\kappa_{I}$},\\\ G_{u}\,\cup\,\accentset{\bullet}{H}_{u}&\mbox{ if
$u\in\kappa_{I}$},\\\ \end{array}\right.$

(where $\accentset{\bullet}{H}_{u}=\\{\accentset{\bullet}{h}:h\in H_{u}\\}$ is
a copy of $H_{u}$ which is disjoint from $G_{u}$),

(2.5) $X=\displaystyle\dot{\bigcup}_{u\in\kappa}X_{u},$

if $u\notin\kappa_{I}$ then $\leq_{u}\,=\,\preceq_{u}$, if $u\in\kappa_{I}$
then $\leq_{u}$ extends $\preceq_{u}$ to $X_{u}$ by letting

(2.6) $\accentset{\bullet}{a}<_{u}\accentset{\bullet}{b}$ and
$x<_{u}\accentset{\bullet}{a}<_{u}y$ if $a,b\in H_{u}$, $x,y\in G_{u}$,
$a\prec_{u}b$, $x\prec_{u}a\preceq_{u}y$,

for $v\in\kappa$, $\rho_{v}:X\to X$ is defined by

(2.7)
$\begin{array}[]{lll}\rho_{v}(x)&=&\left\\{\begin{array}[]{ll}\varsigma_{u\to
v}(x)&\mbox{ if $x\in G_{u}$ and $u<_{\kappa}v$},\\\ x&\mbox{ if $x\in G_{u}$
and $u\geq_{\kappa}v$},\\\ \end{array}\right.\\\
\rho_{v}(\accentset{\bullet}{x})&=&\left\\{\begin{array}[]{ll}\varsigma_{u\to
v}(x)&\mbox{ if $\accentset{\bullet}{x}\in\accentset{\bullet}{H}_{u}$ and
$\kappa_{I}\ni u<_{\kappa}v$},\\\ \accentset{\bullet}{x}&\mbox{ if
$\accentset{\bullet}{x}\in\accentset{\bullet}{H}_{u}$ and $\kappa_{I}\ni
u\geq_{\kappa}v$},\\\ \end{array}\right.\end{array}$

for $x\in X_{u}$ and $y\in X_{v}$,

$x<y$ iff $\rho_{u\vee v}(x)<_{u\vee v}\rho_{u\vee v}(y)$ or $\rho_{u\vee
v}(x)=\rho_{u\vee v}(y)$ and $u<_{\kappa}v$,

for $u\in\kappa_{I}$, $h_{u}:X_{u}\to G_{u}$,

(2.8) $\begin{array}[]{ll}h_{u}(x)=x&\mbox{ if $x\in G_{u}$,}\\\
h_{u}(\accentset{\bullet}{x})=x&\mbox{ if
$\accentset{\bullet}{x}\in\accentset{\bullet}{H}_{u}$,}\\\ \end{array}$

for $x,y\in X_{u}$,

(2.9)
${x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{u}}{y}=\left\\{\begin{array}[]{ll}\accentset{\bullet}{\left({h_{u}(x)}\cdot_{u}{h_{u}(y)}\right)}&\mbox{
if $u\in\kappa_{I}$,
${h_{u}(x)}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{h_{u}(y)}\in H_{u}$ and
$\neg(x,y\in H_{u})$}\\\ {h_{u}(x)}\cdot_{u}{h_{u}(y)}&\mbox{ if
$u\in\kappa_{I}$,
${h_{u}(x)}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{h_{u}(y)}\notin H_{u}$ or
$x,y\in H_{u}$}\\\ x\cdot_{u}y&\mbox{ if $u\notin\kappa_{I}$}\\\
\end{array}\right.,$

for $x\in X_{u}$ and $y\in X_{v}$,

(2.10) ${x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{y}={\rho_{u\vee
v}(x)}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{u\vee v}}{\rho_{u\vee v}(y)},$

for $x\in X$,

(2.11)
$\begin{array}[]{lll}{(\accentset{\bullet}{x})}{{}^{\prime}}&=&\left\\{\begin{array}[]{ll}x^{-1}&\mbox{
\ \ \ if $u\in\kappa_{I}$ and
$\accentset{\bullet}{x}\in\accentset{\bullet}{H}_{u}$}\\\
\end{array}\right.\\\
{x}{{}^{\prime}}&=&\left\\{\begin{array}[]{ll}\accentset{\bullet}{\left(x^{-1}\right)}&\mbox{
if $u\in\kappa_{I}$ and $x\in H_{u}$}\\\ x^{-1}&\mbox{ if $u\in\kappa_{I}$ and
$x\in G_{u}\setminus H_{u}$}\\\ x^{-1}&\mbox{ if $u\in\kappa_{o}$ and $x\in
G_{u}$}\\\ {x^{-1}}_{\downarrow}&\mbox{ if $u\in\kappa_{J}$ and $x\in
G_{u}$}\\\ \end{array}\right.,\end{array}$

for $x,y\in X$,

(2.12)
${x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{y}={({x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{y}{{}^{\prime}}})}{{}^{\prime}},$

$\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}$ is the residual
operation of ${\mathbin{*\mkern-9.0mu\circ}}$,

$t$ is the least element of $\kappa$,  (2.13) $f$ is the residual complement
of $t$,

and is given by

$\begin{array}[]{lll}{t}{{}^{\prime}}&=&\left\\{\begin{array}[]{ll}\accentset{\bullet}{(t^{-1})}&\mbox{
if $u\in\kappa_{I}$}\\\ t^{-1}&\mbox{ if $u\in\kappa_{o}$}\\\
{t^{-1}}_{\downarrow}&\mbox{ if $u\in\kappa_{J}$}\\\
\end{array}\right..\end{array}$

In addition,

$\rho_{v}(x)={v}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{x}\ \ \mbox{for
$v\in\kappa$ and $x\in X$},$

$\mathbf{X}_{\mathcal{X}}$ is odd if $t\in\kappa_{o}$, even with a non-
idempotent falsum if $t\in\kappa_{J}$, and even with an idempotent falsum if
$t\in\kappa_{I}$.

  1. (C)

Items (A) and (B) describe a one-to-one correspondence between the class
containing all odd and even involutive FLe-algebras and the class of bunches
of layer groups: given a bunch of layer groups $\mathcal{X}$ it holds true
that $\mathcal{X}_{({\mathbf{X}}_{\mathcal{X}})}=\mathcal{X}$, and given an
odd or even involutive FLe-chain $\mathbf{X}$ it holds true that
$\mathbf{X}_{(\mathcal{X}_{\mathbf{X}})}=\mathbf{X}$. ∎

##  3\. Reduction of amalgamation in $\mathfrak{I}_{0}$ and $\mathfrak{I}_{1}$
to amalgamation in $\mathfrak{C}$ and some direct systems in
$\mathfrak{A}^{\mathfrak{c}}$

Let $\mathbf{X},\mathbf{Y}\in\mathfrak{I}_{0}$ or
$\mathbf{X},\mathbf{Y}\in\mathfrak{I}_{1}$. Lemma 3.2 characterizes how
‘$\mathbf{X}$ is a subalgebra of $\mathbf{Y}$’ can be seen from the respective
group representations of $\mathbf{X}$ and $\mathbf{Y}$. The proof is an easy
but tedious verification guided by Theorem 2.5.

######  Definition 3.1.

Given two bunches of layer groups
${\mathcal{X}}=\langle\textbf{{G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}},\textbf{{H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}\rangle_{\langle\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}},\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}},\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}},\leq_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}\rangle}$
and
${\mathcal{Y}}=\langle\textbf{{G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}},\textbf{{H${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}\rangle_{\langle\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}},\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}},\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}},\leq_{\kappa^{\scaleto{(\mathbf{Y})}{3pt}}}\rangle}$,
we say that $\mathcal{X}$ is a sub-bunch of $\mathcal{Y}$, in notation
$\mathcal{X}\leq\mathcal{Y}$, if the following conditions hold.

  1. (S1)

$\min_{\leq_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}}\kappa^{\scaleto{(\mathbf{X})}{4pt}}=\min_{\leq_{\kappa^{\scaleto{(\mathbf{Y})}{3pt}}}}\kappa^{\scaleto{(\mathbf{Y})}{4pt}}$.

  2. (S2)

$\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}}\subseteq\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}}$,
$\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}\subseteq\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}$,
$\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}\subseteq\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}$,
and
$\leq_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}\,\subseteq\,\leq_{\kappa^{\scaleto{(\mathbf{Y})}{3pt}}}$.

  3. (S3)

If $u\in\kappa^{\scaleto{(\mathbf{X})}{4pt}}$ then
$\textbf{{G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}}\leq\textbf{{G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}}$,
if $u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$ then
$\textbf{{H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}}\leq\textbf{{H${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}}$,
if $u\in\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}$ then the lower cover of the
unit element of G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$ coincides with the
lower cover of the unit element of G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$.

  4. (S4)

For $u,v\in\kappa^{\scaleto{(\mathbf{X})}{4pt}}$,
$u<_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}v$, $\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}:G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}\to
G_{v}^{\scaleto{(\mathbf{Y})}{4pt}}$ is an extension of $\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}:G_{u}^{\scaleto{(\mathbf{X})}{4pt}}\to
G_{v}^{\scaleto{(\mathbf{X})}{4pt}}$.

######  Lemma 3.2.

Let

$\mathbf{X}=(X,\leq_{X},\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu},\mathbin{\rightarrow_{\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu}}},t,f),$
$\mathbf{Y}=(Y,\leq_{Y},{\mathbin{*\mkern-9.0mu\circ}},\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}},\rm{t},\rm{f})$

be algebras both in $\mathfrak{I}^{\mathfrak{c}}_{0}$ or in
$\mathfrak{I}^{\mathfrak{c}}_{1}$ along with their respective group
representations

${\mathcal{X}}=\langle\textbf{{G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}},\textbf{{H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}\rangle_{\langle\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}},\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}},\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}},\leq_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}\rangle},$
${\mathcal{Y}}=\langle\textbf{{G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}},\textbf{{H${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}\rangle_{\langle\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}},\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}},\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}},\leq_{\kappa^{\scaleto{(\mathbf{Y})}{3pt}}}\rangle}.$

Then $\mathbf{X}\leq\mathbf{Y}$ if and only if $\mathcal{X}\leq\mathcal{Y}$.

###### Proof.

Referring to Theorem 2.5, let
${\kappa^{\scaleto{(\mathbf{X})}{4pt}}}=\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$
and
${\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}=\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}$,
denote the monoidal operation of the layer group
G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$ by $\cdot_{u}$ and the monoidal
operation of the layer group G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$ by
$\diamond_{u}$. Now, $\mathbf{X}\leq\mathbf{Y}$ is equivalent to the following
six conditions:

  1. (B1)

$X\subseteq Y$,

  2. (B2)

$\leq_{X}\,\subseteq\,\leq_{Y}$,

  3. (B3)

$\mathbin{\circ\mkern-7.0mu\cdot\mkern 2.0mu}$ is the restriction of
${\mathbin{*\mkern-9.0mu\circ}}$ $X$,

  4. (B4)

$\mathbin{\rightarrow_{\mathbin{\circ\mkern-7.0mu\cdot\mkern 2.0mu}}}$ is the
restriction of $\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}$ to
$X$,

  5. (B5)

$t={\rm t}$,

  6. (B6)

$f={\rm f}$.

\- By Theorem 2.5,
$\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}={\kappa^{\scaleto{(\mathbf{X})}{4pt}}}=\\{u\geq_{X}t:u\mbox{
is idempotent}\\}$ and
$\leq_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}\,=\,\leq_{X}$ over
${\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$. Hence
$\min_{\leq_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}}(\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}})=t$.
Likewise follows
$\min_{\leq_{\kappa^{\scaleto{(\mathbf{Y})}{3pt}}}}(\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}})=\rm
t$. Therefore, condition (S1) is equivalent to 5.

(I) Assume $\mathbf{X}\leq\mathbf{Y}$.

(S2): Let, for instance, $u\in\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}$; the
proof of the other two cases is similar. By Table 2.3, either $u=t$ and
$\mathbf{X}\in\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\neg\mathfrak{id}}$ in
which case $u=\rm t$ and
$\mathbf{Y}\in\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\neg\mathfrak{id}}$
(since being even and idempotent inherit to subalgebras) thus yielding
$u\in\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}$ by Table 2.3, or
$u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}_{J}$ is strictly positive and
idempotent and its residual complement is not idempotent (in $\mathbf{X}$)
hence $u$ is strictly positive and idempotent and its residual complement is
not idempotent in $\mathbf{Y}$ either (since $\mathbf{X}\leq\mathbf{Y}$) thus
yielding $u\in\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}$ by Table 2.3. Finally,
$\leq_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}\,\subseteq\,\leq_{\kappa^{\scaleto{(\mathbf{Y})}{3pt}}}$
holds since by Theorem 2.5, $\leq_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}$ is
the restriction of $\leq_{X}$ to
${\kappa^{\scaleto{(\mathbf{X})}{4pt}}}=\\{u\geq_{X}t:u\mbox{ is
idempotent}\\}$ and $\leq_{\kappa^{\scaleto{(\mathbf{Y})}{3pt}}}$ is the
restriction of $\leq_{Y}$ to
${\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}=\\{u\geq_{Y}\rm t:u\mbox{ is
idempotent}\\}$, and both idempotency and the underlying ordering inherits to
subalgebras.

(S3): Let $u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$. Then
$u\in{\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}$ by (S2). Since
$\mathbf{X}\leq\mathbf{Y}$, we denote the operations of $\mathbf{X}$ by
${\mathbin{*\mkern-9.0mu\circ}}$,
$\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}$,
${}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}$, too.
If $u\not\in\kappa_{I}$ then by (2.2),
$G_{u}^{\scaleto{(\mathbf{X})}{4pt}}=\\{x\in
X:{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}=u\\}$
and $G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}=\\{x\in
Y:{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}=u\\}$,
hence
$\textbf{{G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}}\leq\textbf{{G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}}$
is obvious from $\mathbf{X}\leq\mathbf{Y}$.
If $u\in\kappa_{I}$ then by (2.2),
$G_{u}^{\scaleto{(\mathbf{X})}{4pt}}=\\{x\in
X:{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}=u\\}\setminus\\{{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}:x\in
X,{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}=u,\,{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}<x\\}$
and $G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}=\\{x\in
Y:{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}=u\\}\setminus\\{{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}:x\in
Y,{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}=u,\,{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}<x\\}$.
If $a\in G_{u}^{\scaleto{(\mathbf{X})}{4pt}}$ then $a\in X$,
${a}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{a}=u$
holds. Since $X\subseteq Y$, $a\in Y$ holds, too, and thus assuming $a\notin
G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}$ would mean that there exists $x\in Y$
such that
${x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}=u$,
${x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}<x$,
and
$a={x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}$.
But then
$x\overset{(\ref{EzLeSzainVerZ})}{=}{({x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{u})}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{u}={({{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}})}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{u}={{a}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{u}$
is in $X$ since so are $a$ and $u$, yielding $a\notin
G_{u}^{\scaleto{(\mathbf{X})}{4pt}}$, a contradiction.
If $u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$ then by (2.2),
$H_{u}^{\scaleto{(\mathbf{X})}{4pt}}=\\{x\in
X:{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}=u,\,{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}{{}^{\prime}}}<x\\}$
and $H_{u}^{\scaleto{(\mathbf{Y})}{4pt}}=\\{x\in
Y:{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}=u,\,{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}{{}^{\prime}}}<x\\}$,
hence
$\textbf{{H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}}\leq\textbf{{H${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}}$
is obvious from $\mathbf{X}\leq\mathbf{Y}$.
The lower cover of the unit element of G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$
is ${u}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}$.
Indeed for any $t\leq x<u$ it holds true by (2.1) that
${x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}<u$ and
hence it is not is the universe of G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$.
For any
${u}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}<x\leq
t$ it holds true that
$t\leq{x}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}<u$,
and since
${{x}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{{x}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}={\left({{x}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{({x}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}})}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}\right)}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}={\left({{x}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{x}\right)}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}={x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}$,
the previous case applies. Likewise follows that the lower cover of the unit
element of G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$ is
${u}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}$, and
hence they are equal since $\mathbf{X}\leq\mathbf{Y}$.

(S4): for $u,v\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$,
$u<_{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}v$, and $x\in G_{u}$ it holds true
by Theorem 2.5 that $\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}(x)={x}\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{v}\overset{\ref{ReSTrict}}{=}{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{v}=\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}(x)$.

(II) Assume that conditions (S1)-(S4) are satisfied.

1: By (S2),

(3.1)
${\kappa^{\scaleto{(\mathbf{X})}{4pt}}}:=\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}\subseteq\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}=:{\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}$

follows. By (S3) it holds true that
$\textbf{{G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}}\leq\textbf{{G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}}$
if
$u\in\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$,
and
$\textbf{{H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}}\leq\textbf{{H${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}}$
if $u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$, hence by (2.4),

(3.2) $X_{u}\subseteq Y_{u}$ holds for
$u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$.

Therefore,
$X\overset{(\ref{EZazX})}{=}\displaystyle\dot{\bigcup}_{u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}}X_{u}\overset{(\ref{ReSzeKAPPAnak})}{\subseteq}\displaystyle\dot{\bigcup}_{u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}}Y_{u}\subseteq\displaystyle\dot{\bigcup}_{u\in{\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}}Y_{u}\overset{(\ref{EZazX})}{=}Y$.

2: From the first two lines in (S3),
${\preceq_{G}}_{u}\subseteq{\preceq_{K}}_{u}$ holds for
$u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$. Therefore, by Theorem 2.5/(B),
${\leq_{G}}_{u}\subseteq{\leq_{K}}_{u}$ follows if
$u\in\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}$,
see the line before (2.6). If $u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$
then a moment’s reflection using the first two lines in (S3) and (2.6) shows
that ${\leq_{G}}_{u}\subseteq{\leq_{K}}_{u}$ holds in this case, too. Summing
up,

(3.3) for $u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$,
${\leq_{G}}_{u}\subseteq{\leq_{K}}_{u}$ holds.

Now, let $x,y\in X$ such that $x\in G_{u}$, $y\in G_{v}$,
$u,v\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$. Then by ((B)) $x\leq_{X}y$
equivalent to

(3.4) $\rho_{G,u\vee v}(x)\,{\leq_{G}}_{u\vee v}\,\rho_{G,u\vee v}(y)$ except
if $u>_{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}v$ and $\rho_{G,u\vee
v}(x)=\rho_{G,u\vee v}(y)$.

Using the first two lines in (S3) it is immediate from (2.7) that for every
$w\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$,

(3.5) for every $w\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$, ${\rho_{K}}_{w}$
is an extension of ${\rho_{G}}_{w}$.

Hence it follows, using (3.3) and (3.4) that

$\rho_{K,u\vee v}(x)\,{\leq_{K}}_{u\vee v}\,\rho_{K,u\vee v}(y)$ except if
$u>_{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}v$ and $\rho_{K,u\vee
v}(x)=\rho_{K,u\vee v}(y)$,

which implies $x\leq_{Y}y$.

3: Referring to the first two lines in (S3), it follows from (2.8) that

(3.6) ${h_{K}}_{u}$ is an extension of ${h_{G}}_{u}$.

For $u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$ and $x,y\in X_{u}$, it
follows that
$u\overset{(\ref{ReSzeKAPPAnak})}{\in}{\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}$
and $x,y\overset{(\ref{XuRESZEyU})}{\in}Y_{u}$, and

$\begin{array}[]{lll}{x}\mathbin{\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu}_{u}}{y}&\overset{(\ref{uPRODigy})}{=}&\left\\{\begin{array}[]{ll}\accentset{\bullet}{\left({{h_{G}}_{u}(x)}\cdot_{u}{{h_{G}}_{u}(y)}\right)}&\mbox{
if $u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$,
${{h_{G}}_{u}(x)}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{h_{G}}_{u}(y)}\in
H_{u}^{\scaleto{(\mathbf{X})}{4pt}}$ and $\neg(x,y\in
H_{u}^{\scaleto{(\mathbf{X})}{4pt}})$}\\\
{{h_{G}}_{u}(x)}\cdot_{u}{{h_{G}}_{u}(y)}&\mbox{ if
$u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$,
${{h_{G}}_{u}(x)}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{h_{G}}_{u}(y)}\notin
H_{u}^{\scaleto{(\mathbf{X})}{4pt}}$ or $x,y\in
H_{u}^{\scaleto{(\mathbf{X})}{4pt}}$}\\\ x\cdot_{u}y&\mbox{ if
$u\notin\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$}\\\ \end{array}\right.\\\
&\overset{(\ref{ExTENSionIZACO}),\ref{MindenkiResZe}}{=}&\left\\{\begin{array}[]{ll}\accentset{\bullet}{\left({{h_{K}}_{u}(x)}\cdot_{u}{{h_{K}}_{u}(y)}\right)}&\mbox{
if $u\in\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}$,
${{h_{K}}_{u}(x)}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{h_{K}}_{u}(y)}\in
H_{u}^{\scaleto{(\mathbf{Y})}{4pt}}$ and $\neg(x,y\in
H_{u}^{\scaleto{(\mathbf{Y})}{4pt}})$}\\\
{{h_{K}}_{u}(x)}\cdot_{u}{{h_{K}}_{u}(y)}&\mbox{ if
$u\in\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}$,
${{h_{K}}_{u}(x)}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{h_{K}}_{u}(y)}\notin
H_{u}^{\scaleto{(\mathbf{Y})}{4pt}}$ or $x,y\in
H_{u}^{\scaleto{(\mathbf{Y})}{4pt}}$}\\\ x\cdot_{u}y&\mbox{ if
$u\notin\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}$}\\\ \end{array}\right.\\\
&\overset{(\ref{uPRODigy})}{=}&{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{u}}{y}\end{array}$

Summing up,

(3.7) for $u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$,
$\mathbin{\circ\mkern-7.0mu\cdot\mkern 2.0mu}_{u}$ is the restriction of
$\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{u}}$ to $X_{u}$

Now, let $x,y\in X$ such that $u,v\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$,
$x\in X_{u}$, $y\in X_{v}$. Since $\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu}$ is commutative, we can safely assume
$u\leq_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}v$. Therefore,
${x}\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{y}\overset{(\ref{EgySzeruTe})}{=}{{\rho_{G}}_{v}(x)}\mathbin{\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu}_{v}}{{\rho_{G}}_{v}(y)}\overset{(\ref{RoKAVAext})\,(\ref{ReSZsZoRZas})}{=}{{\rho_{K}}_{v}(x)}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{v}}{{\rho_{K}}_{v}(y)}\overset{(\ref{EgySzeruTe})}{=}{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{y}$
holds.

4: Referring to (2.12), it is sufficient to prove that
${}^{{\prime^{\mkern-4.0mu{}^{{}_{\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu}}}}}}$ is extension of
${}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}$. Since
by the first row of (S3) the inverse operation in
G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$ is extension of the inverse operation
in G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$, for $x\in X$,

(3.8)
$\begin{array}[]{lll}{(\accentset{\bullet}{x})}^{{\prime^{\mkern-4.0mu{}^{{}_{\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu}}}}}}&\overset{(\ref{SplitNega})}{=}&\left\\{\begin{array}[]{ll}\
x^{-1_{G_{u}}}\ \ \ \ \ \ \ =\ \
x^{-1_{G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}}}&\mbox{ \ \ \ \ if
$u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$ and
$\accentset{\bullet}{x}\in\accentset{\bullet}{H}_{u}^{\scaleto{(\mathbf{X})}{4pt}}$}\\\
\end{array}\right.\\\
{x}^{{\prime^{\mkern-4.0mu{}^{{}_{\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu}}}}}}&\overset{(\ref{SplitNega})}{=}&\left\\{\begin{array}[]{llll}\accentset{\bullet}{\left(x^{-1_{G_{u}}}\right)}&=&\accentset{\bullet}{\left(x^{-1_{G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}}}\right)}&\mbox{
if $u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$ and $x\in
H_{u}^{\scaleto{(\mathbf{X})}{4pt}}$}\\\
x^{-1_{G_{u}}}&=&x^{-1_{G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}}}&\mbox{ if
$u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$ and $x\in G_{u}\setminus
H_{u}^{\scaleto{(\mathbf{X})}{4pt}}$}\\\
x^{-1_{G_{u}}}&=&x^{-1_{G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}}}&\mbox{ if
$u\in\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}}$ and $x\in G_{u}$}\\\
{x^{-1_{G_{u}}}}_{\downarrow_{G_{u}}}&=&{x^{-1_{G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}}}}_{\downarrow_{G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}}}&\mbox{
if $u\in\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}$ and $x\in G_{u}$}\\\
\end{array}\right..\end{array}$

In the last row we also used that
$\downarrow_{G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}}$ is extension of
$\downarrow_{G_{u}}$, as it readily follows from the last condition in (S3).
Referring to the first three items in (S2) and first two items in (S3), it
follows, respectively, that

$\left\\{\begin{array}[]{lll}u\in\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}&\mbox{
and
}&\accentset{\bullet}{x}\in\accentset{\bullet}{H}_{u}^{\scaleto{(\mathbf{Y})}{4pt}}\\\
u\in\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}&\mbox{ and }&x\in
H_{u}^{\scaleto{(\mathbf{Y})}{4pt}}\\\
u\in\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}&\mbox{ and }&x\in
G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}\setminus
H_{u}^{\scaleto{(\mathbf{Y})}{4pt}}\\\
u\in\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}}&\mbox{ and }&x\in
G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}\\\
u\in\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}&\mbox{ and }&x\in
G_{u}^{\scaleto{(\mathbf{Y})}{4pt}}\\\ \end{array}\right.,$

where in the middle row we also used the second condition from (S3).
Therefore, by (2.12), the expression in (3.8) is equal to

$\left\\{\begin{array}[]{ll}{(\accentset{\bullet}{x})}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}&\mbox{
if $u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$ and
$\accentset{\bullet}{x}\in\accentset{\bullet}{H}_{u}^{\scaleto{(\mathbf{X})}{4pt}}$}\\\
{x}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}&\mbox{
if $u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$ and $x\in G_{u}$}\\\
\end{array}\right..$

6: Obvious, since the falsum constant is the residual complement of the unit
element (see (2.13)), $t=\rm t$ (see 5), and
${}^{{\prime^{\mkern-4.0mu{}^{{}_{\mathbin{\circ\mkern-7.0mu\cdot\mkern
2.0mu}}}}}}$ is extension of
${}^{{\prime^{\mkern-4.0mu{}^{{}_{{\mathbin{*\mkern-9.0mu\circ}}}}}}}$ (see
the proof of 4). ∎

Corollary 3.3 shows how ‘$\mathbf{X}$ is embeddable into $\mathbf{Y}$’ can be
seen from the respective group representations of $\mathbf{X}$ and
$\mathbf{Y}$. Since by Theorem 2.5 the universe of the skeleton and the
universe of the layer groups are subsets of the universe of the algebra, any
mapping $\iota$ from $\mathbf{X}$ to $\mathbf{Y}$ induces a mapping
$\iota_{\kappa}$ from the skeleton of $\mathbf{X}$, and a mapping $\iota_{u}$
in every layer $u$, from the $u^{\rm th}$-layer group
G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$ of $\mathbf{X}$.

######  Corollary 3.3.

Assume the hypotheses of Lemma 3.2. Then $\iota:\mathbf{X}\to\mathbf{Y}$ is an
embedding if and only if the following conditions hold.

  1. (E1)

$\iota_{\kappa}:=\iota_{|_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}}$ is an
embedding555Injective order preserving mapping. of the skeleton of
$\mathcal{X}$ into the skeleton of $\mathcal{Y}$, which preserves the least
element and the partition,

  2. (E2)

For every $u\in\kappa^{\scaleto{(\mathbf{X})}{4pt}}$,
$\iota_{u}:=\iota_{|_{\textbf{{G${}^{\scaleto{(\mathbf{X})}{3pt}}_{u}$}}}}$ is
an ($o$-group) embedding of the $u^{\rm th}$-layer group
G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$ of $\mathcal{X}$ into the
$\iota_{\kappa}(u)^{\rm th}$-layer group
G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{\iota_{\kappa}(u)}$ of $\mathcal{Y}$ 666A
consequence of this can be phrased as “amalgamation must be done layerwise”.
such that

    1. -

for every $u,v\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$, $u<v$ the diagram in
Fig. 5 commutes,

Figure 5.

    2. -

if $u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$ then $\iota_{u}$ maps
H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$ into
H${}^{\scaleto{(\mathbf{Y})}{4pt}}_{\iota(u)}$,

    3. -

if $u\in\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}$ then $\iota_{u}$ preserves
the cover of the unit element $u$ of G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$.

###### Proof.

For the last statement we mention that preserving the (upper) cover of the
unit element is equivalent to preserving the lower cover of it, since these
are the inverses of one another. ∎

##  4\. Extension of direct systems

For the main construction in the proof of Theorem 5.3 we shall need the
following lemma, which we have not been able to find in the literature. It
states the extendability of direct systems over chains to larger chains. The
proof of Lemma 4.1 works for direct systems of algebras of any kind over
chains. Here we state it only for abelian $o$-groups.

######  Lemma 4.1.

  1. (1)

Let $\langle\alpha,\leq\rangle$ be a subalgebra of the totally ordered set
$\langle\beta,\leq\rangle$. Every direct system $\langle\mathbf{A}_{u},f_{u\to
v}\rangle$ of abelian $o$-groups over $\alpha$ can be extended to a direct
system $\langle\mathbf{B}_{u},g_{u\to v}\rangle$ of abelian $o$-groups over
$\beta$, by letting for $s\in\beta$,

(4.1) $\mathbf{B}_{s}=\underset{\underset{i\in\alpha,i\leq
s}{\longrightarrow}}{\lim}\,\mathbf{A}_{i},$

and for $i,j\in\beta$, $i<j$,

(4.2) $g_{i\to j}=\left\\{\begin{array}[]{ll}f_{i\to j}&\mbox{if
$i\in\alpha$}\hfill\mbox{(extension)}\\\ \phi_{i\to j}&\mbox{if
$i\in\beta\setminus\alpha$, $j\in\alpha$}\\\ id_{i\to j}&\mbox{if
$i,j\in\beta\setminus\alpha$ and $\nexists w\in\alpha$ such that $i<w<j$}\\\
\pi_{w\to j}\circ\phi_{i\to w}&\mbox{if $i,j\in\beta\setminus\alpha$ and
$\exists w\in\alpha$ such that $i<w<j$}\\\ \end{array}\right..$

  2. (2)

It holds true that

(4.3)
$\mathbf{B}_{s}=\left\\{\begin{array}[]{ll}\underset{\underset{i\in\beta,i\leq
s}{\longrightarrow}}{\lim}\,\mathbf{A}_{i}&\mbox{if $s\in\alpha$}\\\
\underset{\underset{i\in\beta,i<s}{\longrightarrow}}{\lim}\,\mathbf{A}_{i}&\mbox{if
$s\in\beta\setminus\alpha$}\\\ \end{array}\right.$

###### Proof.

The first line of (4.3) holds since $\\{i\in\beta,i\leq s\\}$ is cofinal in
$\\{i\in\alpha,i\leq s\\}$ if $s\in\alpha$ and hence their direct limits
coincide, whereas the second line of (4.3) holds since for
$s\in\beta\setminus\alpha$, $\\{i\in\alpha,i\leq s\\}=\\{i\in\alpha,i<s\\}$.
This ends the proof of 2.

It is immediate from the definition of the direct limit that for $s\in\alpha$,
$\mathbf{B}_{s}\cong\mathbf{A}_{s}$ (extension), so for $s\in\alpha$ we shall
identify $\mathbf{B}_{s}$ with $\mathbf{A}_{s}$, and thus for every
$i\in\alpha$, $i\leq s$ it holds true that $\pi_{i\to s}$, the canonical
homomorphism from $\mathbf{A}_{i}$ to $\mathbf{B}_{s}$, is equal to $f_{i\to
s}$. Hence for $i\in\alpha$, we shall write $\pi_{i\to s}$ instead of $f_{i\to
s}$ in the sequel, and rewrite the direct system property as $\pi_{j\to
k}\circ\pi_{i\to j}=\pi_{i\to k}$ $(i,j,k\in\alpha)$. From the property of
canonical homomorphisms, this identity also holds for $i,j\in\alpha$ and
$k\in\beta\setminus\alpha$, hence it holds true that

(4.4) $\pi_{j\to k}\circ\pi_{i\to j}=\pi_{i\to k}\ \
(i,j\in\alpha,k\in\beta).$

For $\beta\setminus\alpha\ni u<v\in\alpha$ let $\phi_{u\to v}$ denote the
unique homomorphism from $\mathbf{B}_{u}$ to $\mathbf{A}_{v}$ guaranteed by
the universal property based on the family $\\{f_{i\to v}:i\in\alpha,i\leq
v\\}$. Then it holds true that

(4.5) $\phi_{j\to k}\circ\pi_{i\to j}=\pi_{i\to k}\ \
(i,k\in\alpha,j\in\beta\setminus\alpha).$

The fourth row of (4.2) is well-defined. Indeed, if $i<z<j$ holds for some
$z\in\alpha$ then since $\alpha$ is a chain we may assume $w<z$. Then for any
$k\in\alpha$, $k<i$ it holds true that $(\pi_{w\to z}\circ\phi_{i\to
w})\circ\pi_{k\to i}=\pi_{w\to z}\circ(\phi_{i\to w}\circ\pi_{k\to
i})\overset{(\ref{PhiIsCOMM})}{=}\pi_{w\to z}\circ\pi_{k\to
w}\overset{(\ref{PiIsComm})}{=}\pi_{k\to z}$, and hence by the unicity in the
universal property we obtain

(4.6) $\pi_{w\to z}\circ\phi_{i\to w}=\phi_{i\to z}.$

Now, $\pi_{w\to j}\circ\phi_{i\to w}\overset{(\ref{PiIsComm})}{=}(\pi_{z\to
j}\circ\pi_{w\to z})\circ\phi_{i\to z}=\pi_{z\to j}\circ(\pi_{w\to
z}\circ\phi_{i\to z})\overset{(\ref{MegKerulom})}{=}\pi_{z\to
j}\circ\phi_{i\to z}$ follows so we are done. It remains to verify that
$\langle\mathbf{B}_{i},g_{i\to j}\rangle$ is a direct system over $\beta$,
that is, to prove

$g_{j\to k}\circ g_{i\to j}=g_{i\to k}$

for $i,j,k\in\beta$, $i<j<k$ (in case of coincidence of any of the indices it
is straightforward).

  * -

If $i,j\in\alpha$ then $g_{j\to k}\circ g_{i\to
j}\overset{(\ref{eLSo3})}{=}\pi_{j\to k}\circ\pi_{i\to
j}\overset{(\ref{PiIsComm})}{=}\pi_{i\to k}\overset{(\ref{eLSo3})}{=}g_{i\to
k}$.

  * -

If $i,k\in\alpha$ and $j\in\beta\setminus\alpha$ then $g_{j\to k}\circ g_{i\to
j}\overset{(\ref{eLSo3})}{=}\phi_{j\to k}\circ\pi_{i\to
j}\overset{(\ref{PhiIsCOMM})}{=}\pi_{i\to k}\overset{(\ref{eLSo3})}{=}g_{i\to
k}$.

  * -

If $i\in\alpha$, $j,k\in\beta\setminus\alpha$ and $\exists w\in\alpha$ such
that $i<w<j$ then $g_{j\to k}\circ g_{i\to
j}\overset{(\ref{eLSo3})}{=}(\pi_{w\to k}\circ\phi_{j\to w})\circ\pi_{i\to
j}=\pi_{w\to k}\circ(\phi_{j\to w}\circ\pi_{i\to
j})\overset{(\ref{PhiIsCOMM})}{=}\pi_{w\to k}\circ\pi_{i\to
w}\overset{(\ref{PiIsComm})}{=}\pi_{i\to k}\overset{(\ref{eLSo3})}{=}g_{i\to
k}$.

  * -

Assume $i\in\alpha$, $j,k\in\beta\setminus\alpha$ and $\nexists w\in\alpha$
such that $j<w<k$. Now $\mathbf{B}_{j}\cong\mathbf{B}_{k}$ follows from the
last condition and (4.1), denote the related isomorphism by $id_{j\to k}$.
From the definition of direct limits, $id_{j\to k}\circ\pi_{z\to j}=\pi_{z\to
k}$ holds for any $z\in\alpha$, $z\leq j$. In particular, $id_{j\to
k}\circ\pi_{i\to j}=\pi_{i\to k}$. Therefore $g_{j\to k}\circ g_{i\to
j}\overset{(\ref{eLSo3})}{=}id_{j\to k}\circ\pi_{i\to j}=\pi_{i\to
k}\overset{(\ref{eLSo3})}{=}g_{i\to k}$.

  * -

If $j,k\in\alpha$ and $i\in\beta\setminus\alpha$ then $g_{j\to k}\circ g_{i\to
j}\overset{(\ref{eLSo3})}{=}\pi_{j\to k}\circ\phi_{i\to
j}\overset{(\ref{MegKerulom})}{=}\phi_{i\to
k}\overset{(\ref{eLSo3})}{=}g_{i\to k}$.

  * -

If $j\in\alpha$ and $i,k\in\beta\setminus\alpha$ then $g_{j\to k}\circ g_{i\to
j}\overset{(\ref{eLSo3})}{=}\pi_{j\to k}\circ\phi_{i\to
j}\overset{(\ref{eLSo3})}{=}g_{i\to k}$.

∎

Call $\langle\mathbf{B}_{u},g_{u\to v}\rangle_{\beta}$ the closure of
$\langle\mathbf{A}_{u},f_{u\to v}\rangle_{\alpha}$ to $\beta$. To simplify
notation, and since it is an extension, we shall denote the closure of
$\langle\mathbf{A}_{u},f_{u\to v}\rangle_{\alpha}$ to $\beta$ by
$\langle\mathbf{A}_{u},f_{u\to v}\rangle_{\beta}$.

##  5\. Amalgamation in $\mathfrak{I}^{\mathfrak{c}}_{0}$ and
$\mathfrak{I}^{\mathfrak{c}}_{1}$

The amalgamation property will be investigated in
$\mathfrak{J}^{\mathfrak{c}}_{\mathfrak{0}}$,
$\mathfrak{J}^{\mathfrak{c}}_{\mathfrak{1}}$,
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$,
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}$, and
$\mathfrak{I}^{\mathfrak{sl}}_{\mathfrak{0},\mathfrak{symm}}$ in this section.
As it may be expected from Section 2.3, first we will need some basic facts
about partially ordered abelian groups ($po$-groups). The positive cone of a
$po$-group is the set of its elements greater or equal to the unit element of
the group. Its strict positive cone is its positive cone except the unit
element. If $\leq$ is the ordering of the abelian $po$-group $\mathbf{G}$ then
we also express it by saying that $\leq$ is a partial order of $\mathbf{G}$.
It is readily seen from [4, Theorem 2 in page 13] that

######  Proposition 5.1.

$P$ is the positive cone of an abelian $po$-group $\mathbf{G}$ if and only if
$P\cap P^{-1}$ contains only the unit element and $PP\subseteq P$. $Q$ is its
strict positive cone if and only if $Q\cap Q^{-1}=\emptyset$ and $QQ\subseteq
Q$.

An easy consequence of Proposition 5.1 is

######  Lemma 5.2.

If $P_{1}$ and $P_{2}$ are the positive cones of two partial orders of the
abelian group $\mathbf{G}$, then

  1. i

there exists the smallest partial order of $\mathbf{G}$ which contains both
$P_{1}$ and $P_{2}$ if and only if $P_{1}\cap{P_{2}}^{-1}=\emptyset$.

  2. ii

If this is the case then its positive cone is $P_{1}\cup P_{2}\cup
P_{1}P_{2}$.

###### Proof.

The positive cone of a partial order of $\mathbf{G}$ which extends both
$P_{1}$ and $P_{2}$ must also contain $P_{1}P_{2}$ since the product of two
positive elements must be positive. On the other hand, we claim that
$P_{1}\cup P_{2}\cup P_{1}P_{2}$ is the positive cone of a partial order of
$\mathbf{G}$ if and only if $P_{1}\cap{P_{2}}^{-1}=\\{t\\}$. That is,
$P_{1}\cup P_{2}\cup P_{1}P_{2}$ is closed under multiplication, and
$(P_{1}\cup P_{2}\cup P_{1}P_{2})\cap(P_{1}\cup P_{2}\cup
P_{1}P_{2})^{-1}=\\{t\\}$. Using Proposition 5.1 and that the multiplication
is commutative, $(P_{1}\cup P_{2}\cup P_{1}P_{2})(P_{1}\cup P_{2}\cup
P_{1}P_{2})=P_{1}P_{1}\cup P_{1}P_{2}\cup P_{1}P_{1}P_{2}\cup P_{2}P_{1}\cup
P_{2}P_{2}\cup P_{2}P_{1}P_{2}\cup P_{1}P_{2}P_{1}\cup P_{1}P_{2}P_{2}\cup
P_{1}P_{2}P_{1}P_{2}\subseteq P_{1}\cup P_{2}\cup P_{1}P_{2}$. In addition,
$(P_{1}\cup P_{2}\cup P_{1}P_{2})\cap(P_{1}\cup P_{2}\cup
P_{1}P_{2})^{-1}=(P_{1}\cup P_{2}\cup P_{1}P_{2})\cap(P_{1}^{-1}\cup
P_{2}^{-1}\cup(P_{1}P_{2})^{-1})=(P_{1}\cap P_{1}^{-1})\cup(P_{1}\cap
P_{2}^{-1})\cup(P_{1}\cap(P_{1}P_{2})^{-1})\cup(P_{2}\cap
P_{1}^{-1})\cup(P_{2}\cap
P_{2}^{-1})\cup(P_{2}\cap(P_{1}P_{2})^{-1})\cup(P_{1}P_{2}\cap
P_{1}^{-1})\cup(P_{1}P_{2}\cap
P_{2}^{-1})\cup(P_{1}P_{2}\cap(P_{1}P_{2})^{-1})=\\{t\\}\cup(P_{1}\cap
P_{2}^{-1})\cup(P_{1}\cap(P_{1}P_{2})^{-1})\cup(P_{1}\cap
P_{2}^{-1})^{-1}\cup\\{t\\}\cup(P_{2}\cap(P_{1}P_{2})^{-1})\cup(P_{1}\cap(P_{1}P_{2})^{-1})^{-1}\cup(P_{2}\cap(P_{1}P_{2})^{-1})^{-1}\cup(P_{1}P_{2}\cap(P_{1}P_{2})^{-1})$

Now, if $P_{1}\cap{P_{2}}^{-1}=\\{t\\}$ then the latest is equal to
$\\{t\\}\cup\\{t\\}\cup(P_{1}\cap(P_{1}P_{2})^{-1})\cup\\{t\\}\cup\\{t\\}\cup(P_{2}\cap(P_{1}P_{2})^{-1})\cup(P_{1}\cap(P_{1}P_{2})^{-1})^{-1}\cup(P_{2}\cap(P_{1}P_{2})^{-1})^{-1}\cup(P_{1}P_{2}\cap(P_{1}P_{2})^{-1})=\\{t\\}\cup(P_{1}\cap(P_{1}P_{2})^{-1})\cup(P_{1}\cap(P_{1}P_{2})^{-1})^{-1}\cup(P_{2}\cap(P_{1}P_{2})^{-1})\cup(P_{2}\cap(P_{1}P_{2})^{-1})^{-1}\cup(P_{1}P_{2}\cap(P_{1}P_{2})^{-1})$,
and here all terms in the brackets are equal to $\\{t\\}$, we check only,
e.g., $P_{1}\cap(P_{1}P_{2})^{-1}=\\{t\\}$. Let $a\in
P_{1}\cap(P_{1}P_{2})^{-1}$. Then $a=a_{1}^{-1}a_{2}^{-1}$ where $a,a_{1}\in
P_{1}$ and $a_{2}\in P_{2}$. Therefore, $a_{2}=a^{-1}a_{1}^{-1}\in P_{1}^{-1}$
holds, and it implies $a_{2}\in P_{2}\cap P_{1}^{-1}$, hence $a_{2}=t$ holds.
It follows that $a=a_{1}^{-1}\in P_{1}\cap P_{1}^{-1}$. Hence $a=t$. All the
other terms can be handled similarly.

On the other hand, if $\\{t\\}\cup(P_{1}\cap
P_{2}^{-1})\cup(P_{1}\cap(P_{1}P_{2})^{-1})\cup(P_{1}\cap
P_{2}^{-1})^{-1}\cup\\{t\\}\cup(P_{2}\cap(P_{1}P_{2})^{-1})\cup(P_{1}\cap(P_{1}P_{2})^{-1})^{-1}\cup(P_{2}\cap(P_{1}P_{2})^{-1})^{-1}\cup(P_{1}P_{2}\cap(P_{1}P_{2})^{-1})=\\{t\\}$
then $P_{1}\cap P_{2}^{-1}$ cannot exceed $\\{t\\}$, and since $P_{1}$ and
$P_{2}$ are positive cones, both $P_{1}$ and $P_{2}^{-1}$ contain $\\{t\\}$.
Hence $P_{1}\cap P_{2}^{-1}=\\{t\\}$. ∎

The major steps of the construction of Theorem 5.3 below are illustrated in
Figures 6-8. to ease the understanding.

Figure 6. The V-formation, the V-formation layerwise

Figure 7. Closures to $\kappa^{\scaleto{(\mathbf{W})}{4pt}}$, extension of the
induced embeddings to $\kappa^{\scaleto{(\mathbf{W})}{4pt}}$

Figure 8. Layerwise amalgamation in $\mathfrak{A}^{\mathfrak{l}}$, in
$\mathfrak{A}^{\mathfrak{c}}$, and amalgamation in
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$ and in
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}$

######  Theorem 5.3.

  1. (1)

$\mathfrak{J}^{\mathfrak{c}}_{\mathfrak{0}}$ and
$\mathfrak{J}^{\mathfrak{c}}_{\mathfrak{1}}$ fail the amalgamation property.

  2. (2)

$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$ and
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}$ have the
amalgamation property

  3. (3)

$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$ and
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}$ fail the strong
amalgamation property

###### Proof.

(1) Denote by $\mathbbm{1}$ the trivial (one-element) group and
$\leq\,=\\{\langle 0,0\rangle,\langle 0,1\rangle,\langle 1,1\rangle\\}$. For
any discrete abelian $o$-group G1, the involutive FLe-chains given by

$\langle\\{\mathbbm{1},\textbf{{G${}_{1}$}}\\},\emptyset,\varsigma_{0\to
1}\rangle_{\langle\\{0\\},\\{1\\},\emptyset,\leq\rangle}$ and
$\langle\\{\textbf{{G${}_{1}$}}\\},\emptyset,\emptyset\rangle_{\langle\emptyset,\\{1\\},\emptyset,\langle
1,1\rangle\\}\rangle\rangle}$

are from $\mathfrak{J}^{\mathfrak{c}}_{\mathfrak{0}}$ or
$\mathfrak{J}^{\mathfrak{c}}_{\mathfrak{1}}$, respectively, see the table in
Theorem 2.5. Note that in both cases $1$ is in the $\kappa_{J}$ part of the
partition, and in the $\kappa_{J}$ part of the partition the related abelian
$o$-groups are discrete (c.f. Definition 2.3). If we had amalgam for any
V-formation using algebras of this form, then according to Corollary 3.3, on
the $1$-layer it would yield an amalgam with an abelian $o$-group $\mathbf{G}$
which is discrete, too, (since the induced embedding on the skeletons
preserves the partition and therefore $\mathbf{G}$ is in the
$\kappa_{J}$-layer of its layer group representation, too) and with normal and
positive embeddings (since in a $\kappa_{J}$-layer the related embeddings must
preserve the cover of the unit element, too, c.f. the last item in Corollary
3.3). However, it has been shown in [9, Lemma 12] that
$\mathfrak{A}^{\mathfrak{c}\mathfrak{d}}$ with normal and positive embeddings
does not have the amalgamation property, a contradiction.

(2) Let
$\mathbf{X},\mathbf{Y},\mathbf{Z}\in\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}$
(or
$\mathbf{X},\mathbf{Y},\mathbf{Z}\in\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$)
and consider the V-formation in Fig. 9.

Figure 9.

Let

$\mbox{
${\mathcal{X}}=\langle\textbf{{G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}},\textbf{{H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}\rangle_{\langle\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}},\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}},\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}},\leq_{\kappa^{\scaleto{(\mathbf{X})}{3pt}}}\rangle}$},$
${\mathcal{Y}}=\langle\textbf{{G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}},\textbf{{H${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}\rangle_{\langle\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}},\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}},\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}},\leq_{\kappa^{\scaleto{(\mathbf{Y})}{3pt}}}\rangle}$,
${\mathcal{Z}}=\langle\textbf{{G${}^{\scaleto{(\mathbf{Z})}{4pt}}_{u}$}},\textbf{{H${}^{\scaleto{(\mathbf{Z})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{Z})}{4pt}}\rangle_{\langle\kappa_{o}^{\scaleto{(\mathbf{Z})}{4pt}},\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}},\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}},\leq_{\kappa^{\scaleto{(\mathbf{Z})}{3pt}}}\rangle}$,

be the respective layer group representations of $\mathbf{X}$, $\mathbf{Y}$,
and $\mathbf{Z}$ (in the extended sense, see Remark 2.4). Without loss of
generality, we may assume that $\iota_{1}$ and $\iota_{2}$ are inclusion maps.
Hence its induced mappings will be inclusion maps, too, see below. By
Corollary 3.3,

  1. (C1)

the induced (order preserving) inclusion maps, see Fig. 10,

Figure 10.

preserve the least element and the partition of $\kappa$,

  2. (C2)

denoting the induced ($o$-group) inclusion maps of $\iota_{1}$ and $\iota_{2}$
for every $u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$ by $\iota_{1,u}$ and
$\iota_{2,u}$, respectively,

    1. (a)

the diagram in Fig. 11 commutes for every
$u,v\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$, $u<v$,

Figure 11.

    2. (b)

if $u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$ then $\iota_{1,u}$ and
$\iota_{1,u}$ preserve the layer subgroups, that is,
$\iota_{1,u}(\textbf{{H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}})\subseteq\textbf{{H${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}}$,
$\iota_{2,u}(\textbf{{H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}})\subseteq\textbf{{H${}^{\scaleto{(\mathbf{Z})}{4pt}}_{u}$}}$,

Notice that the last condition of Corollary 3.3 can be omitted here since
$\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}=\emptyset$ holds by the
$\mathfrak{symm}$ subclass restriction for $\mathbf{X}$.

We shall construct an amalgam of $\mathtt{V}$ in
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}$ (on in
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$) as follows.
There exists a strong amalgamation of Fig. 10 in $\mathfrak{C}$ as shown in
Fig. 12.

Figure 12.

To simplify notation, we shall assume that $\nu_{1}$, and $\nu_{2}$ are
inclusion maps, that is,

(5.1)
${\kappa^{\scaleto{(\mathbf{X})}{4pt}}}={\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}\cap{\kappa^{\scaleto{(\mathbf{Z})}{4pt}}},\
{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}={\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}\cup{\kappa^{\scaleto{(\mathbf{Z})}{4pt}}}.$

Let

(5.2)
$\kappa_{o}^{\scaleto{(\mathbf{W})}{4pt}}=\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{o}^{\scaleto{(\mathbf{Z})}{4pt}}$,
$\kappa_{J}^{\scaleto{(\mathbf{W})}{4pt}}=\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}}$,
$\kappa_{I}^{\scaleto{(\mathbf{W})}{4pt}}=\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}}$.

Let

$\begin{array}[]{lllll}\langle\textbf{{G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}&\mbox{be
the closure of
}&\langle\textbf{{G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}&\mbox{to}&{\kappa^{\scaleto{(\mathbf{W})}{4pt}}},\\\
\langle\textbf{{G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}&\mbox{be
the closure of
}&\langle\textbf{{G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}&\mbox{to}&{\kappa^{\scaleto{(\mathbf{W})}{4pt}}},\\\
\langle\textbf{{G${}^{\scaleto{(\mathbf{Z})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{Z})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}&\mbox{be
the closure of
}&\langle\textbf{{G${}^{\scaleto{(\mathbf{Z})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{Z})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{Z})}{4pt}}}&\mbox{to}&{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}.\\\
\end{array}$

We extend the family
$\\{\iota_{1,u}:u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}\\}$ of embeddings
to a family $\\{\iota_{1,u}:u\in{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}\\}$ of
embeddings as follows. For $v\in{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$, let
$\iota_{1,v}$ be the unique homomorphism mapping
G${}^{\scaleto{(\mathbf{X})}{4pt}}_{v}$ to
G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{v}$, guaranteed by the universal property
based on the family of homomorphism

$\\{\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}\circ\iota_{1,u}:G_{u}^{\scaleto{(\mathbf{X})}{4pt}}\to
G_{v}^{\scaleto{(\mathbf{Y})}{4pt}}\ |\
{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}\ni
u\leq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v\\}.$

Clearly, whenever $v\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$, thus defined
$\iota_{1,v}$ coincides with the $\iota_{1,v}$ in (C2), so this definition is
an extension, and thus defined $\iota_{1,v}$’s make the left square of the
diagram in Fig. 11 commute for every
${\kappa^{\scaleto{(\mathbf{X})}{4pt}}}\ni
u<_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v$. Moreover, by Lemma 4.1, thus
defined $\iota_{1,v}$’s make the left square of the diagram in Fig. 11 commute
for every ${\kappa^{\scaleto{(\mathbf{W})}{4pt}}}\ni
u<_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v$. In addition, $\iota_{1,v}$ is an
embedding, too, since the unique homomorphism which makes the left square of
the diagram in Fig. 11 commute for every
${\kappa^{\scaleto{(\mathbf{X})}{4pt}}}\ni
u<_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v$ is an embedding by Lemma 2.2,
since elements of $\\{\iota_{1,u}:{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}\ni
u<_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v\\}$ are all embeddings. Likewise we
extend the family
$\\{\iota_{2,u}:u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}\\}$ of embeddings
in (C2) to a family
$\\{\iota_{2,u}:u\in{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}\\}$ of embeddings.
Summing up, the diagram in Fig. 11 commutes for every
$u,v\in{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$,
$u<_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v$.

For every $u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$ consider the free product
in $\mathfrak{A}^{\mathfrak{l}}$ of G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$
and G${}^{\scaleto{(\mathbf{Z})}{4pt}}_{u}$ with
G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$ amalgamated: let

(5.3)
$\textbf{{L${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}}=\textbf{{G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}}\ast_{\textbf{{G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}}}\textbf{{G${}^{\scaleto{(\mathbf{Z})}{4pt}}_{u}$}},$

equipped with (abelian $\ell$-group) embeddings $\iota_{3,u}$ and
$\iota_{4,u}$ according to Fig. 13.

Figure 13.

For $u,v\in{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$,
$u<_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v$ let $\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}$ be the unique (abelian $\ell$-group)
homomorphism which makes the two squares in the middle of Fig. 14 commute.

Figure 14.

To see that $\varsigma_{u\to v}^{\scaleto{(\mathbf{W})}{4pt}}$ is well-
defined, that is, to infer the existence and the unicity of $\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}$ from the universal property described in
(P2) we have to show $\iota_{3,v}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}\circ\iota_{1,u}=\iota_{4,v}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{Z})}{4pt}}\circ\iota_{2,u}$. Using that $\iota_{1,u}$,
$\iota_{2,u}$, $\iota_{1,v}$, $\iota_{2,v}$, are embeddings, (S4) implies
$\iota_{1,v}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}=\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}\circ\iota_{1,u}$ and
$\iota_{2,v}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}=\varsigma_{u\to
v}^{\scaleto{(\mathbf{Z})}{4pt}}\circ\iota_{2,u}$, and hence
$\iota_{3,v}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}\circ\iota_{1,u}=\iota_{3,v}\circ\iota_{1,v}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}\overset{\ref{commuteREF}}{=}\iota_{4,v}\circ\iota_{2,v}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{X})}{4pt}}=\iota_{4,v}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{Z})}{4pt}}\circ\iota_{2,u}$, so we are done.

Next we prove that
$\langle\textbf{{L${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle$ is a direct system of abelian
$\ell$-groups over $\kappa^{\scaleto{(\mathbf{W})}{4pt}}$. To see that the
$\varsigma_{u\to v}^{\scaleto{(\mathbf{W})}{4pt}}$’s satisfy the direct system
property, see (G1), we proceed as follows. Referring to the unicity in the
universal property (P2), to prove

$\varsigma_{u\to v}^{\scaleto{(\mathbf{W})}{4pt}}=\varsigma_{v\to
w}^{\scaleto{(\mathbf{W})}{4pt}}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}$

for $u,v,w\in{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$,
$u<_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v<_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}w$,
it suffices to prove that $\varsigma_{v\to
w}^{\scaleto{(\mathbf{W})}{4pt}}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}$ makes the lower two squares in Fig. 15
commute.

Figure 15.

That is, $\varsigma_{v\to w}^{\scaleto{(\mathbf{W})}{4pt}}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\circ\iota_{3,u}=\iota_{3,w}\circ\varsigma^{\scaleto{(\mathbf{Y})}{4pt}}_{u\to
w}$ and $\varsigma_{v\to w}^{\scaleto{(\mathbf{W})}{4pt}}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\circ\iota_{4,u}=\iota_{4,w}\circ\varsigma_{u\to
w}^{\scaleto{(\mathbf{Z})}{4pt}}$ hold. We prove the first one, the other one
can be proven analogously. Using that the upper four little squares in Fig. 15
commute, $\varsigma_{v\to w}^{\scaleto{(\mathbf{W})}{4pt}}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\circ\iota_{3,u}=\varsigma_{v\to
w}^{\scaleto{(\mathbf{W})}{4pt}}\circ\iota_{3,v}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}=j_{1,w}\circ\varsigma^{\scaleto{(\mathbf{Y})}{4pt}}_{v\to
w}\circ\varsigma_{u\to
v}^{\scaleto{(\mathbf{Y})}{4pt}}\overset{\ref{(G1)}}{=}j_{1,w}\circ\varsigma^{\scaleto{(\mathbf{Y})}{4pt}}_{u\to
w}$ holds, so we are done.

Our next aim is to derive a direct system
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
of abelian $o$-groups from the direct system
$\langle\textbf{{L${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
of abelian $\ell$-groups such that for every
$u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$ the diagram in Fig. 16 commutes.

Figure 16.

The aim will be achieved by changing only the underlying orderings. Let
$\mathcal{P}$ be the set of objects
$\\{P_{u}:u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}\\}$ where $P_{u}$ is a
partial order which extends the lattice order of
L${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ such that
$\langle\textbf{{L${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$,
where the ordering of L${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ is replaced by
$P_{u}$, is a direct system of abelian $po$-groups. It has just been shown
that $\mathcal{P}$ is nonempty. For $p,q\in\mathcal{P}$,
$p=\\{P_{u}:u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}\\}$,
$q=\\{Q_{u}:u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}\\}$, we let $p\preceq q$
if for every $u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$, $P_{u}\subseteq
Q_{u}$. Then $\preceq$ is a partial ordering of $\mathcal{P}$, and since the
union of chains of partial orders (ordered by inclusion) is a partial order,
too, the union of any chain in $\langle\mathcal{P},\preceq\rangle$ is in
$\mathcal{P}$. It follows from Zorn’s lemma that
$\langle\mathcal{P},\preceq\rangle$ has a greatest element
$m=\\{\leq^{\scaleto{(\mathbf{W})}{4pt}}_{u}:u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}\\}$.
For every $u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$, denote the related
positive cone by $P^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ and the strict positive
cone by $Q^{\scaleto{(\mathbf{W})}{4pt}}_{u}$. For
$u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$, replace the ordering of
L${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ by
$\leq^{\scaleto{(\mathbf{W})}{4pt}}_{u}$, and denote the obtained abelian
$po$-group by G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$. By the definition of
$\mathcal{P}$,
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
is a direct system of abelian $po$-groups. We claim that
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
is, in fact, a direct system of abelian $o$-groups. Its proof amounts to
proving that for every $i\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$,
$\leq^{\scaleto{(\mathbf{W})}{4pt}}_{i}$ is a total order on
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{i}$. Assume, for reductio ad absurdum,
that there is $s\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$ such that

$\leq^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ is not total.

Our plan is to construct an element of $\mathcal{P}$ which is larger than $m$,
thus obtaining a contradiction. To that end first we show that in the direct
system
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
the preimage of any strictly positive element is strictly positive. More
formally,

$\bullet$ for $u,v\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$,
$u\leq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v$ it holds true that

$(\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}})^{-1}(Q^{\scaleto{(\mathbf{W})}{4pt}}_{v})\subseteq
Q^{\scaleto{(\mathbf{W})}{4pt}}_{u}.$

Consider, for $u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$,

$\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{u}=\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}\cup\\{t^{\scaleto{(\mathbf{W})}{4pt}}_{u}\\},$

where
$\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}=\left\\{x_{u}\in\textit{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}:\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(x_{u})\in
Q^{\scaleto{(\mathbf{W})}{4pt}}_{v}\mbox{ for some
$\kappa^{\scaleto{(\mathbf{W})}{4pt}}\ni
v\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}u$}\right\\}.$

$-$ Notice that since the case $v=u$ is included in the definition of
$\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ it holds true, for
$u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$, that

$P^{\scaleto{(\mathbf{W})}{4pt}}_{u}\subseteq\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{u}.$

$-$ Next we claim that for $u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$,
$\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ (as a positive cone) makes the
group reduct of G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ a $po$-group. By
Proposition 5.1 it suffices to prove, that

  1. (i)

$\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}\cap{\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}}^{-1}=\emptyset$
and

  2. (ii)

$\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}\subseteq\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$.

(i) Assume
$a\in\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}\cap{\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}}^{-1}$.
Then $a,a^{-1}\in\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$. There exist
$j,k\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}u$ such that $\varsigma_{u\to
j}^{\scaleto{(\mathbf{W})}{4pt}}(a)$ and $\varsigma_{u\to
k}^{\scaleto{(\mathbf{W})}{4pt}}(a^{-1})$ are strictly positive (in their
respective algebras). Without loss of generality we may assume
$j\leq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}k$. It follows that
$\varsigma_{u\to k}^{\scaleto{(\mathbf{W})}{4pt}}(a)=\varsigma_{j\to
k}^{\scaleto{(\mathbf{W})}{4pt}}(\varsigma_{u\to
j}^{\scaleto{(\mathbf{W})}{4pt}}(a))$ is positive since $\varsigma_{u\to
j}^{\scaleto{(\mathbf{W})}{4pt}}(a)$ is positive and $\varsigma_{j\to
k}^{\scaleto{(\mathbf{W})}{4pt}}$ is a positive homomorphism. Since
$\varsigma_{j\to k}^{\scaleto{(\mathbf{W})}{4pt}}(a)$ is positive and
$\varsigma_{j\to k}^{\scaleto{(\mathbf{W})}{4pt}}(a^{-1})$ is strictly
positive, $\varsigma_{j\to
k}^{\scaleto{(\mathbf{W})}{4pt}}(aa^{-1})=\varsigma_{j\to
k}^{\scaleto{(\mathbf{W})}{4pt}}(a)\varsigma_{j\to
k}^{\scaleto{(\mathbf{W})}{4pt}}(a^{-1})$ is strictly positive, a
contradiction, since unit elements are mapped into unit elements.
(ii) Let $a,b\in\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$. Then there exist
$v,w\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}u$ such that $\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(a)$ and $\varsigma_{u\to
w}^{\scaleto{(\mathbf{W})}{4pt}}(b)$ are strictly positive. Without loss of
generality we may assume $v\leq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}w$. It
follows that $\varsigma_{u\to
w}^{\scaleto{(\mathbf{W})}{4pt}}(a)=\varsigma_{v\to
w}^{\scaleto{(\mathbf{W})}{4pt}}(\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(a))$ is positive since $\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(a)$ is positive and $\varsigma_{v\to
w}^{\scaleto{(\mathbf{W})}{4pt}}$ is a positive homomorphism. Since
$\varsigma_{u\to w}^{\scaleto{(\mathbf{W})}{4pt}}(a)$ is positive and
$\varsigma_{u\to w}^{\scaleto{(\mathbf{W})}{4pt}}(b)$ is strictly positive,
$\varsigma_{u\to w}^{\scaleto{(\mathbf{W})}{4pt}}(ab)=\varsigma_{u\to
w}^{\scaleto{(\mathbf{W})}{4pt}}(a)\varsigma_{u\to
w}^{\scaleto{(\mathbf{W})}{4pt}}(b)$ is strictly positive, ensuring
$ab\in\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$.

$-$ Finally, we claim that if
$x_{u}\in\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ then $\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(x_{u})\in\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$
holds for $v\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}u$. Indeed, the case
$v=u$ being obvious, we may assume
$v>_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}u$. Since
$x_{u}\in\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$, either
$x_{u}=t^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ in which case the statement
obviously holds, or there exists
$w\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}u$ such that $\varsigma_{u\to
w}^{\scaleto{(\mathbf{W})}{4pt}}(x_{u})\in
Q^{\scaleto{(\mathbf{W})}{4pt}}_{w}$. If
$w\leq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v$ then $\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(x_{u})=\varsigma_{w\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(\varsigma_{u\to
w}^{\scaleto{(\mathbf{W})}{4pt}}(x_{u}))\in\varsigma_{w\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(Q^{\scaleto{(\mathbf{W})}{4pt}}_{w})\subseteq
P^{\scaleto{(\mathbf{W})}{4pt}}_{v}\subseteq\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$
follows since
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
is a direct system and
$P^{\scaleto{(\mathbf{W})}{4pt}}_{v}\subseteq\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$,
respectively, whereas if $w>_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}v$ then
$\varsigma_{v\to w}^{\scaleto{(\mathbf{W})}{4pt}}(\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(x_{u}))=\varsigma_{u\to
w}^{\scaleto{(\mathbf{W})}{4pt}}(x_{u})\in
Q^{\scaleto{(\mathbf{W})}{4pt}}_{w}$ implies $\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(x_{u})\in\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{v}\subseteq\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$.

We have just proved that
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
modified by extending, for $u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$, the
positive cone $P^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ of
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ by
$\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ is a direct system of abelian
$po$-group. Since $m$ is maximal, for
$u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$,
$P^{\scaleto{(\mathbf{W})}{4pt}}_{u}=\bar{P}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$
and hence
$Q^{\scaleto{(\mathbf{W})}{4pt}}_{u}=\bar{Q}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$
follow. Therefore, the preimage of any strictly positive element is strictly
positive, so our claim is settled.

We are ready to construct an element of $\mathcal{P}$ which is strictly larger
than $m$.

$\bullet$ First we show that $P^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ can be
extended into a total order of the group reduct of
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$. By [4, Corollary 13 on page 39] any
partial order of an abelian group can be extended to a total order of the
group if and only if the group is torsion-free. But
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ is torsion-free since
L${}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ is an abelian $\ell$-group,
$\ell$-groups are known to be torsion-free (see e.g. [8, Corollary 0.1.2.]),
and G${}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ shares its multiplication with
L${}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$. Hence there exists an extension
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ of
$P^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ which makes the group reduct of
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ an abelian $o$-group.

For $\kappa^{\scaleto{(\mathbf{W})}{4pt}}\ni
v>_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$ let

(5.4) $\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}=\varsigma_{s\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}).$

$\bullet$ We claim that for all $\kappa^{\scaleto{(\mathbf{W})}{4pt}}\ni
v\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$,
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ (as a positive cone) makes
the group reduct of G${}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ a $po$-group.

(i) Let
$a\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}\cap({\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}})^{-1}$.
Then $\varsigma_{s\to v}^{\scaleto{(\mathbf{W})}{4pt}}(a_{s})=a$ and
$\varsigma_{s\to v}^{\scaleto{(\mathbf{W})}{4pt}}(b_{s})=a^{-1}$, for some
$a_{s},b_{s}\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$. Now
$\varsigma_{s\to v}^{\scaleto{(\mathbf{W})}{4pt}}({b_{s}}^{-1})=a$ follows,
hence ${b_{s}}^{-1}\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$, too.
Since $b_{s},{b_{s}}^{-1}\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$
and $\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ is a positive cone, it
holds true that $b_{s}=t^{\scaleto{(\mathbf{W})}{4pt}}_{s}$. Hence
$a^{-1}=t^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ and therefore,
$a=t^{\scaleto{(\mathbf{W})}{4pt}}_{v}$.
(ii) Let $a,b\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$. Then
$\varsigma_{s\to v}^{\scaleto{(\mathbf{W})}{4pt}}(a_{s})=a$ and
$\varsigma_{s\to v}^{\scaleto{(\mathbf{W})}{4pt}}(b_{s})=b$, for some
$a_{s},b_{s}\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$. Since
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ is a positive cone, it holds
true that $a_{s}b_{s}\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ and
hence $ab=\varsigma_{s\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(a_{s})\varsigma_{s\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(b_{s})=\varsigma_{s\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(a_{s}b_{s})\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$.

$\bullet$ Next, we claim that
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
modified by replacing, for $v\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$,
the partial order $\leq^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ of
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ by the partial order given by (the
positive cone) $\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ is a direct
system of abelian $po$-groups. We need to prove the positivity of the
$\varsigma_{u\to v}^{\scaleto{(\mathbf{W})}{4pt}}$’s.
\- If $u,v<_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$ then $\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}$ is positive since the ordering of
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ and
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ have not been changed in the direct
system
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$.
\- Assume $u<_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$ and
$v\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$. Since
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}\supseteq
P^{\scaleto{(\mathbf{W})}{4pt}}_{s}$, $\varsigma_{u\to
s}^{\scaleto{(\mathbf{W})}{4pt}}$ is positive so we may assume
$v>_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$. Because of the definition of
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ in (5.4), $\varsigma_{s\to
v}^{\scaleto{(\mathbf{W})}{4pt}}$ is positive, too. Therefore,
$\varsigma_{u\to v}^{\scaleto{(\mathbf{W})}{4pt}}=\varsigma_{s\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\circ\varsigma_{u\to
s}^{\scaleto{(\mathbf{W})}{4pt}}$, is positive.
\- Assume $u\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$. Let
$a\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$. There exists
$x_{s}\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ such that
$a=\varsigma_{s\to u}^{\scaleto{(\mathbf{W})}{4pt}}(x_{s})$ hence
$\varsigma_{u\to v}^{\scaleto{(\mathbf{W})}{4pt}}(a)=\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(\varsigma_{s\to
u}^{\scaleto{(\mathbf{W})}{4pt}}(x_{s}))=\varsigma_{s\to
v}^{\scaleto{(\mathbf{W})}{4pt}}(x_{s})$ is in
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ by (5.4).

Although we constructed $\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ such
that $\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}\supseteq
P^{\scaleto{(\mathbf{W})}{4pt}}_{s}$, and also
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}\supset
P^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ follows by the indirect assumption (since
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ is total and
$P^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ is not),
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}\supseteq
P^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ does not necessarily hold for
$v>_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$. Therefore, the direct system of
the previous point is not necessarily larger than $m$. Our next aim is to
construct one which is, by considering the partial order generated by
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ and
$P^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ at those indices. To that end

$\bullet$ first we claim that for
$v>_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$,
$(\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v})^{-1}\cap
P^{\scaleto{(\mathbf{W})}{4pt}}_{v}=\\{t^{\scaleto{(\mathbf{W})}{4pt}}_{v}\\}$.
Assume $\\{t^{\scaleto{(\mathbf{W})}{4pt}}_{v}\\}\neq
a\in(\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v})^{-1}\cap
P^{\scaleto{(\mathbf{W})}{4pt}}_{v}$. Since
$a^{-1}\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ there exists
$x_{s}\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$ such that
$a^{-1}=\varsigma_{s\to j}^{\scaleto{(\mathbf{W})}{4pt}}(x_{s})$. Therefore,
$a=\varsigma_{s\to j}^{\scaleto{(\mathbf{W})}{4pt}}(x_{s}^{-1})$ holds. Since
$a\in Q^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ and since the preimage of any
strictly positive element is strictly positive, it follows that $x_{s}^{-1}\in
Q^{\scaleto{(\mathbf{W})}{4pt}}_{s}$. Since
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}\supseteq
P^{\scaleto{(\mathbf{W})}{4pt}}_{s}$, $x_{s}^{-1}$ is strictly positive in
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$, too, contradicting to
$x_{s}\in\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$.

Therefore, by Lemma 5.2, for $v>_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$,
there exists the smallest partial order (denote its positive cone by
$\widehat{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$) of the group reduct of
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ which contains both
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ and
$P^{\scaleto{(\mathbf{W})}{4pt}}_{v}$, and let
$\widehat{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}=\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}$.
Since the homomorphisms of
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
are positive (with respect to the system of its own positive cones
$P^{\scaleto{(\mathbf{W})}{4pt}}_{i}$) and are also positive with respect to
the system of positive cones where for
$v\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$,
$P^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ is replaced by
$\widetilde{P}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$, it is obvious from claim ii
of Lemma 5.2 that the homomorphisms of
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
are positive, too, with respect to the systems of positive cones where for
$v\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$,
$P^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ is replaced by
$\widehat{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$. Therefore,
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
modified by replacing, for $v\geq_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$,
the positive cone $P^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ of
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ by
$\widehat{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ is a direct system of abelian
$po$-groups. Since for $v>_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}s$,
$\widehat{P}^{\scaleto{(\mathbf{W})}{4pt}}_{v}\supseteq
P^{\scaleto{(\mathbf{W})}{4pt}}_{s}$, and since
$\widehat{P}^{\scaleto{(\mathbf{W})}{4pt}}_{s}\supset
P^{\scaleto{(\mathbf{W})}{4pt}}_{s}$, it contradicts to the maximality of $m$
in $\mathcal{P}$.

We have shown that
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
is a direct system of abelian $o$-groups. Obviously, for every
$u\in\kappa^{\scaleto{(\mathbf{W})}{4pt}}$ the diagram in Fig. 16 commutes and
all its arrows are embeddings, since the universes of the respective algebras
and the mappings have not been changed.

For $u\in\kappa_{I}^{\scaleto{(\mathbf{W})}{4pt}}$ let

(5.5)
$\textbf{{H${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}}=\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},$

and let

$\mathcal{W}=\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\textbf{{H${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\langle\kappa_{o}^{\scaleto{(\mathbf{W})}{4pt}},\kappa_{J}^{\scaleto{(\mathbf{W})}{4pt}},\kappa_{I}^{\scaleto{(\mathbf{W})}{4pt}},\leq_{\kappa^{\scaleto{(\mathbf{W})}{3pt}}}\rangle}.$

First we verify that $\mathcal{W}$ is a bunch of layer groups.

  * •

We have already seen that
$\langle{\kappa^{\scaleto{(\mathbf{W})}{4pt}}},\leq_{\kappa^{\scaleto{(\mathbf{W})}{3pt}}}\rangle$
is a totally ordered set.

  * •

Since by (C1) it holds true that
$\min{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}=\min{\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}=\min{\kappa^{\scaleto{(\mathbf{Z})}{4pt}}}$,
and since by the construction
${\kappa^{\scaleto{(\mathbf{W})}{4pt}}}={\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}\cup{\kappa^{\scaleto{(\mathbf{Z})}{4pt}}}$,
it follows that ${\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$ has a least element,
too, and
$\min{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}=\min{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}=\min{\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}=\min{\kappa^{\scaleto{(\mathbf{Z})}{4pt}}}$.

  * •

We check that (5.2) defines a partition of
${\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$. By (5.2) and (5.1),
$\kappa_{o}^{\scaleto{(\mathbf{W})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{W})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{W})}{4pt}}=(\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{o}^{\scaleto{(\mathbf{Z})}{4pt}})\cup(\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}})\cup(\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}})=(\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}})\cup(\kappa_{o}^{\scaleto{(\mathbf{Z})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}})=\kappa^{\scaleto{(\mathbf{Y})}{4pt}}\,\cup\,\kappa^{\scaleto{(\mathbf{Z})}{4pt}}={\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$.
These three are pairwise disjoint, too. Assume, on the contrary, that $u$ is
in, say,
$\kappa_{I}^{\scaleto{(\mathbf{W})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{W})}{4pt}}$.
Then by distributivity,
$u\overset{(\ref{PaRTitiOn})}{\in}(\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}})\cap(\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}})=(\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}})\cup(\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}})\cup(\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}})\cup(\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}})=\emptyset\cup(\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}})\cup(\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}})\cup\emptyset\subseteq{\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}\cap{\kappa^{\scaleto{(\mathbf{Z})}{4pt}}}={\kappa^{\scaleto{(\mathbf{X})}{4pt}}}=\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}$.
If
$u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}\cup\kappa_{o}^{\scaleto{(\mathbf{X})}{4pt}}$
then
$u\in\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}\cup\kappa_{o}^{\scaleto{(\mathbf{Y})}{4pt}}$
and
$u\in\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}}\cup\kappa_{o}^{\scaleto{(\mathbf{Z})}{4pt}}$
holds by (C1), and hence $u\notin\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}$ and
$u\notin\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}}$ follows, respectively, a
contradiction to
$u\in(\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}})\cup(\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}})$,
whereas if $u\in\kappa_{J}^{\scaleto{(\mathbf{X})}{4pt}}$ then
$u\in\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}$ and
$u\in\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}}$ holds by (C1), and hence
$u\notin\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}$ and
$u\notin\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}}$ follows, a contradiction to
$u\in(\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{Z})}{4pt}})\cup(\kappa_{I}^{\scaleto{(\mathbf{Z})}{4pt}}\cap\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}})$,
too.

  * •

For $u\in{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$,
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ given by (5.3), and for
$u\in\kappa_{I}^{\scaleto{(\mathbf{W})}{4pt}}$,
H${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ given by (5.5) are abelian $o$-groups
and obviously
$\textbf{{H${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}}\leq\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}}$
hold for $u\in\kappa_{I}^{\scaleto{(\mathbf{W})}{4pt}}$.

  * •

We have already seen that for $u,v\in{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$,
$u<v$, $\varsigma_{u\to v}^{\scaleto{(\mathbf{W})}{4pt}}$ is a homomorphism
mapping G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ to
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$.

  * •

We have already seen that
$\langle\textbf{{G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$}},\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}\rangle_{\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$
is a direct system of abelian $o$-groups.

  * •

As for (G2), it is obvious that for
$v\in\kappa_{I}^{\scaleto{(\mathbf{W})}{4pt}}$, $\varsigma_{u\to
v}^{\scaleto{(\mathbf{W})}{4pt}}$ maps G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$
into H${}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$, since it maps into
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$, and
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ and
H${}^{\scaleto{(\mathbf{W})}{4pt}}_{v}$ coincide.

Finally, we prove that the involutive FLe-algebra $\mathbf{W}$ of
$\mathcal{W}$ is an amalgam of $\mathtt{V}$ in
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}$ (or in
$\in\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$).

Figure 17. Amalgam of $\mathtt{V}$

That $\mathbf{Y}$ is embeddable into $\mathbf{W}$ is verified, using Corollary
3.3, as follows. By (5.2), the inclusion map is and order embedding from
${\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}$ into
${\kappa^{\scaleto{(\mathbf{W})}{4pt}}}$ which preserves the least element and
the partition. Second, as shown above, for every
$u\in{\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}$, there exists an embedding
$\iota_{3,u}$ of G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$ into
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ such that for every
$u,v\in{\kappa^{\scaleto{(\mathbf{Y})}{4pt}}}$, $u<v$ the top left square in
Fig. 15 is commutative. Third, if
$u\in\kappa_{I}^{\scaleto{(\mathbf{Y})}{4pt}}$ then
H${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$ is mapped into
H${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$, since
$\textbf{{H${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}}\leq\textbf{{G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$}}$
and G${}^{\scaleto{(\mathbf{Y})}{4pt}}_{u}$ is mapped into
G${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ which is equal to
H${}^{\scaleto{(\mathbf{W})}{4pt}}_{u}$ by (5.5). The last item in Corollary
3.3 does not apply since
$\mathbf{Y}\in\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}\cup\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$
implies $\kappa_{J}^{\scaleto{(\mathbf{Y})}{4pt}}=\emptyset$, so we are done.
Analogously we can verify that $\mathbf{Z}$ is embeddable into $\mathbf{W}$,
too.

Let $x\in X$. Since by (2.5) and (2.2), the universe of any algebra in
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}\cup\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$
is the disjoint union of the universes of its layer groups and layer
subgroups, there exists a unique $u\in{\kappa^{\scaleto{(\mathbf{X})}{4pt}}}$
such that $x$ is an element of G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$ or
there exists a unique $u\in\kappa_{I}^{\scaleto{(\mathbf{X})}{4pt}}$ such that
$x$ is an element of
$\accentset{\bullet}{\textbf{{H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$}}}$. In
the former case
$\iota_{3}(\iota_{1}(x))=\iota_{3,u}(\iota_{1,u}(x))\overset{Fig.\ref{amalgamationTwiceFIG}}{=}\iota_{4,u}(\iota_{2,u}(x))=\iota_{4}(\iota_{2}(x))$
holds and we are done. In the latter case, referring to the row before (2.2),
there exist an element $y$ of H${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$ such
that
$x={y}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{X}}{{u}^{{\prime^{\mkern-4.0mu{}^{{}_{\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{X}}}}}}}}$,
and hence
$\iota_{3}(\iota_{1}(x))=\iota_{3}(\iota_{1}({y}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}^{{\prime^{\mkern-4.0mu{}^{{}_{\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{X}}}}}}}}))={\iota_{3}(\iota_{1}(y))}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{Z}}{{\iota_{3}(\iota_{1}(u))}^{{\prime^{\mkern-4.0mu{}^{{}_{\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{Z}}}}}}}}$
follows. Since $y$ and $u$ are elements in
G${}^{\scaleto{(\mathbf{X})}{4pt}}_{u}$, just like in the previous case, the
latest is equal to
${\iota_{4}(\iota_{2}(y))}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{Z}}{{\iota_{4}(\iota_{2}(u))}^{{\prime^{\mkern-4.0mu{}^{{}_{\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{Z}}}}}}}}=\iota_{4}(\iota_{2}({y}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{u}^{{\prime^{\mkern-4.0mu{}^{{}_{\mathbin{{\mathbin{*\mkern-9.0mu\circ}}_{X}}}}}}}}))=\iota_{4}(\iota_{2}(x))$,
and we are done, too.

(3) It is known that the strong amalgamation property fails in
$\mathfrak{A}^{\mathfrak{c}}$ [2]. Consider a V-formation
$\mathtt{V}=\langle\mathbf{A},\mathbf{B}_{1},\mathbf{B}_{2},\iota_{1},\iota_{2}\rangle$
in $\mathfrak{A}^{\mathfrak{c}}$ for which no strong amalgam exists in
$\mathfrak{A}^{\mathfrak{c}}$.

$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$: Assume that
$\mathtt{V}$ (where $\mathbf{A},\mathbf{B}_{1},\mathbf{B}_{2}$ are viewed as
idempotent-symmetric odd involutive FLe-chains, and $\iota_{1},\iota_{2}$ are
viewed as embeddings in
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$) has a strong
amalgam $\langle\mathbf{C},\iota_{3},\iota_{4}\rangle$ in
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$. Then, by (E2) in
Corollary 3.3, $\iota_{3}$ and $\iota_{4}$ induces a strong amalgam
$\langle\mathbf{D},\iota_{3}|_{D},\iota_{4}|_{D}\rangle$ of $\mathbf{B}_{1}$
and $\mathbf{B}_{2}$ with common subgroup $\mathbf{A}$ in
$\mathfrak{A}^{\mathfrak{c}}$, where $\mathbf{D}$ is the layer group of
$\mathbf{C}$ corresponding to the unit element of $\mathbf{C}$. Contradiction.

The $\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}$ case can be
treated similarly. Let $\leq\,=\\{\langle 0,0\rangle,\langle
0,1\rangle,\langle 1,1\rangle\\}$ and consider the involutive FLe-chains
$X_{\mathbf{A}}$, $X_{\mathbf{B}_{1}}$, $X_{\mathbf{B}_{2}}$, given by the
group representations
$\langle\\{\mathbbm{1},\textbf{{A}}\\},\emptyset,\varsigma_{0\to
1}\rangle_{\langle\emptyset,\emptyset,\\{0,1\\},\leq\rangle}$,
$\langle\\{\mathbbm{1},\textbf{{B${}_{1}$}}\\},\emptyset,\varsigma_{0\to
1}\rangle_{\langle\emptyset,\emptyset,\\{0,1\\},\leq\rangle}$, and
$\langle\\{\mathbbm{1},\textbf{{B${}_{2}$}}\\},\emptyset,\varsigma_{0\to
1}\rangle_{\langle\emptyset,\emptyset,\\{0,1\\},\leq\rangle}$, respectively.
These are even since the unit element $0$ is in the third partition element,
c.f. the last sentence before item C of Theorem 2.5. These are idempotent-
symmetric because the second partition element is empty. By Corrollary 3.3,
using the identity map between the trivial groups, $\iota_{1}$ and $\iota_{2}$
can be extended to embeddings of the V-formation $\mathtt{V}_{1}$ in Fig. 18.

Figure 18.

Again, if we assume the existence of a strong amalgam $\langle
X_{\mathbf{C}},\iota_{3},\iota_{4}\rangle$ of $\mathtt{V}_{1}$ in
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}$ then by Corollary
3.3, the $1$-level group of $X_{\mathbf{C}}$ along with the induced embeddings
of $\iota_{3}$ and $\iota_{4}$ (on the $1$-level) would yield a strong amalgam
of $\mathtt{V}$ in $\mathfrak{A}^{\mathfrak{c}}$, a contradiction. ∎

As pointed out in [13], totally ordered Sugihara monoids are either odd or
even888There, the term ‘disconnected’ is used for ‘even’, and the term
‘connected’ is used for ‘odd’.: Indeed, ${t}{{}^{\prime}}=f$ and since $f$ is
idempotent,
$t={f}{{}^{\prime}}={f}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{f}\geq
f$ holds on the one hand. Now, $f<x<t$ would imply $f<{x}{{}^{\prime}}<t$, and
hence
${x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{x}{{}^{\prime}}}\geq{\min(x,{x}{{}^{\prime}})}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{\min(x,{x}{{}^{\prime}})}=\min(x,{x}{{}^{\prime}})>f$
contradicts residuation. Moreover, in Sugihara monoids every element is
idempotent, hence obviously, the residual complement of a positive idempotent
element is idempotent, too, thus ensuring that totally ordered Sugihara
monoids belong to either
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$ or
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{1},\mathfrak{symm}}$. Denote by
$\mathbbm{1}$ the trivial (one-element) group. Since the group representation
of totally ordered even or odd Sugihara monoids is of the form
$\langle\mathbbm{1}_{u},\mathbbm{1}_{u},\varsigma^{u\to
v}\rangle_{\langle\emptyset,\emptyset,\kappa,\leq_{\kappa}\rangle}$ or
$\langle\mathbbm{1}_{u},\mathbbm{1}_{u},\varsigma^{u\to
v}\rangle_{\langle\\{t\\},\emptyset,\kappa\setminus\\{t\\},\leq_{\kappa}\rangle},$
respectively [12], in every layer the groups in the induced V-formation are
trivial, and the construction of Theorem 5.3 produces the trivial group as
their amalgamation in every layer. Hence the amalgamation, as done in this
paper, of a V-formation of totally ordered even or odd Sugihara monoids is
also a totally ordered even or odd Sugihara monoid, respectively. Summing up,
a by-product of the construction in Theorem 5.3 is Corollary 5.4 (cf. [13],
where the same conclusion is reached by quantifier elimination in first-order
theories).

######  Corollary 5.4.

The classes of odd and even totally ordered Sugihara monoids have the
amalgamation property.

Recall that $\mathfrak{J}^{\mathfrak{sl}}_{\mathfrak{0}}$ and
$\mathfrak{I}^{\mathfrak{sl}}_{\mathfrak{0},\mathfrak{symm}}$ are the classes
of the semilinear members of $\mathfrak{J}_{\mathfrak{0}}$ and
$\mathfrak{I}_{\mathfrak{0},\mathfrak{symm}}$, respectively. These are
varieties and are generated by $\mathfrak{J}^{\mathfrak{c}}_{\mathfrak{0}}$
and $\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$,
respectively. Indeed, $\mathfrak{J}^{\mathfrak{sl}}_{\mathfrak{0}}$ is clearly
a variety and the class of its chains is
$\mathfrak{J}^{\mathfrak{c}}_{\mathfrak{0}}$. By Theorem 2.5,
$\\{{x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x}:x\in
X\\}$ is the set of positive idempotent elements of the involutive FLe-algebra
$\mathbf{X}$, hence idempotent-symmetry of involutive FLe-algebras can be
captured by posing
${{({x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x})}{{}^{\prime}}}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{({x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x})}{{}^{\prime}}}={({x}\mathbin{\mathbin{\rightarrow_{{\mathbin{*\mkern-9.0mu\circ}}}}}{x})}{{}^{\prime}}$
or its equivalent simpler variant
${{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{x}{{}^{\prime}}}}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{x}{{}^{\prime}}}}={x}\mathbin{{\mathbin{*\mkern-9.0mu\circ}}}{{x}{{}^{\prime}}}$.
Therefore, also $\mathfrak{I}^{\mathfrak{sl}}_{\mathfrak{0},\mathfrak{symm}}$
is a variety, and the class of its chains is
$\mathfrak{I}^{\mathfrak{c}}_{\mathfrak{0},\mathfrak{symm}}$. Finally, a
variety of residuated lattices is semilinear if and only if it is generated by
its chains [11, Proposition].
As shown in [14, Corollary 44 and Theorem 49], respectively, any variety of
commutative pointed residuated lattices has the congruence extension property,
and a variety of semilinear residuated lattices satisfying the congruence
extension property has the amalgamation property if the class of its chains
has the amalgamation property. Therefore,
$\mathfrak{I}^{\mathfrak{sl}}_{\mathfrak{0},\mathfrak{symm}}$ has the
amalgamation property. A strengthening of the amalgamation property is the
transferable injections property. A class of algebras $\mathfrak{U}$ has the
transferable injections property if whenever
$\mathbf{A},\mathbf{B},\mathbf{C}\in\mathfrak{U}$, $\iota_{1}$ is an embedding
of $\mathbf{A}$ into $\mathbf{B}$, and $\iota_{2}$ is a homomorphism from
$\mathbf{A}$ into $\mathbf{C}$, there exist an algebra
$\mathbf{D}\in\mathfrak{U}$, a homomorphism $\iota_{3}$ from $\mathbf{B}$ into
$\mathbf{D}$, and an embedding $\iota_{4}$ from $\mathbf{C}$ into $\mathbf{D}$
such that $\iota_{3}\circ\iota_{1}=\iota_{4}\circ\iota_{2}$. As shown in [14,
Corollary 44], any variety of commutative pointed residuated lattices has the
amalgamation property iff it has the transferable injections property.

######  Corollary 5.5.

The variety $\mathfrak{I}^{\mathfrak{sl}}_{\mathfrak{0},\mathfrak{symm}}$ has
the transferable injections property.

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