# Structurally complete finitary extensions of positive Łukasiewicz logic

Paolo Aglianò  DIISM, Università di Siena, Siena, Italy<EMAIL_ADDRESS>Francesco Manfucci  DIISM, Università di Siena, Siena, Italy
<EMAIL_ADDRESS>
###### Abstract.

In this paper we study $\mathcal{MV}^{+}$, i.e. the positive fragment of
Łukasiewicz Multi-Valued Logic $\mathcal{MV}$. In particular we describe all
the finitary extensions of $\mathcal{MV}^{+}$ that are structurally complete
and all the axiomatic extensions of $\mathcal{MV}^{+}$ that are hereditarily
structurally complete. Examples of hereditarily structurally complete finitary
extensions and non hereditarily structurally complete finitary extensions are
provided.

##  1\. Introduction

A large class of substructural logics (i.e. logics that lack some structural
rules) is given by the extensions of the Full Lambek Calculus $\mathcal{FL}$
(see [27] p. 76). All extensions of $\mathcal{FL}$ have a primitive connective
$\mathbf{0}$ that denotes falsum; so if $\mathcal{L}$ is an extension of
$\mathcal{FL}$ it makes sense to study its positive fragment
$\mathcal{L}^{+}$. The language of $\mathcal{L}^{+}$ is obtained by the
lnguage of $\mathcal{L}$ by deleting $\mathbf{0}$ from the signature; the
valid derivations in $\mathcal{L}^{+}$ are the valid derivations in
$\mathcal{L}$, that contains only $0$-free formulas.

An important subfamily of substructural logics over $\mathcal{FL}$ consists of
logics that satisfy both exchange and weakening but lack contraction; in this
case the primitive connectives can be taken as
$\vee,\wedge,\rightarrow,\mathbf{0},\mathbf{1}$ where of course $\mathbf{1}$
represents truth. The minimal substructural logic of this kind is denoted by
$\mathcal{FL}_{ew}$ and hence all the substructural logics lacking contraction
are an extension of it. Probably Intuitionistic Logic $\mathcal{IL}$ is the
most famous of them but there are many others such as the monoidal logic
$\mathcal{MTL}$ [25], Hajek’s Basic Logic $\mathcal{BL}$ [31], Dummet’s Logic
$\mathcal{GL}$ [31], Product Logic $\mathcal{PL}$ [31] and, last but not
least, Łukasiewicz’s Many Valued logic $\mathcal{MV}$ [37].

All extensions of $\mathcal{FL}$ are algebraizable with an equivalent
algebraic semantics (in the sense of Blok-Pigozzi[17]) that is at least a
quasivariety of algebras; by the same token all positive fragments are
algebraizable as well and moreover the machinery of algebrization takes a very
transparent form in both cases. The equivalent algebraic semantics of
$\mathcal{FL}$ and $\mathcal{FL}^{+}$ are the variety of
$\mathsf{FL}$-algebras [27] and the variety of residuated lattices [33]
respectively. Algebraizability in Blok-Pigozzi’s fashion entails that the
deducibility relation of an algebraizable logic is characterized by means of
the algebraic equational consequence of its equivalent algebraic semantics.
Therefore, interesting properties of algebraizable logics can be studied via
algebraic means.

Hoops are a particular variety of residuated monoids related to logic which
were defined in an unpublished manuscript by Büchi and Owens, inspired by the
work of on partially ordered monoids in [19]. Hoops indeed correspond to
residuated commutative monoids whose order is the inverse divisibility
ordering, i.e. $a\leq b$ if and only if there exists $c$ such that $a=bc$. The
first systematic study of hoops has been carried out by Ferreirim in her PhD
thesis [26] and in her works with Blok [16]; the connection between hoops and
substructural logics has been investigated in [6]. A subvariety of hoops, the
variety $\mathsf{WH}$ of Wajsberg hoops, plays a special role in this
framework. From the algebraic point of view, they can be used to describe
subdirectly irreducible hoops, and the whole variety of hoops can be obtained
as the join of iterated powers of the variety of Wajsberg hoops, in the sense
defined in [16]. Wajsberg hoops also have a peculiar connection with lattice-
ordered abelian groups (abelian $\ell$-groups for short). In fact, the variety
of Wajsberg hoops is generated by its totally ordered members, that are, in
loose terms, either negative cones of abelian $\ell$-groups, or intervals in
abelian $\ell$-groups [8]. However the most relevant property of Wajsberg
hoops in this context is that they are the equivalent algebraic semantics of
the positive fragment of Łukasiewicz logic $\mathcal{MV}$. This will allow us
to reduce logical questions, such as the structural completeness of its
finitary extensions, to a purely algebraic matter.

##  2\. Preliminaries

###  2.1. Universal algebra

Let $\mathsf{K}$ be a class of algebras; we denote by
${\mathbf{I}},{\mathbf{H}},{\mathbf{P}},{\mathbf{S}},{\mathbf{P}}_{u}$ the
class operators sending $\mathsf{K}$ in the class of all isomorphic copies,
homomorphic images, direct products, subalgebras and ultraproducts of members
of $\mathsf{K}$. The operators can and will be composed in the obvious way;
for instance ${\mathbf{S}}{\mathbf{P}}(\mathsf{K})$ denotes all algebras that
are embeddable in a direct product of members of $\mathsf{K}$; moreover there
are relations among the classes resulting from applying operators in a
specific order, for instance
${\mathbf{P}}{\mathbf{S}}(\mathsf{K})\subseteq{\mathbf{S}}{\mathbf{P}}(\mathsf{K})$
and ${\mathbf{H}}{\mathbf{S}}{\mathbf{P}}(\mathsf{K})$ is the largest class we
can obtain composing the operators. We will use all the known relations
without further notice; for those and for any other unexplained algebraic
notion we refer the reader to [20] for a textbook treatment.

If $\tau$ is a type of algebras, an equation is a pair $p,q$ of $\tau$-terms
(i.e. elements of the absolutely free algebra ${\textbf{T}}_{\tau}(\omega)$)
that we write suggestively as $p\approx q$; a universal sentence in $\tau$ is
$\Sigma\Rightarrow\Gamma$ where $\Sigma,\Gamma$ are finite sets of equations;
a universal sentence is a quasiequation if $|\Gamma|=1$ and it is negative if
$\Delta=\emptyset$. Clearly an equation is a quasiequation in which
$\Sigma=\emptyset$. An equation $p(x_{1},\dots,x_{n})\approx
q(x_{1},\dots,x_{n})$ is valid in A (and we write
${\textbf{A}}\vDash\Sigma\Rightarrow\Delta$) if for all $a_{1},\dots,a_{n}\in
A$, $p(a_{1},\dots,a_{n})=q(a_{1},\dots,a_{n})$; if $\Sigma$ is a set of
equations then ${\textbf{A}}\vDash\Sigma$ if ${\textbf{A}}\vDash\sigma$ for
all $\sigma\in\Sigma$. A universal sentence is valid in A (and we write
${\textbf{A}}\vDash\Sigma\Rightarrow\Delta$) if whenever
${\textbf{A}}\vDash\Sigma$ there is a $\varepsilon\approx\delta\in\Delta$ with
${\textbf{A}}\vDash\varepsilon\approx\delta$; in other words a universal
sentence can be understood as the formula
$\forall\mathbf{x}(\bigwedge\Sigma\rightarrow\bigvee\Delta)$. An equation or a
universal sentence is valid in a class $\mathsf{K}$ if it is valid in all
algebras in $\mathsf{K}$.

A class of algebras is a variety if it is closed under
${\mathbf{H}},{\mathbf{S}}$ and ${\mathbf{P}}$, a quasivariety if it is closed
under ${\mathbf{I}}$,${\mathbf{S}}$,${\mathbf{P}}$ and ${\mathbf{P}}_{u}$ and
a universal class if it is closed under
${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}$. The following facts were
essentially discovered by A. Tarski, J. Łòs and A. Lyndon in the pioneering
phase of model theory; for proof of this and similar statements the reader can
consult [21].

######  Lemma 2.1.

Let $\mathsf{K}$ be any class of algebras. Then:

  1. (1)

$\mathsf{K}$ is a universal class if and only if
${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}(\mathsf{K})=\mathsf{K}$ if and only
if it is the class of all algebras in which a set of universal sentences is
valid;

  2. (2)

$\mathsf{K}$ is a quasivariety if and only if
${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}{\mathbf{P}}_{u}(\mathsf{K})=\mathsf{K}$
if and only if it is the class of all algebras in which a set of
quasiequations is valid;

  3. (3)

$\mathsf{K}$ is a variety if and only if
${\mathbf{H}}{\mathbf{S}}{\mathbf{P}}(\mathsf{K})=\mathsf{K}$ if and only if
it is the class of all algebras in which a set of equations is valid.

We will often write ${\mathbf{V}}$ for ${\mathbf{H}}{\mathbf{S}}{\mathbf{P}}$
and ${\mathbf{Q}}$ for ${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}{\mathbf{P}}_{u}$.
It is clear that both ${\mathbf{V}}$ and ${\mathbf{Q}}$ are closure operators
on classes of algebras of the same type; this implies among other things that
for a given variety $\mathsf{V}$ the class of subvarieties of $\mathsf{V}$ is
a complete lattice which we denote by $\Lambda(\mathsf{V})$. If $\mathsf{Q}$
is a quasivariety that is not a variety then $\Lambda(\mathsf{Q})$ is still a
lattice but it is not necessarily complete (in particular does not need to
have maximum). The class of subquasivarieties of a quasivariety $\mathsf{Q}$
is a complete lattice denoted by $\Lambda_{q}(\mathsf{Q})$.

For the definition of free algebras in a class $\mathsf{K}$ on a set $X$ of
generators in symbols ${\textbf{F}}_{\mathsf{K}}(X)$, we refer again to [20].
We merely observe that every free algebra on a class $\mathsf{K}$ belongs to
${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}(\mathsf{K})$. It follows that every free
algebra in $\mathsf{K}$ is free in
${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}(\mathsf{K})$ and therefore for any
quasivariety $\mathsf{Q}$,
${\textbf{F}}_{\mathsf{Q}}(X)={\textbf{F}}_{{\mathbf{V}}(\mathsf{Q})}(X)$.

Let ${\textbf{B}},({\textbf{A}}_{i})_{i\in I}$ be algebras in the same
signature; we say that B embeds in $\prod_{i\in I}{\textbf{A}}_{i}$ if
${\textbf{B}}\in{\mathbf{I}}{\mathbf{S}}(\prod_{i\in I}{\textbf{A}}_{i})$. Let
$p_{i}$ be the $i$-th projection, or better, the composition of the
isomorphism and the $i$-th projection, from B to ${\textbf{A}}_{i}$; the
embedding is subdirect if for all $i\in I$,
$p_{i}({\textbf{B}})={\textbf{A}}_{i}$ and in this case we will write

${\textbf{B}}\leq_{sd}\prod_{i\in I}{\textbf{A}}_{i}.$

An algebra B is subdirectly irreducible if it is nontrivial and for any
subdirect embedding

${\textbf{B}}\leq_{sd}\prod_{i\in I}{\textbf{A}}_{i}.$

there is an $i\in I$ such that B and ${\textbf{A}}_{i}$ are isomorphic. It can
be shown that A is subdirectly irreducible if and only if the congruence
lattice $\operatorname{Con}({\textbf{A}})$ of A has a unique minimal element
different from the trivial congruence. If $\mathsf{V}$ is a variety we denote
by $\mathsf{V}_{si}$ the class of subdirectly irreducible algebras in
$\mathsf{V}$.

######  Theorem 2.2.

  1. (1)

(Birkhoff [15]) Every algebra can be subdirectly embedded in a product of
subdirectly irreducible algebras. So if ${\textbf{A}}\in\mathsf{V}$, then A
can be subdirectly embedded in a product of members of $\mathsf{V}_{si}$.

  2. (2)

(Jónsson’s Lemma [34]) Suppose that $\mathsf{K}$ is a class of algebras such
that $\mathsf{V}(\mathsf{K})$ is congruence distributive; then
$\mathsf{V}_{si}\subseteq{\mathbf{H}}{\mathbf{S}}{\mathbf{P}}_{u}(\mathsf{K})$.

If $\mathsf{Q}$ is a quasivariety and ${\textbf{A}}\in\mathsf{Q}$, a
$\mathsf{Q}$-congruence of A is a congruence $\theta$ such that
${\textbf{A}}/\theta\in\mathsf{Q}$; clearly $\mathsf{Q}$-congruences form an
algebraic lattice $\textrm{Con}_{\mathsf{Q}}({\textbf{A}})$. An algebra
${\textbf{A}}\in\mathsf{Q}$ is $\mathsf{Q}$-irreducible if
$\textrm{Con}_{\mathsf{Q}}({\textbf{A}})$ has a unique minimal element; since
clearly $\textrm{Con}_{\mathsf{Q}}({\textbf{A}})$ is a meet subsemilattice of
$\operatorname{Con}({\textbf{A}})$, any subdirectly irreducible algebra is
$\mathsf{Q}$-irreducible in any quasivariety $\mathsf{Q}$ to which it belongs.
For a quasivariety $\mathsf{Q}$ we denote by $\mathsf{Q}_{ir}$ the class of
$\mathsf{Q}$-irreducible algebras in $\mathsf{Q}$. We have the equivalent of
Birkhoff’s and Jónsson’s results for quasivarieties:

######  Theorem 2.3.

Let $\mathsf{Q}$ be any quasivariety.

  1. (1)

(Mal’cev [38]) Every ${\textbf{A}}\in\mathsf{Q}$ is subdirectly embeddable in
a product of algebras in $\mathsf{Q}_{ir}$.

  2. (2)

(Czelakowski-Dziobiak [23]) If $\mathsf{Q}={\mathbf{Q}}(\mathsf{K})$, then
$\mathsf{Q}_{ir}\subseteq{\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}(\mathsf{K})$.

###  2.2. The Blok-Pigozzi connection

In what follows by a logic $\mathcal{L}$ we mean a substitution invariant
consequence relation $\vdash_{\mathcal{L}}$ on the set of terms
${\textbf{T}}_{\tau}(\omega)$ (also called _algebra of formulas_) of some
algebraic language $\tau$. A clause in $\mathcal{L}$ is a formal expression
$\Sigma\Rightarrow\Delta$ where $\Sigma,\Delta$ are finite set of formulas of
$\mathcal{L}$; a clause is a rule if $\Delta=\\{\delta\\}$. A rule is an axiom
if $\Sigma=\emptyset$.

If the equivalence between formulas (i.e. provable equivalence according to
$\vdash$) is a congruence on the algebra of formulas, then we can form the
Lindenbaum-Tarski algebra as the quotient of provably equivalent formulas of
the algebra of formulas. If the quasivariety $\mathsf{Q}_{\mathcal{L}}$
generated by the Lindembaum-Tarski algebra satisfies further conditions, then
it is the class of algebraic models of $\mathcal{L}$ and is algebraizable with
equivalent algebraic semantics $\mathsf{Q}_{\mathcal{L}}$; Blok and Pigozzi
[17] formalized this connection and gave necessary and sufficient conditions
for a logic to be algebraizable. Essentially, one needs a finite set of
equations $\tau=\\{\delta_{i}\approx\varepsilon_{i}:i=1,\dots,n\\}$ in the
language of $\mathsf{Q}_{\mathcal{L}}$ and a finite set of formulas of
$\mathcal{L}$ in two variables
$\Delta(x,y)=\\{\varphi_{1}(x,y),\dots,\varphi_{m}(x,y)\\}$ that allow to
transform equations, quasiequations and universal sentences in
$\mathsf{Q}_{\mathcal{L}}$ into formulas, rules and clauses of $\mathcal{L}$
and viceversa; moreover this transformation must respect both the consequence
relation and the semantical consequence. That is to say, for all sets of
formulas $\Gamma$ of $\mathcal{L}$ and formulas
$\varphi\in{\textbf{T}}_{\tau}(\omega)$

$\Gamma\vdash\varphi\quad\text{if and only
if}\quad\\{\delta_{i}(\Gamma)\approx\varepsilon_{i}(\Gamma):i=1,\dots,n\\}\vDash_{\mathsf{Q}_{\mathcal{L}}}\\{\delta_{i}(\varphi)\approx\varepsilon_{i}(\varphi):i=1,\dots,n\\}$

where of course $\delta_{i}(\Gamma)\approx\varepsilon_{i}(\Gamma)$ is a
shorthand for $\delta_{i}(\psi)\approx\varepsilon_{i}(\psi)$ for all
$\psi\in\Gamma$ and also

$\mathsf{Q}_{\mathcal{L}}\vDash(x\approx
y)\Leftrightarrow(\bigwedge_{i=1}^{n}\bigwedge_{j=1}^{m}(\delta_{i}(\varphi_{j}(x,y))\approx\varepsilon_{i}(\varphi_{j}(x,y))).$

A quasivariety $\mathsf{Q}$ is a quasivariety of logic if it is the equivalent
quasivariety semantics for some logic $\mathsf{L}_{\mathsf{Q}}$; the Galois
connection between algebraizable logics and quasivarieties of logic is given
by

$\mathcal{L}_{\mathsf{Q}_{\mathcal{L}}}=\mathcal{L}\qquad\qquad\mathsf{Q}_{\mathcal{L}_{\mathsf{Q}}}=\mathsf{Q}.$

###  2.3. Structural completeness in algebra and logic

An extension of $\mathcal{L}$ over the language $\tau$ is a logic
$\mathcal{L}^{\prime}$ over the same language such that
$\Sigma\vdash_{\mathcal{L}}\delta$ implies
$\Sigma\vdash_{\mathcal{L}^{\prime}}\delta$. A finitary extension of
$\mathcal{L}$ is an extension of $\mathcal{L}$ obtained by adding a set of
rules to the calculus of $\mathcal{L}$; an axiomatic extension of
$\mathcal{L}$ is an extension of $\mathcal{L}$ obtained by adding a set of
axioms to the calculus of $\mathcal{L}$. The class of the (finitary,
axiomatic) extension of the logic $\mathcal{L}$ is never empty, since
$\mathcal{L}$ clearly belongs to it. Therefore we can define closure operators
in a standard way and hence complete lattices; so $\textrm{Th}(\mathcal{L})$,
$\textrm{Th}_{f}(\mathcal{L})$ and $\textrm{Th}_{a}(\mathcal{L})$ will be the
lattice of all the extensions, finitary extensions and axiomatic extensions of
$\mathcal{L}$ respectively. Via the Blok-Pigozzi connection we get:

######  Theorem 2.4.

Let $\mathcal{L}$ be an algebraizable logic with equivalent algebraic
semantics $\mathsf{Q}_{\mathcal{L}}$. Then

  1. (1)

the lattice $\textrm{Th}_{a}(\mathcal{L})$ of the axiomatic extensions of
$\mathcal{L}$ is dually isomorphic with $\Lambda(\mathsf{Q})$;

  2. (2)

the lattice $\textrm{Th}_{f}(\mathcal{L})$ of the finitary extensions of
$\mathcal{L}$ is dually isomorphic with $\Lambda_{q}(\mathsf{Q})$.

An algebraizable logic $\mathcal{L}$ is tabular if it is the logic of a finite
algebra; in other words $\mathsf{Q}_{\mathcal{L}}$ is a finitely generated
quasivariety, i.e. $\mathsf{Q}_{\mathcal{L}}={\mathbf{Q}}({\textbf{A}})$ for
some finite algebra A. A logic is locally tabular if, for any finite $k$,
there exist only finitely many pairwise nonequivalent formulas in
$\mathcal{L}$ built from the variables $x_{1},\dots,x_{k}$. It is clear that
an algebraizable logic is locally tabular if and only if
$\mathsf{Q}_{\mathcal{L}}$ is a locally finite quasivariety.

A clause $\Sigma\Rightarrow\Delta$ is admissible in $\mathcal{L}$ if every
substitution that makes the premises into a theorem of $\mathcal{L}$, also
makes at least one of the conclusions in $\Delta$ a theorem of $\mathcal{L}$.
In particular a rule is admissible in $\mathcal{L}$ if, when added to its
calculus, it does not produce any new theorem. A clause
$\Sigma\Rightarrow\Delta$ is derivable in $\mathcal{L}$ if
$\Sigma\vdash\delta$ for some $\delta\in\Delta$. A logic $\mathcal{L}$ is
structurally complete if every admissible rule of $\mathcal{L}$ is derivable
in $\mathcal{L}$; a logic is hereditarily structurally complete if every
finitary extension of $\mathcal{L}$ is structurally complete. Determining
structural completeness of a logic is in general a very deep and challenging
problem; here we will use only the parts of the theory that are necessary but
for an extensive treatment of the subject we direct the reader to [10].

A quasivariety $\mathsf{Q}$ is structural if for every subvariety
$\mathsf{Q}^{\prime}\subseteq\mathsf{Q}$,
${\mathbf{H}}(\mathsf{Q}^{\prime})={\mathbf{H}}(\mathsf{Q})$ implies
$\mathsf{Q}^{\prime}=\mathsf{Q}$.

######  Theorem 2.5.

[13]For a quasivariety $\mathsf{Q}$ the following are equivalent:

  1. (1)

$\mathsf{Q}$ is structural;

  2. (2)

${\mathbf{Q}}({\textbf{F}}_{\mathsf{Q}}(\omega)=\mathsf{Q}$.

For any quasivariety $\mathsf{Q}$, we define the structural core of
$\mathsf{Q}$ as the smallest $\mathsf{Q}^{\prime}\subseteq\mathsf{Q}$ such
that ${\mathbf{H}}(\mathsf{Q})={\mathbf{H}}(\mathsf{Q}^{\prime})$. The
structural core of a quasivariety always exists:

######  Corollary 2.6.

For any quasivariety $\mathsf{Q}$,
${\mathbf{Q}}(\mathsf{F}_{\mathsf{Q}}(\omega))$ is structural and it is the
structural core of $\mathsf{Q}$.

###### Proof.

${\mathbf{Q}}(\mathsf{F}_{\mathsf{Q}}(\omega))$ is structural by Theorem 2.5;
if $\mathsf{Q}^{\prime}\subseteq\mathsf{Q}$ is such that
${\mathbf{H}}(\mathsf{Q}^{\prime})={\mathbf{H}}(\mathsf{Q})$, then clearly
${\textbf{F}}_{\mathsf{Q}}(\omega)\in\mathsf{Q}^{\prime}$ from which the
thesis follows. ∎

It follows at once that a quasivariety $\mathsf{Q}$ is structural if and only
if it coincides with its structural core. As a consequence the structural
subquasivarieties of a quasivariety $\mathsf{Q}$ are exactly those that
coincide with the structural cores of $\mathsf{Q}^{\prime}$ for some
$\mathsf{Q}^{\prime}\subseteq\mathsf{Q}$; even more, since
${\mathbf{H}}(\mathsf{Q})$ is a variety, the structural subquasivarieties of a
variety $\mathsf{V}$ are exactly the structural cores of $\mathsf{V}^{\prime}$
for some subvariety $\mathsf{V}^{\prime}$ of $\mathsf{V}$. As we will see in
the sequel this observation is particularly useful when the free countably
generated algebra in $\mathsf{V}$ has a reasonable description.

If $\mathsf{Q}$ is a quasivariety and $\mathsf{Q}^{\prime}$ is a
subquasivariety of $\mathsf{Q}$ we say that $\mathsf{Q}^{\prime}$ is
equational in $\mathsf{Q}$ if
$\mathsf{Q}^{\prime}={\mathbf{H}}(\mathsf{Q}^{\prime})\cap\mathsf{Q}$; this is
clearly equivalent to saying that $\mathsf{Q}^{\prime}$ is axiomatized modulo
$\mathsf{Q}$ by a set of equations. A quasivariety $\mathsf{Q}$ is primitive
if each subquasivariety of $\mathsf{Q}^{\prime}$ is equational in
$\mathsf{Q}$. The following lemma is straightforward:

######  Lemma 2.7.

For a quasivariety $\mathsf{Q}$ the following are equivalent;

  1. (1)

$\mathsf{Q}$ is primitive;

  2. (2)

every subquasivariety of $\mathsf{Q}$ is structural.

For locally finite quasivarieties we have a necessary and sufficient condition
due essentially to Gorbunov; an algebra A is weakly projective in a class
$\mathsf{K}$ if for all ${\textbf{B}}\in\mathsf{K}$,
${\textbf{A}}\in{\mathbf{H}}({\textbf{B}})$ implies
${\textbf{A}}\in{\mathbf{I}}{\mathbf{S}}({\textbf{B}})$. An algebra is
projective in $\mathsf{K}$ if it is weakly projective in $\mathsf{K}$ and the
epimorphism and the embedding witnessing weak projectivity compose to the
identity on A.

######  Theorem 2.8.

(see [30]) For a locally finite quasivariety $\mathsf{Q}$ the following are
equivalent:

  1. (1)

$\mathsf{Q}$ is primitive;

  2. (2)

every finite $\mathsf{Q}$-irreducible algebra is weakly projective in
$\mathsf{Q}$;

  3. (3)

every finite $\mathsf{Q}$-irreducible algebra is weakly projective in the
class of finite algebras in $\mathsf{Q}$.

From the Blok-Pigozzi connection we get at once:

######  Theorem 2.9.

Let $\mathcal{L}$ be an algebraizable logic with equivalent algebraic
semantics $\mathsf{Q}_{\mathcal{L}}$. Then

  1. (1)

$\mathcal{L}$ is structurally complete if and only if
$\mathsf{Q}_{\mathcal{L}}$ is structural;

  2. (2)

$\mathcal{L}$ is hereditarily structurally complete if and only if
$\mathsf{Q}_{\mathcal{L}}$ is primitive.

Combining Theorems 2.4 and 2.9 it follows at once that if $\mathcal{L}$ is
algebraizable and $\mathsf{Q}_{\mathcal{L}}$ is primitive, then every finitary
extension of $\mathcal{L}$ is axiomatic.

Some investigations using (part of) the machinery we have described in this
section has already been used to investigate structural completeness in
(quasi)varieties of fuzzy logics (see for instance [35] or [22]). In this note
however we will get more into the details for a specific variety and this will
allow us to characterize all the structurally complete finitary extensions of
positive Łukasiewicz logic.

###  2.4. Hoops as quasivarieties of logics

A commutative integral residuated lattice is an algebra
$\langle A,\vee,\wedge,\cdot,\rightarrow,1\rangle$ such that

  1. (1)

$\langle A,\vee,\wedge,1\rangle$ is a lattice with largest element $1$;

  2. (2)

$\langle A,\cdot,1\rangle$ is a commutative monoid;

  3. (3)

$(\cdot,\rightarrow)$ form a residuated pair w.r.t. the lattice ordering, i.e.
for all $a,b,c\in A$

$a\cdot b\leq c\qquad\text{if and only if}\qquad a\leq b\rightarrow c.$

In what follows, we will often write $xy$ for $x\cdot y$. If we augment the
signature with an extra constant $0$ that is the least element in the lattice
order, then we get $\mathsf{FL}_{ew}$-algebras. Commutative integral
residuated lattices and $\mathsf{FL}_{ew}$-algebras form varieties that have a
very rich structure; for an equational axiomatization and a list of valid
identities we refer the reader to [18].

Varieties of $\mathsf{FL}_{ew}$-algebras and commutative integral residuated
lattices are ideal determined (w.r.t. $1$) in the sense of [11]; this means
that there is a one-to-one correspondence (which is in fact a lattice
isomorphism) between the congruences of an algebra A and certain special
subsets of A. In the present case if A is an $\mathsf{FL}_{ew}$-algebra or a
commutative integral residuated lattice a filter of A is a filter $F$ of the
lattice structure which is also closed under the monoidal operation. If
$\theta\in\operatorname{Con}({\textbf{A}})$ then $1/\theta$ is clearly a
filter of A and it is easily checked that if $F$ is a filter then
$\theta_{F}=\\{(a,b):a\rightarrow b,b\rightarrow a\in
F\\}\in\operatorname{Con}({\textbf{A}})$ and the correspondence is

$\theta\longmapsto 1/\theta\qquad F\longmapsto\theta_{F}.$

If $F$ is a filter we will write ${\textbf{A}}/F$ for
${\textbf{A}}/\theta_{F}$.

Let’s focus on two equations that bear interesting consequences, i.e.,
prelinearity and divisibility:

(prel) $\displaystyle(x\rightarrow y)\vee(y\rightarrow x)\approx 1.$ (div)
$\displaystyle x(x\rightarrow y)\approx y(y\rightarrow x);$

It can be shown (see [18] and [33]) that a subvariety of $\mathsf{FL_{ew}}$
satisfies the prelinearity equation (prel) if and only if any algebra therein
is a subdirect product of totally ordered algebras, and this implies via
Birkhoff’s Theorem that all the subdirectly irreducible algebras are totally
ordered. Such varieties are called representable (or semilinear) and the
subvariety axiomatized by (prel) is the largest subvariety of
$\mathsf{FL_{ew}}$ that is representable; such variety is usually denoted by
$\mathsf{M}\mathsf{T}\mathsf{L}$, since it is the equivalent algebraic
semantics of Esteva-Godo’s Monoidal t-norm based logic [25].

If an algebra in $\mathsf{FL_{ew}}$ satisfies both (prel) and (div) then it is
called a $\mathsf{B}\mathsf{L}$-algebra and the variety of all
$\mathsf{B}\mathsf{L}$-algebras is denoted by $\mathsf{B}\mathsf{L}$. Again
the name comes from logic: the variety of $\mathsf{B}\mathsf{L}$-algebras is
the equivalent algebraic semantics of Hájek’s Basic Logic $\mathcal{BL}$ [31].
A systematic investigation of varieties of $\mathsf{B}\mathsf{L}$-algebras
started with [7] and it is still ongoing (see [2] and the bibliography
therein).

It follows from the definition that given a variety of bounded commutative
integral residuated lattices, the class of its $0$-free subreducts is a class
of residuated lattices; we have a very general result.

######  Lemma 2.10.

Let $\mathsf{V}$ be any subvariety of $\mathsf{FL_{ew}}$; then the class
${\mathbf{S}}^{0}(\mathsf{V})$ of the zero-free subreducts of algebras in
$\mathsf{V}$ is a variety.

###### Proof.

The proof is as in Proposition 1.10 of [6]; it is stated for varieties of
$\mathsf{B}\mathsf{L}$-algebras but it uses only the description of the
congruence filters, that can be used in any subvariety of $\mathsf{FL_{ew}}$
(as the reader can easily check). ∎

This implies at once that if a variety of $\mathsf{FL}_{ew}$-algebras is the
equivalent algebraic semantics of a logic $\mathcal{L}$, then the variety of
its zero-free subreducts is the equivalent algebraic semantics of the positive
fragment $\mathcal{L}^{+}$. A basic hoop is a zero-free subreduct of a
divisible and prelinear $\mathsf{FL}_{ew}$-algebra. Note that in any
$\mathsf{FL}_{ew}$-algebras the prelinearity equation makes the join definable
using $\wedge$ and $\rightarrow$ (see for instance [3]):

$((x\rightarrow y)\rightarrow y)\wedge((y\rightarrow x)\rightarrow x)\approx
x\vee y.$

So basic hoops are often presented in the signature $\wedge,\rightarrow,1$; as
noted in [6] the variety $\mathsf{BH}$ of basic hoops is the equivalent
algebraic semantics of the positive fragment of the logic $\mathcal{BL}$.

A Wajsberg hoop is a basic hoop satisfying the so-called Tanaka’s equation

$(x\rightarrow y)\rightarrow y\approx(y\rightarrow x)\rightarrow x.$

If we add a constant $0$ to the signature that is the least element in the
lattice order then we have Wajsberg algebras; Wajsberg algebras are term
equivalent to $\mathsf{MV}$-algebras (see [8] p. 354 for a detailed
explanation) and the variety of $\mathsf{MV}$-algebras is usually presented as
the equivalent algebraic semantics of Łukasiewicz logic $\mathcal{MV}$. It
follows that the variety $\mathsf{WH}$ of Wajsberg hoops, which is the variety
of zero-free subreducts of Wajsberg algebras, is the equivalent algebraic
semantics of $\mathcal{MV}^{+}$, i.e. the positive fragment of Łukasiewicz
logic.

In $\mathsf{FL}_{ew}$-algebras it is customary to introduce the derived
operation $\neg x:=x\rightarrow 0$; now in a bounded (i.e. with a minimum
element $a$) Wajsberg hoop we can still introduce a negation $\neg
x=x\rightarrow a$ that is of course not a term but rather a polynomial. Now it
is easy to verify that any bounded Wajsberg hoop is polynomially equivalent to
a Wajsberg algebra and we will freely use the expression $\neg x$ letting the
context clear the meaning.

A commutative integral residuated lattice is cancellative if the underlying
monoid is cancellative in the usual sense.

######  Lemma 2.11.

[16] Every cancellative basic hoop is a Wajsberg hoop. A totally ordered
Wajsberg hoop is either cancellative or bounded.

This allows us to show that the connection between Wajsberg hoops and Wajsberg
algebras is even stricter. For instance the operator
${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}$ on Wajsberg hoops has been studied
in [7] using the results about Wajsberg algebras appeared in [28]; while we
maintain that it should be clear why we can do this (and in [7] no explanation
was given), maybe some clarification is useful. Wajsberg algebras are
polynomially equivalent to bounded Wajsberg hoops; it is easy to see that if
${\mathbf{O}}$ is a class operator that is a composition of
${\mathbf{I}},{\mathbf{H}},{\mathbf{S}},{\mathbf{P}},{\mathbf{P}}_{u}$,
${\textbf{A}},{\textbf{B}}$ are Wajsberg algebras and
${\textbf{A}}_{0},{\textbf{B}}_{0}$ are their Wajsberg hoop reducts, then
${\mathbf{O}}({\textbf{A}})\subseteq{\mathbf{O}}({\textbf{B}})$ if and only if
${\mathbf{O}}({\textbf{A}}_{0})={\mathbf{O}}({\textbf{B}}_{0})$. This allows
us to consider bounded Wajsberg hoops as they were Wajsberg algebras. Since a
totally ordered Wajsberg hoop is either bounded or cancellative we can use
results about Wajsberg algebras and integrate them with the cancellative case.

##  3\. Some useful tools

###  3.1. Wajsberg chains

Bounded Wajsberg hoops have a canonical representation. Let G be a lattice
ordered abelian group; by [39], if $u$ is a strong unit of G we can construct
a bounded Wajsberg hoop
$\Gamma({\textbf{G}},u)=\langle[0,u],\rightarrow,\cdot,0,u\rangle$ where
$ab=\textrm{max}\\{a+b-u,0\\}$ and $a\rightarrow b=\textrm{min}\\{u-a+b,u\\}$.
The main result of [39] is that any bounded Wajsberg hoop can be presented in
this way (really there is a categorical equivalence between the category
abelian $\ell$-groups with strong unit and the category of bounded Wajsberg
hoops). Let now $\mathbb{Z}\times_{l}\mathbb{Z}$ denote the lexicographic
product of two copies of $\mathbb{Z}$. In other words, the universe is the
Cartesian product and the group operations are defined componentwise and the
ordering is the lexicographic ordering (w.r.t. the natural ordering of
$\mathbb{Z}$); then $\mathbb{Z}\times_{l}\mathbb{Z}$ is a totally ordered
abelian group and we can apply $\Gamma$ to it. A Wajsberg chain is a totally
ordered Wajsberg hoop. Let’s define some useful Wajsberg chains:

  1. $\bullet$

the finite Wajsberg chain with $n+1$ elements
${\textbf{\L}}_{n}=\Gamma(\mathbb{Z},n)$;

  2. $\bullet$

the infinite finitely generated Wajsberg chain
${\textbf{\L}}_{n}^{\infty}=\Gamma(\mathbb{Z}\times_{l}\mathbb{Z},(n,0))$;

  3. $\bullet$

the infinite finitely generated Wajsberg chain
${\textbf{\L}}_{n,k}=\Gamma(\mathbb{Z}\times_{l}\mathbb{Z},(n,k))$;

  4. $\bullet$

the unbounded Wajsberg chain ${\textbf{C}}_{\omega}$; this can be regarded
either as the free monoid on one generator, where the product is the monoid
product and $a^{l}\rightarrow a^{m}=a^{\textrm{max}(l-m,0)}$ or, equivalently,
as the negative cone of $\mathbb{Z}$ with the operations defined in the
obvious way.

We observe that ${\textbf{\L}}_{n}^{\infty}={\textbf{\L}}_{n,0}$; moreover the
proof of the following is a simple verification:

######  Lemma 3.1.

  1. (1)

For $n,m\in\mathbb{N}$,
${\textbf{\L}}_{n}\in{\mathbf{I}}{\mathbf{S}}({\textbf{\L}}_{m})$ if and only
if ${\textbf{\L}}_{n}\in{\mathbf{I}}{\mathbf{S}}({\textbf{\L}}_{m}^{\infty})$
if and only if $n\mathrel{|}m$.

  2. (2)

For $n,r,j\in\mathbb{N}$,
${\textbf{\L}}_{n}\in{\mathbf{I}}{\mathbf{S}}({\textbf{\L}}_{r,j})$ if and
only if $n\mathrel{|}\gcd\\{r,j\\}$.

  3. (3)

If A is a cancellative Wajsberg chain and $a\in A\setminus\\{1\\}$, then $a$
generates a subalgebra of A isomorphic with ${\textbf{C}}_{\omega}$.

In [8] it has been shown that every proper variety of Wajsberg hoops is
generated by a finite number of chains of the type described above. More
precisely a presentation $P$ is a triple $(I,J,K)$ $I,J$ are finite subsets of
$\mathbb{N}\setminus\\{0\\}$ and $K\subseteq\\{\omega\\}$. We say that the
presentation $P=(I,J,K)$ is reduced if

  1. $\bullet$

$I\cup J\cup K\neq\emptyset$;

  2. $\bullet$

if $J\neq\emptyset$ then $K=\emptyset$;

  3. $\bullet$

no $m\in I$ divides any $m^{\prime}\in(I\setminus\\{m\\})\cup J$;

  4. $\bullet$

no $n\in J$ divides any $n^{\prime}\in J\setminus\\{m\\}$.

For any reduced presentation $P=(I,J,K)$ we define a set of Wajsberg hoops
$\mathsf{K}_{P}$ in the following way:

  1. $\bullet$

if $P=(I,J,\emptyset)$ then $\mathsf{K}_{P}=\\{{\textbf{\L}}_{i}:i\in
I\\}\cup\\{{\textbf{\L}}_{j}^{\infty}:j\in J\\}$,

  2. $\bullet$

if $P=(I,\emptyset,\\{\omega\\})$ then
$\mathsf{K}_{P}=\\{{\textbf{\L}}_{i}:i\in
I\\}\cup\\{{\textbf{C}}_{\omega}\\}$.

Then we set $\mathsf{V}(P)={\mathbf{V}}(\mathsf{K}_{P})$ and
$\mathsf{Q}(P)={\mathbf{Q}}(\mathsf{K}_{P})$.

######  Theorem 3.2.

[8] The proper subvarieties of Wajsberg hoops are in one-to-one correspondence
with the reduced presentations via

$P\longmapsto\mathsf{V}(P).$

###  3.2. The construction of ${\textbf{B}}_{\Delta}$

Any proper subvariety of Wajsberg hoops is axiomatizable (modulo Wajsberg
hoops) by an equation in a single variable; this is essentially Theorem 4.4 in
[8] and its proof uses the functional characterization of free Wajsberg hoops
which we will be using later in this paper. From that, it follows that for any
proper subvariety $\mathsf{V}$ of Wajsberg hoops
${\mathbf{Q}}({\textbf{F}}_{\mathsf{V}}(\omega))={\mathbf{Q}}({\textbf{F}}_{\mathsf{V}}(x))$.
Indeed,
$\mathsf{V}={\mathbf{H}}{\mathbf{Q}}({\textbf{F}}_{\mathsf{V}}(\omega))={\mathbf{H}}{\mathbf{Q}}({\textbf{F}}_{\mathsf{V}}(x))$
because $\mathsf{V}$ can be axiomatized in one variable and, since
${\mathbf{Q}}({\textbf{F}}_{\mathsf{V}}(\omega))$ is structural, this means
that it has to be the smallest quasivariety that generates $\mathsf{V}$, so
${\mathbf{Q}}({\textbf{F}}_{\mathsf{V}}(\omega))\subseteq{\mathbf{Q}}({\textbf{F}}_{\mathsf{V}}(x))$;
the other inclusion is trivial.

In this section we will give an alternative description of
${\textbf{F}}_{\mathsf{V}}(x)$ where $\mathsf{V}$ is a proper subvariety of
Wajsberg hoops.

######  Lemma 3.3.

Let A be a totally ordered Wajsberg hoop, assume that A is one-generated, then
${\textbf{A}}\in{\mathbf{V}}({\textbf{\L}}_{n}^{\infty})$ if and only if one
of the following holds:

  1. (1)

there exists $1\leq k|n$ such that ${\textbf{A}}\cong{\textbf{L}}_{k}$;

  2. (2)

there exist $1\leq k|n$ and $0\leq h<k$ with $k,h$ relatively prime such that
${\textbf{A}}\cong{\textbf{\L}}_{k,h}$;

  3. (3)

${\textbf{A}}\cong C_{\omega}$.

###### Proof.

As A is totally ordered, it is either bounded or cancellative. If it is
cancellative then since it is one-generated, it must be ${\textbf{A}}\cong
C_{\omega}$; if it is bounded, then it can be seen as an
$\mathsf{MV}$-algebra, so, by [24] (Theorem 1.8), either 1. or 2. holds.

Conversely if ${\textbf{A}}\cong C_{\omega}$, then clearly
${\textbf{A}}\in{\mathbf{V}}({\textbf{\L}}_{n}^{\infty})$; in the other cases
we appeal again to Theorem 1.8 in [24]. ∎

From now on, given a finite subset $X$ of $\mathbb{N}$, we will denote by
$X{\downarrow}$ the set of all the divisors of elements of $X$.

######  Lemma 3.4.

Let A be a totally ordered Wajsberg hoop, assume that A is one generated and
let $P=(I,J,\emptyset)$ be a reduced presentation with $J\neq\emptyset$. Then
${\textbf{A}}\in\mathsf{V}(P)$ if and only if one of the following holds:

  1. (1)

there exists $k\in I{\downarrow}\cup J{\downarrow}$ such that
${\textbf{A}}\cong{\textbf{\L}}_{k}$;

  2. (2)

there exist $k\in J{\downarrow}$ and $0\leq h<k$ with $k,h$ relatively prime
such that ${\textbf{A}}\cong{\textbf{\L}}_{k,h}$;

  3. (3)

${\textbf{A}}\cong C_{\omega}$.

###### Proof.

The “only if” part is exactly as in Lemma 3.3.

If ${\textbf{A}}\cong{\textbf{\L}}_{k}$ for some $k\in I{\downarrow}\cup
J{\downarrow}$, then, if $k\in I{\downarrow}$ there exists $i\in I$ such that
${\textbf{A}}\in{\mathbf{V}}({\textbf{\L}}_{i})\subseteq\mathsf{V}(I,J,\emptyset)$;
if $k\in J{\downarrow}$, then by Lemma 3.3 there exists a $j\in J$ such that
${\textbf{A}}\in{\mathbf{V}}({\textbf{\L}}_{j}^{\infty})\subseteq\mathsf{V}(I,J,\emptyset)$.

If ${\textbf{A}}\cong{\textbf{\L}}_{k,h}$ for some $k\in J{\downarrow}$ and
$0\leq h<k$ with $h,k$ relatively prime, then by Lemma 3.3 there exists $j\in
J$ such that
${\textbf{A}}\in{\mathbf{V}}({\textbf{\L}}_{j}^{\infty})\subseteq\mathsf{V}(I,J,\emptyset)$.
Finally, if ${\textbf{A}}\cong C^{\omega}$ clearly
${\textbf{A}}\in{\mathbf{V}}({\textbf{\L}}_{j}^{\infty})$ for every $j\in
J{\downarrow}$, so, since $J\neq\emptyset$,
${\textbf{A}}\in\mathsf{V}(I,J,\emptyset)$. ∎

######  Remark 3.5.

If ${\textbf{A}}\cong{\textbf{\L}}_{k}$, then $h$ generates A if and only if
$h,k$ are relatively prime.

If ${\textbf{A}}\cong{\textbf{\L}}_{k,h}$ with $k\neq 1$ and $h<k$ then there
exists a unique $g_{k,h}\in{\textbf{A}}$ with $g_{k,h}\leq\neg g_{k,h}$ such
that $a$ generates A if and only if $a=g_{k,h}$ or $a=\neg g_{k,h}$. Moreover
$g_{k,1}=(1,0),g_{k,k-1}=(1,1)$ and, if $h\neq 1,h\neq k-1$, then
$g_{k,h}=(r,s)$ with $1<r<\frac{k}{2}$. If $k=1$ we get that
${\textbf{\L}}_{1,0}$ is generated by $g_{1,0}=(0,1)$, but this time $\neg
g_{1,0}=(1,-1)$ generates a subalgebra of ${\textbf{\L}}_{1,0}$ isomorphic to
$C_{\omega}$.

Note that in all these cases, we can use the operation $\neg$ because all the
algebras are bounded; in particular, since the generator of
${\textbf{\L}}_{k,h}$ has always order $2$, we can write $0$ as $g_{k,h}^{2}$,
so for every $a\in{\textbf{\L}}_{k,h}$ we get $\neg a=a\rightarrow
g_{k,h}^{2}$.

Now we fix a reduced triple $P=(I,J,K)$ and let

$\displaystyle\Delta_{I}=\\{(k,h,2):0\leq h<k\in I{\downarrow},\text{$k,h$
relatively prime}\\}$ $\displaystyle\Delta_{J}=\\{(k,h,i):i\in\\{0,1\\},h<k\in
J{\downarrow},\text{$k,h$ relatively prime}\\}$
$\displaystyle\Delta_{K}=\left\\{\begin{array}[]{ll}\emptyset,&\hbox{if
$J\neq\emptyset$;}\\\ \\{(0,0,3)\\},&\hbox{otherwise.}\end{array}\right.$

Moreover for any $k,h$ we define

$\displaystyle{\textbf{A}}^{0}_{k,h}={\textbf{A}}^{1}_{k,h}:={\textbf{\L}}_{k,h}$
$\displaystyle{\textbf{A}}^{2}_{k,h}={\textbf{\L}}_{k}$
$\displaystyle{\textbf{A}}^{3}_{k,h}={\textbf{C}}_{\omega}.$

If $\Delta=\Delta_{I}\cup\Delta_{J}\cup\Delta_{K}$ we let
${\textbf{A}}_{\Delta}=\prod_{(k,h,i)\in\Delta}{\textbf{A}}^{i}_{k,h}$;
moreover we denote by $c$ the generator of ${\textbf{C}}_{\omega}$. We want to
define a $\overline{g}\in{\textbf{A}}_{\Delta}$ by cases; for any
$(k,h,i)\in\Delta$

$\displaystyle\text{if $J=\emptyset$,}\ \
\overline{g}(k,h,i)=\left\\{\begin{array}[]{ll}h,&\hbox{if $i=2$;}\\\
c,&\hbox{if $i=3$.}\end{array}\right.$ $\displaystyle\text{if
$J\neq\emptyset$,}\ \
\overline{g}(k,h,i)=\left\\{\begin{array}[]{ll}g_{k,h},&\hbox{if $i=0$;}\\\
\neg g_{k,h},&\hbox{if $i=1$;}\\\ h,&\hbox{if $i=2$.}\end{array}\right.$

######  Theorem 3.6.

Let $P=(I,J,K)$ be any reduced triple and let $\overline{g}$ and
${\textbf{A}}_{\Delta}$ as above. If ${\textbf{B}}_{\Delta}$ is the subalgebra
of A generated by $\overline{g}$, then
${\textbf{B}}_{\Delta}\cong{\textbf{F}}_{\mathsf{V}(P)}(x)$.

###### Proof.

First we observe that ${\textbf{A}}_{\Delta}\in\mathsf{V}(P)$ and so does
${\textbf{B}}_{\Delta}$. Suppose that $p(x)\approx q(x)$ is an equation that
fails in $\mathsf{V}(P)$; then it must fail in some one-generated totally
ordered algebra ${\textbf{C}}\in\mathsf{V}(P)$ and such algebra is either
bounded or cancellative.

First, let us show that we only need to discuss the case in which $p(x)\approx
q(x)$ fails in a generator of C. Suppose that the equation fails in some
$x\in{\textbf{C}}$, then, if we call ${\textbf{C}}^{\prime}$ the subalgebra of
C generated by $x$, we have that $p(x)\approx q(x)$ fails in the generator of
${\textbf{C}}^{\prime}$, that is still an algebra in $\mathsf{V}(P)$.

Suppose that $J=\emptyset$. If C is bounded, then it cannot be infinite, as
${\textbf{\L}}_{n}^{\infty}\notin\mathsf{V}(P)$ for any $n\in\mathbb{N}$.
Hence it must be equal to ${\textbf{\L}}_{k}$ for some $k\in I{\downarrow}$;
this implies that $p(x)\approx q(x)$ fails in $g(k,h,2)$ for some $h$ and, as
above, fails in ${\textbf{B}}_{\Delta}$. This covers the case $K=\emptyset$
and half of the case $K\neq\emptyset$. To conclude, if C is cancellative, then
the equation must fail in ${\textbf{C}}_{\omega}$ and hence (if $c$ is the
generator of ${\textbf{C}}_{\omega}$), $p(c)\neq q(c)$; this implies that
$p(\overline{g}(0,0,3))\neq p(\overline{g}(0,0,3))$, so $p(\overline{g})\neq
q(\overline{g})$ and $p(x)\approx q(x)$ fails in ${\textbf{B}}_{\Delta}$.

Suppose now that $J\neq\emptyset$ (and thus $K=\emptyset$). By Lemma 3.4 we
have only three possibilities.

If ${\textbf{C}}\cong{\textbf{\L}}_{k}$ for some $k\in I{\downarrow}\cup
J\downarrow$; if $k\in I{\downarrow}$ then $p(x)\approx q(x)$ fails in some
generator of ${\textbf{\L}}_{k}$, so it fails in some $\overline{g}(k,h,2)$
for some $h$ and eventually fails in ${\textbf{B}}_{\Delta}$. If $k\in
J{\downarrow}$ again $p(x)\approx q(x)$ fails in some generator $h$ of
${\textbf{\L}}_{k}$; now $h$ and $k$ must be relatively prime and thus
$p(x)\approx q(x)$ fails in ${\textbf{\L}}_{k,h}$. By Remark 3.5 it fails
either in $g_{h,k}$ or $\neg g_{h,k}$, hence $p(x)\approx q(x)$ fails either
in $\overline{g}(h,k,0)$ or $\overline{g}(h,k,1)$. In any case $p(x)\approx
q(x)$ fails in ${\textbf{B}}_{\Delta}$.

If ${\textbf{C}}\cong{\textbf{\L}}_{h,k}$ the argument is similar to the one
above, but easier. Finally if ${\textbf{C}}\cong{\textbf{C}}_{\omega}$, since
$\overline{g}(1,0,1)=\neg g_{1,0}=(1,-1)$ generates a subalgebra of
${\textbf{\L}}_{1,0}$ isomorphic to $C^{\omega}$, for sure
$p(\overline{g}(1,0,1))\neq q(\overline{g}(1,0,1))$ and so again $p(x)\approx
q(x)$ fails in B.

We have thus proved that every equation in one variable that fails in
$\mathsf{V}(P)$ fails in ${\textbf{B}}_{\Delta}$. At last, let us show that
this is sufficient to say that
${\textbf{B}}_{\Delta}\cong{\textbf{F}}_{\mathsf{V}(P)}(x)$. Indeed, since
${\textbf{B}}_{\Delta}$ is in $\mathsf{V}(P)$ and it is one-generated, we know
that we have a surjective homomorphism $\varphi$ from
${\textbf{F}}_{\mathsf{V}(P)}(x)$ to ${\textbf{B}}_{\Delta}$; suppose now that
$\varphi$ is not injective, this means that Ker($\varphi$) is non-trivial, so
there exist two terms $p,q\in{\textbf{F}}_{\mathsf{V}(P)}(x)$ such that $p\neq
q$ but $\varphi(p)=\varphi(q)$, thus $p(x)\not\approx q(x)$ in
${\textbf{F}}_{\mathsf{V}(P)}(x)$ but $p(x)\approx q(x)$ in
${\textbf{B}}_{\Delta}$. Hence $\varphi$ must be an isomorphism. ∎

###  3.3. Wajsberg functions

A McNaughton function over the $n$-cube is a continuous function
$f:[0,1]^{n}\to[0,1]$ such that there exist finitely many linear functions
$f_{1},\dots,f_{k}$, where each $f_{i}$ is of the form
$f_{i}=a_{i}^{0}x_{0}+a_{i}^{1}x_{1}+\dots+a_{i}^{n}x_{n}+b_{i}$ with
$a_{i}^{0}\dots a_{i}^{n},b_{i}$ integers, and such that for any
$v\in[0,1]^{n}$ there exists $i\in\\{1,\dots k\\}$ with $f(v)=f_{i}(v)$. A
McNaughton function $f(x_{1},\dots,x_{n})$ is a Wajsberg function if
$f(1,1,\dots,1)=1$.

######  Theorem 3.7.

[8] For each $n$, the free $n$-generated Wajsberg hoop
${\textbf{F}}_{\mathsf{WH}}(n)$ is isomorphic to the algebra of all Wajsberg
functions over the $n$-cube, where the operations are defined pointwise.

So we can always identify an $n$-ary term in the language of Wajsberg hoops
with a Wajsberg function over the $n$-cube. Conversely, given a Wajsberg
function over the $n$-cube, we can associate to it an equivalence class of
Wajsberg terms (where the equivalence is of course mutual provability in the
theory). With the usual abuse of notation we will identify the class with any
of its representatives, i.e. given a Wajsberg function $f$ we will denote by
$\widehat{f}$ the Wajsberg term which is a representative to the equivalence
class corresponding to $f$.

In [8] the authors used this representation to give an easy way to axiomatize
all proper subvarieties of Wajsberg hoops. Let $(I,J,K)$ be a reduced triple,
we define two finite subsets $\mathcal{I},\mathcal{J}$ of rational points of
$[0,1]$ as

  1. $\bullet$

if $K\neq\emptyset$ then $\mathcal{J}=\\{1\\}$;

  2. $\bullet$

if $K=\emptyset$ then $\mathcal{J}=\\{v\in[0,1]:\text{den}(v)\in
J{\downarrow}\\}$;

  3. $\bullet$

$\mathcal{I}=\\{u\in[0,1]:\text{den}(u)\in
I{\downarrow}\\}\backslash\mathcal{J}$.

Given a reduced triple $(I,J,K)$ an $(I,J,K)$-comb is any
$\alpha\in{\textbf{F}}_{\mathsf{WH}}(x)$ such that

  1. (1)

for every $v\in\mathcal{J}$, there exists a neighborhood $V$ of $v$ such that
$\alpha=1$ on $V$;

  2. (2)

for every $u\in\mathcal{I}$, $\alpha(u)=1$;

  3. (3)

for every $u\in\mathcal{I}$ there exists $v\in\mathcal{I}$ such that
den($v$)$|$den($u$) and $\alpha$ is not identically $1$ on any neighborhood of
$v$;

  4. (4)

if $d\notin(I\cup J){\downarrow}$, then there exists $0\leq h<d$ with
$\alpha(h/d)\neq 1$.

######  Theorem 3.8.

[8] Let $P=(I,J,K)$ be a reduced triple and let
$\alpha(x)\in{\textbf{F}}_{\mathsf{WH}}(x)$. Then the identity $\alpha(x)=1$
axiomatizes $\mathsf{V}(P)$ relative to $\mathsf{WH}$ if and only if $\alpha$
is an $(I,J,K)$-comb.

This is a very powerful result, in that it gives a procedure that allows to
axiomatize every proper subvariety of Wajsberg hoops, and combs are quite easy
to construct. Next, we have a very useful lemma.

######  Lemma 3.9.

Let $p(x)\approx q(x)$ be an identity in the language of Wajsberg hoops and
let $f,g$ be Wajsberg functions such that $p=\widehat{f}$ and $q=\widehat{g}$.
Then for any $n,k\in\mathbb{N}$ with $k\leq n$

  1. (1)

if $f(\frac{k}{n})=g(\frac{k}{n})$, then $p(k)=q(k)$ where
$k\in{\textbf{\L}}_{n}$;

  2. (2)

if $f(x)=g(x)$ in a neighborhood of $1$, then ${\textbf{C}}_{\omega}\vDash
p(x)\approx q(x)$;

  3. (3)

if $f(x)=g(x)$ in a neighborhood of $\frac{k}{n}$, then $p(c)=q(c)$ for any
$c\in{\textbf{\L}}_{n,h}$ such that $c/\textrm{Rad}({\textbf{\L}}_{n,h})=k$.

The proof of Lemma 3.9 can be extracted from the proof of Theorem 3.3 in [8],
by setting $\kappa=1$.

Next, we want to have an easy way to construct Wajsberg functions and force
them to have certain fixed values (see [32]). If $0=t_{0}<t_{1}<\dots<t_{k}=1$
and $x_{0},\dots,x_{k}\in[0,1]$, then we denote by
$f=L(t_{0},x_{0};\dots;t_{k},x_{k})$ the continuous picewise linear function
$f:[0,1]\to[0,1]$ such that $f(t_{i})=x_{i}$ and $f$ is linear on each
interval $[t_{i},t_{i+1}]$; in other words, $f$ is the linear interpolation
between the nodes $(t_{0},x_{0}),\dots,(t_{k},x_{k})$. Clearly it is possible
to play with the variables in order to make $f$ a Wajsberg function.

Let’s see an example. If we want to find a $(2,\emptyset,\emptyset)$-comb we
need to take a function $f$ that has value $1$ only in
$\\{0,\frac{1}{2},1\\}$, so we can take

$f=L(0,1;\frac{1}{4},0;\frac{1}{2},1;\frac{3}{4},0;1,1).$

This function has integer coefficients in every interval and $f(1)=1$, so it
is a Wajsberg function and the identity $f(x)\approx 1$ axiomatizes the
subvariety of Wajsberg hoops generated by ${\textbf{\L}}_{2}$.

Now we will heavily use Wajsberg functions and lemma 3.9 to prove a couple of
fundamental results. From now on, for any reduced triple $P=(I,J,K)$,
${\textbf{B}}_{\Delta}$ will be the algebra constructed from $P$ following the
directions in Section 3.2.

######  Theorem 3.10.

Let $P=(I,J,K)$ be a reduced triple and let $a\in I$; then ${\textbf{\L}}_{a}$
is embeddable in ${\textbf{B}}_{\Delta}$.

###### Proof.

If $1\in I$, then $P=(\\{1\\},\emptyset,K)$ and if $K=\emptyset$, then
${\textbf{B}}_{\Delta}\cong{\textbf{\L}}_{1}$. So suppose that
$K\neq\emptyset$; then by Theorem 3.6 ${\textbf{B}}_{\Delta}$ is isomorphic
with the subalgebra of ${\textbf{L}}_{1}\times{\textbf{C}}_{\omega}$ generated
by $(0,c)$. Consider the Wajsberg function

$f(x)=L(0,0;\dfrac{1}{2},1;1,1);$

then it is easy to check that $\widehat{f}=(x\rightarrow x^{2})\rightarrow x$.
Now $f(0)=0$ and $f(1)=1$ in a neighborhood of $1$, hence by lemma 3.9
$f(\overline{g})=(0,1)$ and thus it generates a subalgebra of
${\textbf{B}}_{\Delta}$ isomorphic with ${\textbf{L}}_{1}$.

Now suppose that $1\notin I$; we fix an $a\in I$ and we let $m$ be the product
of all the elements of $I\cup J$. Then we consider the Wajsberg function

$f(x)=L(0,1;\dfrac{1}{a}-\dfrac{1}{2m},1;\dfrac{1}{a},0;\dfrac{1}{a}+\dfrac{1}{2m},1;1,1).$

Now clearly $f(x)=1$ in any neighborhood of $\frac{n}{m}$ where $n\leq m$ and
$\frac{n}{m}\neq\frac{1}{a}$ and moreover $f(\dfrac{1}{a})=0$. Since $(I,J,K)$
is reduced, $i$ does not divide any element of $I\cup J$ and so by Lemma 3.9

$f(\overline{g}(k,h,i))=\left\\{\begin{array}[]{ll}0,&\hbox{if
$(k,h,i)=(a,1,2)$;}\\\ 1,&\hbox{otherwise.}\end{array}\right.$

Hence $\overline{g}\vee f(\overline{g})$ generates a subalgebra of
${\textbf{B}}_{\Delta}$, isomorphic to the one generated by
$\overline{g}(a,1,2)$. But the latter is a generator of ${\textbf{\L}}_{a}$ so
${\textbf{\L}}_{a}$ is embeddable in ${\textbf{B}}_{\Delta}$, as wished. ∎

######  Theorem 3.11.

Let $P=(I,\emptyset,{\textbf{C}}_{\omega})$; then ${\textbf{C}}_{\omega}$ is
embeddable in ${\textbf{B}}_{\Delta}$.

###### Proof.

If $I=\emptyset$, then ${\textbf{B}}_{\Delta}\cong{\textbf{C}}_{\omega}$.
Otherwise let $m$ be the product of all elements of $I$ and consider the
Wajsberg function

$f(x)=L(1,1;\dfrac{m-1}{m},1;\dfrac{m}{m+1},\dfrac{m}{m+1};1,1).$

Again it is easy to see that $\widehat{f}(x)=x^{m}\rightarrow x^{m+1}$ and
that $f(\frac{n}{m})=1$ for $\dfrac{n}{m}\neq 1$ so that
$f(\overline{g}(k,h,i))=1$ whenever $i=2$. In a neighborhood of $1$ we have
that $f(x)=x$, so $f(\overline{g}(0,0,3))=c$, which generates
${\textbf{C}}_{\omega}$; thus $f(\overline{g})$ generates a subalgebra of
${\textbf{B}}_{\Delta}$ that is isomorphic with ${\textbf{C}}_{\omega}$ and
thus ${\textbf{C}}_{\omega}$ is embeddable in ${\textbf{B}}_{\Delta}$. ∎

##  4\. Structurally complete extensions

By Theorem 2.4 and 2.9 classifying all the structurally complete finitary
(axiomatic) extensions of positive Łukasiewicz’s Logic amounts to describing
all the structural quasivarieties (varieties) of Wajsberg hoops.

###  4.1. Structural subvarieties

The radical of a Wajsberg chain A, in symbols $\textrm{Rad}({\textbf{A}})$, is
the intersection of the maximal filters of A; it is easy to see that
$\textrm{Rad}({\textbf{A}})$ is cancellative and A is cancellative if and only
if $\textrm{Rad}({\textbf{A}})={\textbf{A}}$. We say that a bounded Wajsberg
hoop A has rank $n$, if
${\textbf{A}}/\textrm{Rad}({\textbf{A}})\cong{\textbf{\L}}_{n}$. For any
bounded Wajsberg hoop the divisibility index $d_{\textbf{A}}$ of A, is the
maximum $k$ such that ${\textbf{\L}}_{k}$ is embeddable in A if any, otherwise
$d_{\textbf{A}}=\infty$.

Here is a summary of the main results about the rank and the divisibility
index; the proofs are either trivial or can be found in [7] or [28].

######  Lemma 4.1.

For any $n,k\geq 1$

  1. (1)

${\textbf{\L}}_{n}$ is simple and
${\textbf{\L}}_{n}\in{\mathbf{I}}{\mathbf{S}}({\textbf{\L}}_{k})$ if and only
if ${\textbf{\L}}_{n}\in{\mathbf{I}}{\mathbf{S}}({\textbf{\L}}_{k}^{\infty})$
if and only if $n\mathrel{|}k$.

  2. (2)

${\textbf{\L}}_{n}$ has rank $n$ and divisibility index $n$.

  3. (3)

For any $k\geq 0$, ${\textbf{\L}}_{n,k}$ is subdirectly irreducible,
${\textbf{\L}}_{n,k}$ has rank $n$ and $d_{{\textbf{\L}}_{n,k}}=\gcd(n,k)$; in
particular $d_{{\textbf{\L}}^{\infty}_{n}}=n$.

  4. (4)

If A has rank $n$, then
${\textbf{A}}\in{\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}({\textbf{\L}}_{n,k})$
if and only if $d_{\textbf{A}}$ divides $\gcd(n,k)$.

  5. (5)

If A has rank $n$, then
${\textbf{\L}}_{n,k}\in{\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}({\textbf{A}})$
if and only if $\gcd(n,k)$ divides $d_{\textbf{A}}$.

  6. (6)

If A is a nontrivial totally ordered cancellative hoop then
${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}({\textbf{A}})={\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}({\textbf{C}}_{\omega})$.

  7. (7)

If A is a bounded Wajsberg chain of finite rank $k$, then $d_{\textbf{A}}$
divides $k$, and
${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}({\textbf{A}})={\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}({\textbf{\L}}_{k,d_{\textbf{A}}})$.

  8. (8)

If A is a bounded Wajsberg chain of finite rank $n$, then
${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}({\textbf{A}})={\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}({\textbf{\L}}^{\infty}_{n})$
if and only if $d_{\textbf{A}}=n$.

We have:

######  Theorem 4.2.

[5] Let ${\textbf{A}}_{1},\dots,{\textbf{A}}_{n}$ be Wajsberg chains; if for
all $i\leq n$

  1. $\bullet$

${\textbf{A}}_{i}$ is finite, or

  2. $\bullet$

${\textbf{A}}_{i}$ is cancellative, or

  3. $\bullet$

${\textbf{\L}}_{n}\in{\mathbf{I}}{\mathbf{S}}({\textbf{A}}_{i})$ for all $n$,
or

  4. $\bullet$

A is infinite, bounded and the rank of A is equal to $d_{\textbf{A}}$,

then
${\mathbf{Q}}({\textbf{A}}_{1},\dots,{\textbf{A}}_{n})={\mathbf{V}}({\textbf{A}}_{1},\dots,{\textbf{A}}_{n})$.
Moreover if $n=1$ then the converse holds as well.

This shows that for every reduced presentation $P$,
$\mathsf{V}(P)=\mathsf{Q}(P)$ and henceforth a proper subvariety
$\mathsf{V}(P)$ of Wajsberg hoop is structural if and only if
$\mathsf{V}(P)={\mathbf{Q}}({\textbf{F}}_{\mathsf{V}(P)}(x)$. First we will
consider locally finite varieties.

It is clear from the definition that both ${\textbf{C}}_{\omega}$ and
${\textbf{\L}}_{j}^{\infty}$ contain finitely generated subalgebras that fail
to be finite. Hence from Theorem 3.2 we get that a variety $\mathsf{V}$ of
Wajsberg hoops is locally finite if and only if it is $\mathsf{V}(P)$ where
$P=(I,\emptyset,\emptyset)$ if and only if it is finitely generated.

Now Wajsberg hoops are basic hoops and:

######  Theorem 4.3.

[9] Every finite basic hoop is projective in the class of finite basic hoops.
So if $\mathsf{V}$ is a locally finite variety of basic hoops every finite
algebra is projective.

Therefore:

######  Theorem 4.4.

Every locally finite quasivariety of Wajsberg hoops is a primitive variety.

###### Proof.

Let $\mathsf{Q}$ be a locally finite quasivariety of Wajsberg hoops; then it
is easy to check that $\mathsf{V}={\mathbf{H}}(\mathsf{Q})$ is locally finite
and hence every finite algebra of $\mathsf{V}$ is projective in $\mathsf{V}$.
By Theorem 2.8 $\mathsf{V}$ is primitive and thus every subquasivariety of
$\mathsf{V}$ is equational, i.e. it is a variety. This implies that
$\mathsf{Q}$ is a variety and in fact $\mathsf{Q}=\mathsf{V}$. ∎

Via the Blok-Pigozzi connection we get:

######  Corollary 4.5.

Every locally tabular extension of $\mathcal{MV}^{+}$ is tabular, axiomatic
and hereditarily structurally complete.

###### Proof.

By Theorem 3.2 every proper subvariety of Wajsberg hoops is of the form
$\mathsf{V}(I,J,K)$ for some reduced presentation. It is obvious that
${\textbf{C}}_{\omega}$ is not locally finite and it is easy to see that the
same holds for ${\textbf{\L}}_{n}^{\infty}$ for $n\geq 1$. Therefore if
$\mathsf{V}(I,J,K)$ is locally finite, then $J=K=\emptyset$. But $I$ is a
finite set, so $\mathsf{V}(I,\emptyset,\emptyset)$ is finitely generated. So
every locally finite variety of Wajsberg hoops is finitely generated; so every
locally tabular extension of $\mathcal{MV}^{+}$ is tabular and the rest
follows from Theorem 4.4. ∎

The variety $\mathsf{C}=\mathsf{V}(\emptyset,\emptyset,\\{0\\})$ is the
variety of cancellative hoops; now $\mathsf{C}$ can be shown to be primitive
by a variety of means. The simplest one is probably to observe first that it
is an atom in the lattice of subvarieties $\Lambda(\mathsf{WH})$ [12] hence it
is equationally complete. Then one can quote [14] where it is stated that any
equationally complete congruence modular variety has no proper
subquasivarieties. As $\mathsf{C}$ is congruence distributive (having a
lattice reduct) it has no proper subquasivarieties and hence it is primitive.

Now we can characterize all the structural varieties of Wajsberg hoops. First
we observe that $\mathsf{WH}$ itself is not structural; this is well-known and
can be shown in several ways. The most direct one is probably to observe that
the set $\\{{\textbf{\L}}_{p}:\text{$p$ prime}\\}$ consists of simple algebras
with no proper subalgebras and then invoke Corollary 1 in [1].

######  Theorem 4.6.

Let $P=(I,J,K)$ be a reduced triple such that $\mathsf{V}=\mathsf{V}(I,J,K)$
is a proper subvariety of Wajsberg hoops. Then $\mathsf{V}$ is structural if
and only if either $J=\emptyset$, or $J=\\{1\\}$.

###### Proof.

Suppose that $J\neq\emptyset$ and $J\neq\\{1\\}$. Then there is an $n\in J$
with $n>1$. Let $\mathsf{K}=\\{{\textbf{\L}}_{i}:i\in
I\\}\cup\\{{\textbf{\L}}_{j}:j\in J,j\neq n\\}\cup\\{{\textbf{\L}}_{n,1}\\}$;
Clearly ${\mathbf{Q}}(\mathsf{K})\subseteq\mathsf{Q}(P)=\mathsf{V}(P)$; if
${\textbf{\L}}_{n}^{\infty}\in{\mathbf{Q}}(\mathsf{K})$ then, by Theorem 2.2,
${\textbf{\L}}_{n}^{\infty}\in{\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}(\mathsf{K})$,
since it is subdirectly irreducible. But all the chains in $\mathsf{K}$ are
either finite or their divisibility index is not divided by $n$; hence, by
Lemma 4.1 (3) and (4),
${\textbf{\L}}_{n}^{\infty}\notin{\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}(\mathsf{K})$.
Therefore ${\mathbf{Q}}(\mathsf{K})\subsetneq\mathsf{Q}(P)$; however, as
${\textbf{\L}}_{n,1}\in{\mathbf{V}}({\textbf{\L}}_{n}^{\infty}$),
${\mathbf{H}}({\mathbf{Q}}(\mathsf{K}))=\mathsf{V}(P)$. So $\mathsf{V}(P)$ is
not structural.

For the converse, modulo the results on locally finite varieties above, we
need only to prove that $\mathsf{V}(P)$ is structurally complete whenever
$P=(I,\emptyset,\\{\omega\\})$ or $P=(I,\\{1\\},\emptyset)$. In either case,
by Theorems 3.10 and 3.11, every generator of $\mathsf{V}(P)=\mathsf{Q}(P)$ is
embeddable in ${\textbf{B}}_{\Delta}$. So

$\mathsf{V}(P)=\mathsf{Q}(P)\subseteq{\mathbf{Q}}({\textbf{B}}_{\Delta})={\mathbf{Q}}({\textbf{F}}_{\mathsf{V}(P)}(x))$

and this proves that $\mathsf{V}(P)$ is structural. ∎

By the description of the proper subvarieties of $\mathsf{WH}$ by reduced
triples, we get at once:

######  Corollary 4.7.

A variety of Wajsberg hoops is structural if and only if it is primitive.

And thus, via the Blok-Pigozzi connection:

######  Corollary 4.8.

An axiomatic extension on $\mathcal{MV}^{+}$ is structurally complete if and
only if it is hereditarily structurally complete.

###  4.2. Structural subquasivarieties

For quasivarieties we need to work a little bit more. First we need a lemma
that appears in [29] (Lemma 4.5):

######  Lemma 4.9.

Let $n>1$ and let ${\textbf{D}}_{n}$ be the subalgebra of
${\textbf{\L}}_{n,1}\times{\textbf{\L}}_{n,n-1}$ generated by $((1,0),(1,1))$.
Then ${\textbf{\L}}_{n,1}$ is embeddable in ${\textbf{D}}_{n}$.

Using Lemma 4.9 we can prove:

######  Lemma 4.10.

Let $P=(I,J,\emptyset)$ be a reduced triple. Then for any $j\in J$,
${\textbf{\L}}_{j,1}$ is embeddable in ${\textbf{B}}_{\Delta}$.

###### Proof.

If $J=\emptyset$ we have nothing to prove.

If $1\in J$, then since $P$ is a reduced triple, $P=(I,\\{1\\},\emptyset)$.
Let $m$ be the product of all the elements of $I$ (if $I=\emptyset$ take
$m=1$) and consider the Wajsberg function

$f(x)=L(0,0;\frac{1}{3m},0;\frac{2}{3m},1;1,1).$

This function has value $0$ in a neighborhood of $0$ and has value $1$ for
every $\frac{n}{m}\neq 0$, so, by Lemma 3.9, $f(\overline{g}(k,h,i))=0$ if
$(k,h,i)=(1,0,0)$, otherwise $f(\overline{g}(k,h,i))=1$. Since
$\overline{g}(1,0,0)$ is a generator of ${\textbf{\L}}_{1}^{\infty}$,
$\overline{g}\vee f(\overline{g})$ generates a subalgebra of
${\textbf{B}}_{\Delta}$ isomorphic to ${\textbf{\L}}_{1}^{\infty}$. Now, by
Lemma 4.1,
$\mathsf{Q}({\textbf{\L}}_{1}^{\infty})=\mathsf{Q}({\textbf{\L}}_{1,1})$;
hence ${\textbf{\L}}_{1,1}$ is embeddable into ${\textbf{\L}}_{1}^{\infty}$
and thus into ${\textbf{B}}_{\Delta}$.

Now suppose $1\notin J$ and fix $j\in J$. Let $m$ be the product of every
element of $I\cup J$ and consider the Wajsberg function

$f(x)=L(0,1;\frac{1}{j}-\frac{2}{3m},1;\frac{1}{j}-\frac{1}{3m},0;\frac{1}{j}+\frac{1}{3m},0;\frac{1}{j}+\frac{2}{3m},1;1,1).$

This function has value $0$ in a neighborhood of $\frac{1}{j}$ and has value
$1$ in a neighborhood of $\frac{n}{m}$ when $\frac{n}{m}\neq\frac{1}{j}$. By
Lemma 3.9, $f(\overline{g}(k,h,i))=0$ if $(k,h,i)=(j,1,0)$ or
$(k,h,i)=(j,j-1,0)$. Now let us show that $f(\overline{g}(k,h,i))=1$ in all
the other cases.

Take $(k,h,i)\neq(j,1,0),(j,j-1,0)$. If $i\in\\{0,1\\}$, then
$\overline{g}(k,h,i)=(r,s)$ for some $(r,s)\in{\textbf{\L}}_{k,h}$; notice
that, by construction of $f$, $f(\overline{g}(k,h,i))\neq 1$ only if
$\frac{r}{k}=\frac{1}{j}$, but this happens only if $jr=k$, but this can not
happen because the triple is reduced, so $k$ can not be a multiple of $j$. If
$i=2$, then $\overline{g}(k,h,i)=h$ for some $0\leq h<k$ and $k,h$ relatively
prime; this time $f(\overline{g}(k,h,i))\neq 1$ only if
$\frac{h}{k}=\frac{1}{j}$, that is only if $j=\frac{k}{h}$, but this means
that $h$ divides $k$, which is not possible because $k,h$ are relatively
prime.

Thus, if we consider $\overline{g}\vee f(\overline{g})$, this generates a
subalgebra of ${\textbf{B}}_{\Delta}$ isomorphic to ${\textbf{D}}_{j}$ as
defined in Lemma 4.9; moreover, by the same lemma, we get that
${\textbf{L}}_{j,1}$ is embeddable into ${\textbf{D}}_{j}$ and hence into
${\textbf{B}}_{\Delta}$. ∎

Let $P=(I,J,K)$ be a triple (not necessarily reduced) and let
$\mathsf{Q}[I,J,K]$ be defined in the following way

$\displaystyle\mathsf{Q}[I,\emptyset,K]=\mathsf{Q}(I,\emptyset,K)$
$\displaystyle\mathsf{Q}[I,J,\emptyset]={\mathbf{Q}}(\\{{\textbf{\L}}_{i}:i\in
I\\}\cup\\{{\textbf{\L}}_{j,1}:j\in J\\})\quad\text{if $J\neq\emptyset$}.$

######  Theorem 4.11.

Let $\mathsf{Q}$ be a quasivariety of Wajsberg hoops; then $\mathsf{Q}$ is
structural if and only if it is either
$\mathsf{Q}={\mathbf{Q}}({\textbf{F}}_{\mathsf{WH}}(x))$ or else
$\mathsf{Q}=\mathsf{Q}[P]$ for some reduced triple $P$.

###### Proof.

As the structurally complete subquasivarieties are exactly
${\mathbf{Q}}({\textbf{F}}_{\mathsf{V}}(x))$ for
$\mathsf{V}\subseteq\mathsf{WH}$, it is enough to show that for every reduced
triple $P=(I,J,K)$,
$\mathsf{Q}[P]={\mathbf{Q}}({\textbf{F}}_{\mathsf{V}(P)}(x))$. Moreover if
$\mathsf{V}$ is a subvariety of $\mathsf{WH}$, then
${\mathbf{Q}}({\textbf{F}}_{\mathsf{V}}(x))$ is the structural core of
$\mathsf{V}$, i.e. the smallest subquasivariety $\mathsf{Q}$ of $\mathsf{V}$
such that ${\mathbf{V}}(\mathsf{Q})=\mathsf{V}$. Now for any reduced
presentation $P$, we clearly have that
${\mathbf{V}}(\mathsf{Q}[P])={\mathbf{V}}(P)$; so to get the conclusion it is
enough to prove that
$\mathsf{Q}[P]\subseteq{\mathbf{Q}}({\textbf{F}}_{\mathsf{V}(P)}(x))$.

But Theorem 3.10, 3.11 and Lemma 4.10 show that any generator of
$\mathsf{Q}[P]$ is embeddable in ${\textbf{B}}_{\Delta}$; so

$\mathsf{Q}[P]\subseteq\mathsf{Q}({\textbf{B}}_{\Delta})={\mathbf{Q}}({\textbf{F}}_{\mathsf{V}(P)}(x))$

and the conclusion holds. ∎

Observe that
${\mathbf{Q}}({\textbf{\L}}_{1,1})={\mathbf{Q}}({\textbf{\L}}_{1}^{\infty})$;
so if $P=(I,J,K)$ is such that either $J=\emptyset$ or $J=\\{1\\}$, then
$Q[P]=Q(P)$ and they are all in fact primitive varieties, by Corollary 4.7. We
can make another observation of some relevance based on the results in [23].
It is well known (and easy to prove) that the Wajsberg chains coincide with
the finitely subdirectly irreducible Wajsberg hoops and the variety of
Wajsberg hoops is congruence distributive. Then:

  1. $\bullet$

every structural subquasivariety $\mathsf{Q}$ is generated by finitely
subdirectly irreducible Wajsberg hoops and hence it is relatively congruence
distributive;

  2. $\bullet$

in any structural subquasivariety $\mathsf{Q}$ the finitely
$\mathsf{Q}$-irreducible algebras are finitely subdirectly irreducible in the
absolute sense, i.e. they are all Wajsberg chains;

  3. $\bullet$

hence any algebra ${\textbf{A}}\in\mathsf{Q}$ is subdirectly embeddable in a
product of Wajsberg chains that belong to $\mathsf{Q}$.

This information might be useful for characterizing the structural
subquasivarieties $\mathsf{Q}[P]$ that are primitive (besides the ones that
are already known). We will consider the quasivarieties $\mathsf{Q}[I,J,K]$
where $K=\emptyset$ and $J\neq\emptyset,\\{1\\}$ and in this case we will
write simply $\mathsf{Q}[I,J]$.

######  Lemma 4.12.

Let $(I,J,\emptyset)$ and $(I^{\prime},J^{\prime},\emptyset)$ be two triples
(not necessarily reduced), then
$\mathsf{Q}[I,J]\subseteq\mathsf{Q}[I^{\prime},J^{\prime}]$ if and only if for
every $i\in I$ s.t. $i\neq 1$ and for every $j\in J$, there are $i^{\prime}\in
I^{\prime}$ and $j^{\prime}\in J^{\prime}$ with $i|i^{\prime}$ and
$j|j^{\prime}$.

###### Proof.

If $i=1$, then
${\textbf{\L}}_{1}\in{\mathbf{I}}{\mathbf{S}}({\textbf{\L}}_{n})$ and
${\textbf{\L}}_{1}\in{\mathbf{I}}{\mathbf{S}}({\textbf{\L}}_{n,1})$ for every
$n$. Take now $i\neq 1$ and suppose that
${\textbf{\L}}_{i}\in{\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}(\\{{\textbf{\L}}_{i^{\prime}}:i\
\in I^{\prime}\\}\cup\\{{\textbf{\L}}_{j^{\prime},1}:j^{\prime}\in
J^{\prime}\\})$; then
${\textbf{\L}}_{i}\notin{\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}(\\{{\textbf{\L}}_{j^{\prime},1}:j^{\prime}\in
J^{\prime}\\})$ as ${\textbf{\L}}_{i}$ has divisibility index $i\neq 1$ and
each ${\textbf{\L}}_{j^{\prime},1}$ has divisibility index equal to $1$. On
the other hand no ${\textbf{\L}}_{j,1}$ can belong to
${\mathbf{I}}{\mathbf{S}}{\mathbf{P}}_{u}(\\{{\textbf{\L}}_{i^{\prime}}:i\ \in
I^{\prime}\\})$ for obvious reasons. From here it is straightforward to check
that the conclusion holds. ∎

Now we can give an example of a class of quasivarieties that are not
primitive.

######  Proposition 4.13.

Let $\mathsf{Q}=\mathsf{Q}[I,J]$ with $(I,J,\emptyset)$ reduced triple; if
$I{\downarrow}\cap J{\downarrow}\supsetneq\\{1\\}$, then $\mathsf{Q}$ is not
primitive.

###### Proof.

If $I{\downarrow}\cap J{\downarrow}\supsetneq\\{1\\}$, then there exists
$n\neq 1$ in $I{\downarrow}\cap J{\downarrow}$ and, by the previous lemma,
$\mathsf{Q}[n,n]\subseteq\mathsf{Q}[I,J]$. Now, notice that
$\mathsf{V}(\mathsf{Q}[n,n])=\mathsf{V}(\emptyset,n,\emptyset)=\mathsf{V}(\mathsf{Q}[\emptyset,n])$,
so, by theorem 4.11, the structural core of $\mathsf{Q}[n,n]$ is
$\mathsf{Q}[\emptyset,n]$. But, by the previous lemma,
$\mathsf{Q}[\emptyset,n]\subsetneq Q[n,n]$ so, in particular,
$\mathsf{Q}[n,n]$ is different from its structural core, hence it is not
structural. Therefore $\mathsf{Q}$ contains a quasivariety that is not
structural and this means that it is not primitive. ∎

We can also give an example of a class of primitive quasivarieties.

######  Proposition 4.14.

Let $\mathsf{Q}=\mathsf{Q}[\emptyset,p]$, where $p$ is a prime number; then
$\mathsf{Q}$ is primitive.

###### Proof.

We know that $\mathsf{V}(\mathsf{Q})=\mathsf{V}(\emptyset,p,\emptyset)$ and,
since $\mathsf{Q}$ is structural, this means that every quasivariety strictly
contained in $\mathsf{Q}$ generates a variety that is strictly contained in
$\mathsf{V}(\mathsf{Q})$. So now let’s consider all the subvarieties of
$\mathsf{V}(\emptyset,p,\emptyset)$.

$\mathbf{0}$$\mathsf{V}(1,\emptyset,\emptyset)$$\mathsf{V}(\emptyset,\emptyset,\omega)$$\mathsf{V}(p,\emptyset,\emptyset)$$\mathsf{V}(1,\emptyset,\omega)$$\mathsf{V}(p,\emptyset,\omega)$$\mathsf{V}(\emptyset,1,\emptyset)$$\mathsf{V}(\emptyset,p,\emptyset)$

If we consider a quasivariety $\mathsf{Q}^{\prime}$ that generates a variety
strictly contained in $\mathsf{V}(\emptyset,p,\emptyset)$, then it has to be
contained in the coatoms of the lattice, that are
$\mathsf{V}(p,\emptyset,\omega)$ and $\mathsf{V}(\emptyset,1,\emptyset)$. By
theorem 4.6, we know that these two varieties are structural, hence primitive,
so every quasivariety contained in them is structural. ∎

Unfortunately, we fall short of characterizing all the primitive
quasivarieties, due to our lack of understanding of the lattice of all the
subquasivarieties. The proof of the Proposition 4.14 is based on the fact that
all the quasivarieties strictly contained in $\mathsf{Q}[\emptyset,p]$ are
actually varieties; so we know that the lattice of the subquasivarieties of
$\mathsf{Q}$ is contained in the lattice of the subvarieties of
$\mathsf{V}(\mathsf{Q})$ which clearly is not always the case.

Consider for example the quasivariety
$\mathsf{Q}=\mathsf{Q}[\emptyset,\\{p,q\\}]$. Using Lemma 4.12, we may sketch
what the lattice of the subquasivarieties of $\mathsf{Q}$ looks like:

$\mathbf{0}$$\mathsf{Q}(1,\emptyset,\emptyset)$$\mathsf{Q}(\emptyset,\emptyset,\omega)$$\mathsf{Q}(1,\emptyset,\omega)$$\mathsf{Q}(\emptyset,p,\emptyset)$$\mathsf{Q}(\emptyset,q,\emptyset)$$\mathsf{Q}(\emptyset,\\{p,q\\},\emptyset)$

Now, $\mathsf{Q}(\emptyset,p,\emptyset)$ and
$\mathsf{Q}(\emptyset,q,\emptyset)$ are primitive by Lemma 4.14, but we don’t
know if there is any quasivariety in the intervals
$[\mathsf{Q}(\emptyset,p,\emptyset),\mathsf{Q}(\emptyset,\\{p,q\\},\emptyset)]$
and
$[\mathsf{Q}(\emptyset,q,\emptyset),\mathsf{Q}(\emptyset,\\{p,q\\},\emptyset)]$.
Note that, if there is such a quasivariety, then it cannot be generated by
chains, and that would immediately imply that $\mathsf{Q}$ is not primitive,
because by Theorem 4.11 this quasivariety would not be structural.

Another problem is that, given a quasivariety $\mathsf{Q}$, the lattice of all
the varieties $\mathsf{V}(\mathsf{Q}^{\prime})$, with
$\mathsf{Q}^{\prime}\subseteq\mathsf{Q}$, is almost always strictly contained
in the lattice of the subvarieties of $\mathsf{V}(\mathsf{Q})$. For example
consider $\mathsf{Q}=\mathsf{Q}[\emptyset,pq]$; then
$\mathsf{V}(\mathsf{Q})=\mathsf{V}(\emptyset,pq,\emptyset)$ and we know that
$\mathsf{V}(pq,p,\emptyset)$ is a subvariety of
$\mathsf{V}(\emptyset,pq,\emptyset)$. Now, if a quasivariety generates
$\mathsf{V}(pq,p,\emptyset)$ then it would not be primitive, because it would
contain $\mathsf{Q}[pq,p]$ that is not primitive by Lemma 4.13, but by lemma
4.12 we know that no subquasivariety of $\mathsf{Q}$ can contain
${\textbf{\L}}_{pq}$, so there is no subquasivariety of $\mathsf{Q}$ that can
generate $\mathsf{V}(pq,p,\emptyset)$.

##  5\. Conclusions and future work

What can we say about the fragments of $\mathcal{MV}^{+}$? We will consider
only the fragments containing $\rightarrow$, as they are the algebraizable
ones. The $\\{\rightarrow\\}$-fragment has been studied first in [36]; its
equivalent algebraic semantics is the variety $\mathsf{LBCK}$ of Łukasiewicz
$\mathsf{BCK}$ algebras. We have that:

  1. $\bullet$

every locally finite quasivariety of $\mathsf{LBCK}$-algebras is a primitive
variety [9];

  2. $\bullet$

the only non-locally finite subvariety is the entire variety $\mathsf{LBCK}$
[36];

  3. $\bullet$

$\mathsf{LBCK}$ it is generated as a quasivariety by its finite chains [6];

  4. $\bullet$

every infinite chain contains all the finite chains as subalgebras [36];

  5. $\bullet$

so if $\mathsf{Q}$ is a quasivariety which contains only finitely many chains,
then ${\mathbf{V}}(\mathsf{Q})$ is locally finite, hence primitive;

  6. $\bullet$

otherwise $\mathsf{Q}$ contains infinitely many chains and so
${\mathbf{V}}(\mathsf{Q})=\mathsf{Q}=\mathsf{LBCK}$.

Hence every subquasivariety of $\mathsf{LBCK}$ is a variety and
$\mathsf{LBCK}$ is primitive.

For the other algebraizable fragments, observe that if $\rightarrow$ is
present then $\vee$ is definable and if $\rightarrow$ and $\cdot$ are present,
then $\wedge$ is definable. So the only remaining interesting fragment is the
$\\{\rightarrow,\wedge,1\\}$-fragment that has been considered in [4]. Its
equivalent algebraic semantics is the variety $\mathsf{LBCK}^{\wedge}$ of
$\mathsf{LBCK}$-semilattices; from the results in [4] (and some
straightforward calculations) one can prove that $\mathsf{LBCK}^{\wedge}$ is
primitive.

As far as the future work is concerned, there is a very natural path to
follow. In this paper we have characterized all the structurally complete
finitary extensions of $\mathcal{MV}^{+}$ and in [29] the same has been
basically done for finitary extensions of $\mathcal{MV}$. Hajek’s Basic Logic
$\mathcal{BL}$ [31] has been investigated from the algebraic point of view in
many papers through its equivalent algebraic semantics, that is the variety of
$\mathsf{BL}$-algebras (see for instance [2] and the bibliography therein). In
particular, in [7] (Theorem 3.7) it has been shown that there is a very deep
algebraic connection between $\mathsf{BL}$-algebras (and their positive
subreducts), $\mathsf{MV}$-algebras and Wajsberg hoops. With the knowledge we
have accumulated so far (and using also more general techniques introduced in
[9]) we believe we can tackle the problem of describing the finitary
structurally complete extensions of $\mathcal{BL}$ with some degree of
success.

## References

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