# Implication in sharply paraorthomodular and relatively paraorthomodular
posets

Ivan Chajda, Davide Fazio, Helmut Länger, Antonio Ledda and
Jan Paseka

###### Abstract

In this paper we show that several classes of partially ordered structures
having paraorthomodular reducts, or whose sections may be regarded as
paraorthomodular posets, admit a quite natural notion of implication, that
admits a suitable notion of adjointness. Within this framework, we propose a
smooth generalization of celebrated Greechie’s theorems on amalgams of finite
Boolean algebras to the realm of Kleene lattices.

AMS Subject Classification: 03G12, 03G25, 03G10, 03B47, 03B60, 06A11, 06C15

Keywords: Paraorthomodular poset, relatively paraorthomodular poset, sharply
paraorthomodular poset, orthogonal poset, Kleene poset, weakly Boolean poset,
Kleene lattice, logics of quantum mechanics, orthomodular lattice,
orthomodular poset, amalgam of Kleene lattices, atomic amalgam, adjointness,
Sasaki implication

##  1 Introduction

Janusz Czelakowski’s early contributions concern quantum logic and the
foundation of quantum mechanics. More precisely, he deepened Kochen and
Specker’s results on partial Boolean algebras (PBAs), which are partial
algebras that generalize projection operators over separable Hilbert spaces
[19]. Among other achievements, Czelakowski’s work provided, on the one hand,
a novel characterization of PBAs embeddable into Boolean algebras. On the
other hand, it clarified why PBAs can be regarded as the algebraic semantic of
a logic in its own right. See e.g. [10, 11, 12] for details.
The interest in partial Boolean algebras is motivated by their coping quite
well with indeterminacy arising when non-commuting observables (self-adjoint
linear operators) are considered. In fact, partial Boolean algebras are
endowed with a binary operation $\lor$ yielding the algebraic counterpart of a
partial disjunction which is defined for pairs of commensurable observables.
Therefore, although rooted in a common concrete model (i.e. projections over
Hilbert spaces), Kochen and Specker’s approach to the foundation of quantum
mechanics differs from Birkhoff and von Neumann’s quantum logic [3]. In fact,
the former avoids interpretation difficulties stemmed by the latter, which
assumes that conjunctions/disjunctions of quantum propositions always exist,
even if they concern incompatible observables.
Quoting Czelakowski and Suppes,

> “They (_Birkhoff and von Neumann_) require that the structure of
> experimental propositions be a lattice, and thus that the conjunction of two
> meaningful propositions in the lattice also be in lattice, but.their
> requirement seems too strict when one proposition expresses a possible
> result of measuring position and the other measuring momentum. The fact that
> the closed linear subspaces of a complex separable Hilbert space form such a
> lattice is not sufficient, in my view, to maintain that it expresses the
> more restricted logic of experimental propositions, in spite of the central
> importance of this lattice of subspaces in the formulation of the theory”.

Indeed, as mentioned in [10, 21],

> “What experimental meaning can one attach to the meet and join of two given
> experimental propositions”.

Among classes of partial Boolean algebras, a prominent role is played by
_transitive_ partial Boolean algebras (see e.g. [10]) which, in turn, are
orthomodular posets whose compatibility relation is _regular_ (see e.g. [2,
18, 20] for details).

Let us recall that a bounded poset $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ with
a complementation which is an antitone involution is called orthomodular if it
is orthogonal, i.e. $x\vee y$ exists for $x,y\in P$ whenever $x\leq
y^{\prime}$, and it satisfies the orthomodular law

  1. (OM)

$x\leq y\ \text{implies}\ x\vee(y\wedge x^{\prime})=y$.

Moreover, if $\mathbf{P}$ is lattice-ordered, i.e. it is an _orthomodular
lattice_ , then (OM) can be expressed in the form of an identity

  1. (OI∨)

$x\vee\big{(}(x\vee y)\wedge x^{\prime}\big{)}\approx x\vee y$

and its dual form

  1. (OI∧)

$x\wedge\big{(}(x\wedge y)\vee x^{\prime}\big{)}\approx x\wedge y$.

A different abstract counterpart of the set of quantum events has been
suggested within the so-called unsharp approach to quantum theory. Such
structures (i.e. effect algebras or MV-algebras) represent a suitable
algebraic counterpart of quantum effects, namely bounded self-adjoint linear
operators on a Hilbert space that satisfy the Born’s rule. However, since the
canonical order induced by Born’s rule on the set of effects on a Hilbert
space is not a lattice in general, effect algebras do not lend themselves to a
smooth logico-algebraic investigation.

For this reason, in 2016 Roberto Giuntini, Francesco Paoli and one of the
present authors proposed the notion of paraorthomodular lattice as natural
generalization of the concept of orthomodular lattice. These structures find a
concrete realization in the set of effects equipped whit Olson’s spectral
ordering, see e.g. [15] or [16]. However, although paraorthomodular lattices
allow to overcome difficulties arising from effect algebras framework in a
very natural and physically motivated fashion, they may as well rise
interesting philosophical questions, paralleling, in some sense, what happens
to their “sharp relatives”.
In [5] paraorthomodular posets were introduced as a framework in which the
order–negation reduct of quantum structures ranging from orthomodular
lattices, effect algebras and, of course, paraorthomodular lattices, can be
treated into a common backdrop. In the same work, a preliminary investigation
of sharply paraorthomodular posets was provided. These structures play for
paraorthomodular lattices the role that orthomodular posets possess within the
theory of orthomodular lattices. Therefore, the natural question arises
whether paraorthomodular posets may be regarded as the semantic counterpart of
a logic in its own right.

In this paper we address the issue of showing that several classes of
partially ordered structures having paraorthomodular reducts, or whose
“sections” (in the sense of section 5 below) may be regarded as
paraorthomodular posets, admit a quite natural notion of implication.
In particular, this paper is organized as follows: after dispatching some
preliminary notions in section 1, in section 2 we discuss a possible smooth
notion of implication in sharply paraorthomodular posets. In section 4, we
provide a generalization of celebrated R. Greechie’s theorems [17] on amalgams
of finite Boolean algebras to the realm of Kleene lattices. In sections 5 and
6, we discuss how the theory of paraortomodular posets carries over in two
different degrees of freedom: _relatively_ paraorthomodular posets and
_relatively_ paraorthomodular join-semilattices. Finally, in section 7, we
address the question if there exists a binary connective that forms an adjoint
pair with the implication formerly introduced.

##  2 Basic concepts

Let $\mathbf{P}=(P,\leq)$ be a poset, $A,B\subseteq P$ and $a,b\in P$. We
define $A\leq B$ if and only if $x\leq y$ for all $x\in A$ and all $y\in B$.
Instead of $A\leq\\{b\\}$, $\\{a\\}\leq B$ and $\\{a\\}\leq\\{b\\}$ we simply
write $A\leq b$, $a\leq B$ and $a\leq b$, respectively. The sets

$\displaystyle L(A)$ $\displaystyle:=\\{x\in P\mid x\leq A\\},$ $\displaystyle
U(A)$ $\displaystyle:=\\{x\in P\mid A\leq x\\}$

are called the lower cone and upper cone of $A$, respectively. Instead of
$L(A\cup B)$, $L(A\cup\\{b\\})$, $L(\\{a,b\\})$ and $L\big{(}U(A)\big{)}$ we
simply write $L(A,B)$, $L(A,b)$, $L(a,b)$ and $LU(A)$, respectively.
Analogously, we proceed in similar cases. On $2^{P}$ we introduce four binary
relations $\leq$, $\leq_{1}$, $\leq_{2}$ and $\approx_{2}$ as follows:

$\displaystyle A\leq B$ $\displaystyle\text{ if and only if for every }x\in
A\text{ and for every }y\in B\text{ we have }x\leq y,$ $\displaystyle
A\leq_{1}B$ $\displaystyle\text{ if and only if for every }x\in A\text{ there
exists some }y\in B\text{ with }x\leq y,$ $\displaystyle A\leq_{2}B$
$\displaystyle\text{ if and only if for every }y\in B\text{ there exists some
}x\in A\text{ with }x\leq y,$ $\displaystyle A\approx_{2}B$
$\displaystyle\text{ if and only if }A\leq_{2}B\text{ and }B\leq_{2}A.$

Recall that $A\leq y$ if and only if $A\leq_{1}y$, and similarly $x\leq B$ if
and only if $x\leq_{2}B$.

A unary operation ′ on $P$ is called an antitone involution if

$x,y\in P\text{ and }x\leq y\text{ together imply }y^{\prime}\leq x^{\prime}$

and if it satisfies the identity

$x^{\prime\prime}\approx x.$

Note that if $\mathbf{P}$ has an antitone involution ′ and a bottom element
$0$, then it has also a top element $1$, namely $0^{\prime}$. We will call
$x,y\in P$ orthogonal to each other, shortly $x\perp y$, if $x\leq y^{\prime}$
or, equivalently, $y\leq x^{\prime}$.

By an orthogonal poset we mean a bounded poset $(P,\leq,{}^{\prime},0,1)$ with
an antitone involution satisfying the following condition:

$x,y\in P\text{ and }x\perp y\text{ together imply that }x\vee y\text{
exists}.$

Here and in the following $x\vee y$ denotes the supremum of $x$ and $y$.

The concept of a paraorthomodular poset was introduced in [5] as a
generalization of the concept of a paraorthomodular lattice introduced in
[16]. Recall that a bounded poset $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ with
an antitone involution is called paraorthomodular if it satisfies the
following condition:

  1. (P)

$x,y\in P$, $x\leq y$ and $x^{\prime}\wedge y=0$ together imply $x=y$.

It should be remarked that $x^{\prime}\wedge y=0$ is equivalent to
$L(x^{\prime},y)=0$. The second notation was used in our previous paper [5].
Moreover, if the paraorthomodular poset $\mathbf{P}$ is lattice-ordered then
it is called a _paraorthomodular lattice_.

Given a poset $\mathbf{P}$, an element $y\in P$ is said to be a complement of
$x\in P$ if $L(x,y)=L(P)$ and $U(x,y)=U(P)$. $\mathbf{P}$ is said to be
complemented if each element of $P$ has a complement in $\mathbf{P}$.
Moreover, if $\mathbf{P}$ is a bounded poset with antitone involution which is
orthogonal and ′ is a complementation, then $\mathbf{P}$ is an orthomodular
poset. Namely, as shown e.g. in [18], for a complementation ′, (P) equivalent
to (OM).

An orthogonal paraorthomodular poset is called sharply paraorthomodular (see
[5]). Of course, any paraorthomodular lattice is sharply paraorthomodular. We
call a sharply paraorthomodular poset $\mathbf{P}$ _regular_ provided it
satisfies the following identity, for any $x,y\in P$:

  1. (reg)

$x\land x^{\prime}\leq y\lor y^{\prime}$.111It is worth observing that
paraorthomodular lattices defined in [15] coincide with regular
paraorthomodular lattices in our framework.

Note that, by virtue of orthogonality, (2) is well-defined. We call a
distributive, regular paraorthomodular lattice a _Kleene lattice_.

A poset $\mathbf{P}$ is called distributive if one of the following equivalent
LU-identities is satisfied:

$\displaystyle L\big{(}U(x,y),z\big{)}$ $\displaystyle\approx
LU\big{(}L(x,z),L(y,z)\big{)},$ $\displaystyle U\big{(}L(x,z),L(y,z)\big{)}$
$\displaystyle\approx UL\big{(}U(x,y),z\big{)},$ $\displaystyle
U\big{(}L(x,y),z\big{)}$ $\displaystyle\approx UL\big{(}U(x,z),U(y,z)\big{)},$
$\displaystyle L\big{(}U(x,z),U(y,z)\big{)}$ $\displaystyle\approx
LU\big{(}L(x,y),z\big{)},$ $\displaystyle
L\big{(}U(x_{1},x_{2},\dots,x_{n}),z\big{)}$ $\displaystyle\approx
LU\big{(}L(x_{1},z),L(x_{2},z),\dots,L(x_{n},z)\big{)},$ $\displaystyle
U\big{(}L(x_{1},x_{2},\dots,x_{n}),z\big{)}$ $\displaystyle\approx
UL\big{(}U(x_{1},z),U(x_{2},z),\dots,U(x_{n},z)\big{)}.$

Recall that a Boolean poset is a bounded distributive complemented poset.
Since the complementation in a Boolean poset $\mathbf{P}$ is unique and
antitone $\mathbf{P}$ is a poset with antitone involution.

An orthomodular poset $\mathbf{P}$ is called weakly Boolean [22] if for every
$a,b\in P$ the condition $a\wedge b=a\wedge b^{\prime}=0$ implies $a=0$.

Let $\mathbf{P}=(P,\leq)$ be a poset. In what follows, for every subset $A$ of
$P$ let $\operatorname{Max}A$ and $\operatorname{Min}A$ denote the set of all
maximal and minimal elements of $A$, respectively. We say that $\mathbf{P}$ is

  1. 1.

mub-complete [1] if for every upper bound $x$ of a finite subset $M$ of $P$
there is a minimal upper bound of $M$ below $x$,

  2. 2.

mlb-complete if for every lower bound $x$ of a finite subset $M$ of $P$ there
is a maximal lower bound of $M$ above $x$,

  3. 3.

mlub-complete if it is both mub-complete and mlb-complete.

If $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ is a bounded poset with an antitone
involution then $\mathbf{P}$ is mub-complete if and only if it is mlb-
complete.

A poset $\mathbf{P}$ is said to satisfy the Ascending Chain Condition (ACC) if
it has no infinite ascending chains. The notion of Descending Chain Condition
(DCC) is defined dually. If $\mathbf{P}$ satisfies the ACC then every non-
empty subset of $P$ has at least one maximal element. The dual statement holds
for the DCC.

Evidently, if $\mathbf{P}$ satisfies the ACC then $\mathbf{P}$ is mlb-complete
and the dual statement holds for the DCC. Moreover, every finite poset and
every lattice is mlub-complete.

The bounded poset $\mathbf{P}$ has the maximality property [22] if for every
$a,b\in P$ the set $L(a,b)$ has a maximal element. Clearly, every mlb-complete
poset has the property of maximality. Recall also that every weakly Boolean
orthomodular poset with the property of maximality is a Boolean algebra [22,
Theorem 4.2].

We sometimes extend an operator $*\colon P^{2}\rightarrow 2^{P}$ to a binary
operation on $2^{P}$ by

$A*B:=\bigcup_{x\in A,y\in B}(x*y)$

for all $A,B\in 2^{P}$.

##  3 What implication for sharply paraorthomodular posets?

Let $(P,\leq,{}^{\prime},0,1)$ be an orthogonal mlb-complete poset. If $y\in
P$ and $A\subseteq P$ then we will denote the set $\\{y\vee a\mid a\in A\\}$
by $y\vee A$ provided that all the elements $y\vee a$ exist. Analogously, we
proceed in similar cases. Now we introduce the operator $\rightarrow\colon
P^{2}\rightarrow 2^{P}$ as follows:

  1. (I1)

$x\rightarrow y:=y\vee\operatorname{Max}L(x^{\prime},y^{\prime})$.

Since $\operatorname{Max}L(x^{\prime},y^{\prime})\leq y^{\prime}$,
$x\rightarrow y$ is correctly defined.

It is worth noticing that every orthomodular poset (see e.g. [6], [8], [13]
and [14]) is paraorthomodular, but there are examples of paraorthomodular
posets (or even lattices, indeed) which are not orthomodular, see [16].

######  Example 3.1.

The paraorthomodular posets depicted in Fig. 1 are neither orthomodular nor
sharply paraorthomodular:

$0$$a$$b$$b^{\prime}$$a^{\prime}$$1=0^{\prime}$(a)$0$$a$$b$$c$$d$$d^{\prime}$$c^{\prime}$$b^{\prime}$$a^{\prime}$$1=0^{\prime}$(b)Fig.
1paraorthomodular
posets$0$$a$$b$$c$$c^{\prime}$$b^{\prime}$$a^{\prime}$$1=0^{\prime}$(c)

The posets visualized in Fig. 2 are sharply paraorthomodular:

$0$$a$$b$$a^{\prime}$$b^{\prime}$$1=0^{\prime}$(a)$0$$a$$b$$c$$d$$d^{\prime}$$c^{\prime}$$b^{\prime}$$a^{\prime}$$1=0^{\prime}$(b)Fig.
2sharply paraorthomodular posets

The first one is a lattice, the second one is not. Moreover, the second one is
not orthomodular since $b\leq d^{\prime}$, but $b\vee(d^{\prime}\wedge
b^{\prime})=b\vee b=b\neq d^{\prime}$.

We first list several elementary properties of the operator $\rightarrow$
defined by (I1) in orthogonal posets.

######  Theorem 3.2.

Let $(P,\leq,{}^{\prime},0,1)$ be an orthogonal mlb-complete poset,
$\rightarrow$ defined by (I1) and $x,y\in P$. Then the following hold:

  1. (i)

$y\leq x\rightarrow y$,

  2. (ii)

$x\leq y$ implies $y\rightarrow z\leq_{1}x\rightarrow z$,

  3. (iii)

$x\rightarrow y\approx\left\\{\begin{array}[]{ll}y\vee y^{\prime}&\text{ if
}x\leq y,\\\ y\vee(x^{\prime}\wedge y^{\prime})&\text{ if }x\perp y,\\\
x^{\prime}\vee y&\text{ if }y\leq x,\end{array}\right.$

  4. (iv)

$(x\rightarrow y)\rightarrow y\approx
y\vee\big{(}y^{\prime}\wedge\operatorname{Min}U(x,y)\big{)}$,

  5. (v)

$\big{(}(x\rightarrow y)\rightarrow y\big{)}\rightarrow y\approx
y\vee\Big{(}y^{\prime}\wedge\big{(}y\vee\operatorname{Max}L(x^{\prime},y^{\prime})\big{)}\Big{)}$.

Let us note that if ′ is a complementation (e.g., if $\mathbf{P}$ is
orthomodular) then the first line of (iii) can be written in the form

  1. (iii’)

$x\leq y$ implies $x\rightarrow y=1$.

Implication in orthomodular lattices is usually defined by $x\rightarrow
y:=y\vee(x^{\prime}\wedge y^{\prime})$.

###### Proof of Theorem 3.2.

  1. (i)

follows directly from (I1).

  2. (ii)

Assume $x\leq y$ and $z\in P$. Then $y^{\prime}\leq x^{\prime}$. Let $u\in
y\rightarrow z$. Then there exists some
$w\in\operatorname{Max}L(y^{\prime},z^{\prime})$ with $z\vee w=u$. Since
$y^{\prime}\leq x^{\prime}$ we have $w\in L(x^{\prime},z^{\prime})$. Thus
there exists some $t\in\operatorname{Max}L(x^{\prime},z^{\prime})$ with $w\leq
t$. Now

$u=z\vee w\leq z\vee t\in x\rightarrow z$

proving $y\rightarrow z\leq_{1}x\rightarrow z$.

  3. (iii)

$x\rightarrow y\approx
y\vee\operatorname{Max}L(x^{\prime},y^{\prime})\approx\left\\{\begin{array}[]{ll}y\vee
y^{\prime}&\text{ if }x\leq y,\\\ y\vee(x^{\prime}\wedge y^{\prime})&\text{ if
}x\perp y,\\\ y\vee x^{\prime}=x^{\prime}\vee y&\text{ if }y\leq
x.\end{array}\right.$

  4. (iv)

$\displaystyle(x\rightarrow y)\rightarrow y$
$\displaystyle\approx\bigcup_{z\in x\rightarrow y}(z\rightarrow
y)\approx\bigcup_{z\in
y\vee\operatorname{Max}L(x^{\prime},y^{\prime})}\big{(}y\vee\operatorname{Max}L(z^{\prime},y^{\prime})\big{)}\approx$
$\displaystyle\approx\bigcup_{w\in\operatorname{Max}L(x^{\prime},y^{\prime})}\big{(}y\vee\operatorname{Max}L(y^{\prime}\wedge
w^{\prime},y^{\prime})\big{)}\approx\bigcup_{w\in\operatorname{Max}L(x^{\prime},y^{\prime})}\big{(}y\vee(y^{\prime}\wedge
w^{\prime})\big{)}\approx$ $\displaystyle\approx
y\vee\big{(}y^{\prime}\wedge\operatorname{Min}U(x,y)\big{)}.$

  5. (v)

According to (iv) and (iii) we have

$\displaystyle\big{(}(x\rightarrow y)\rightarrow y\big{)}\rightarrow y$
$\displaystyle\approx\Big{(}y\vee\big{(}y^{\prime}\wedge\operatorname{Min}U(x,y)\big{)}\Big{)}\rightarrow
y\approx$
$\displaystyle\approx\bigcup_{w\in\operatorname{Min}U(x,y)}\Big{(}\big{(}y\vee(y^{\prime}\wedge
w)\big{)}\rightarrow y\Big{)}\approx$
$\displaystyle\approx\bigcup_{w\in\operatorname{Min}U(x,y)}\Big{(}\big{(}y^{\prime}\wedge(y\vee
w^{\prime})\big{)}\vee y\Big{)}\approx$ $\displaystyle\approx
y\vee\Big{(}y^{\prime}\wedge(y\vee\operatorname{Max}L(x^{\prime},y^{\prime})\big{)}\Big{)}.$

∎

If $\mathbf{P}$ is, moreover, sharply paraorthomodular, we can prove a bit
more.

######  Lemma 3.3.

Let $(P,\leq,{}^{\prime},0,1)$ be a sharply paraorthomodular mlb-complete
poset, $\rightarrow$ defined by (I1) and $a,b\in P$. Then the following hold:

  1. (i)

$b^{\prime}\rightarrow b=b$.

  2. (ii)

If $a\perp b$ and $b\wedge b^{\prime}=0$ then $a\rightarrow b=b$ if and only
if $a=b^{\prime}$.

  3. (iii)

$a\rightarrow b=1$ implies $a\leq b$.

###### Proof.

  1. (i)

$b^{\prime}\rightarrow b=b\vee(b\wedge b^{\prime})=b$.

  2. (ii)

Assume $a\perp b$ and $b\wedge b^{\prime}=0$. First suppose $a\rightarrow
b=b$. Then $b\vee\operatorname{Max}L(a^{\prime},b^{\prime})=a\rightarrow b=b$.
Hence $\operatorname{Max}L(a^{\prime},b^{\prime})\leq b$. This shows
$\operatorname{Max}L(a^{\prime},b^{\prime})\leq b\wedge b^{\prime}=0$, i.e.
$\operatorname{Max}L(a^{\prime},b^{\prime})=0$ and therefore
$L(a^{\prime},b^{\prime})=0$, i.e. $a^{\prime}\wedge b^{\prime}=0$ which
together with $a\leq b^{\prime}$ and (P) yields $a=b^{\prime}$. The other
implication follows from (i).

  3. (iii)

Assume $a\rightarrow b=1$. Let
$c\in\operatorname{Max}L(a^{\prime},b^{\prime})$. Then $c\leq b^{\prime}$ and
$b\vee c=1$, i.e. $c^{\prime}\wedge b^{\prime}=0$. Applying (P) we conclude
$c=b^{\prime}$. Hence

$b^{\prime}=c\in\operatorname{Max}L(a^{\prime},b^{\prime})\subseteq
L(a^{\prime},b^{\prime})\subseteq L(a^{\prime})$

and thus $b^{\prime}\leq a^{\prime}$, i.e. $a\leq b$.

∎

As shown above, the operator $\rightarrow$ in sharply paraorthomodular posets
satisfies several nice properties and hence it can be considered as the
connective implication in the logic formalized by means of paraorthomodular
posets (see [5] for the connection of this logic with the logic of quantum
mechanics, see also [13] and [16] for the lattice version). Hence the natural
question arises whether this logic can be based on this connective only.
Contrary to the case of orthomodular posets treated from this point of view in
[8], this is not possible since we cannot derive the order $\leq$ by means of
the implication $\rightarrow$ only. However, we can get a condition on an
orthogonal poset $\mathbf{P}$ equipped with $\rightarrow$ formulated in this
language such that $\mathbf{P}$ becomes a sharply paraorthomodular one.

######  Theorem 3.4.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthogonal mlb-complete poset
and $\rightarrow$ defined by (I1). Then the following are equivalent:

  1. (i)

$\mathbf{P}$ is paraorthomodular.

  2. (ii)

If $x\rightarrow y=1$ then $x\leq y$.

###### Proof.

(i) $\Rightarrow$ (ii):
Assume that $x\rightarrow y=1$. Let
$z\in\operatorname{Max}L(x^{\prime},y^{\prime})$. Then $y\vee z=1$, i.e.,
$y^{\prime}\wedge z^{\prime}=0$. Since $z\leq y^{\prime}$ we have according to
(P) that $z=y^{\prime}$, i.e., $y^{\prime}\leq x^{\prime}$. We conclude $x\leq
y$.
(ii) $\Rightarrow$ (i):
If $x\leq y$ and $x^{\prime}\wedge y=0$ then

$y\rightarrow x=y^{\prime}\vee x=(x^{\prime}\wedge y)^{\prime}=0^{\prime}=1$

which implies $y\leq x$ by (ii) and together with $x\leq y$ yields $x=y$. ∎

In the case of lattices, (I1) reduces to

  1. (I2)

$x\rightarrow y:=y\vee(x^{\prime}\wedge y^{\prime})$.

The next lemma shows that $\rightarrow$ defined by (I2) is antitone in the
first variable provided the poset in question is a lattice.

######  Lemma 3.5.

Let $(L,\vee,\wedge,{}^{\prime})$ be a lattice with an antitone mapping ′ and
$\rightarrow$ defined by (I2). Then $x\leq y$ implies $y\rightarrow z\leq
x\rightarrow z$.

###### Proof.

$x\leq y$ implies $y^{\prime}\leq x^{\prime}$ and hence $y\rightarrow
z=z\vee(y^{\prime}\wedge z^{\prime})\leq z\vee(x^{\prime}\wedge
z^{\prime})=x\rightarrow z$. ∎

##  4 Amalgams of Kleene lattices

In this section we provide a generalization of celebrated R. Greechie’s
theorems [17] on amalgams of finite Boolean algebras to the realm of Kleene
lattices. Specifically, we will show how to obtain sharply paraorthomodular
posets and paraorthomodular lattices by “gluing” together Kleene lattices
pairwise sharing a common atomic subalgebra of at most four elements.
Interestingly enough, we will show that _any_ amalgam of Kleene lattices
$\mathscr{K}$ yields a paraorthomodular poset $\mathbf{K}$. Such a structure
is also a sharply paraorthomodular poset (paraorthomodular lattice) provided
that its building Kleene blocks do not form loops of order 3 (loops of order 3
or 4). Analogous results have been already obtained for (lattice) effect
algebras in terms of pastings of MV algebras, see e.g. [9, 23].

Let $\mathscr{K}=\\{\mathbf{K}_{i}\\}_{i\in I}$ be a family of finite Kleene
lattices such that $|K_{i}|\geq 6$, for any $i\in I$. Moreover, let $K_{i}\cap
K_{j}$ $(i\neq j)$ be either $\\{0,1\\}$ or an atomic Kleene subalgebra of
both $K_{i}$ and $K_{j}$ (on which operations coincide) such that $|K_{i}\cap
K_{j}|\leq 4$. Also, we assume that any $x\in K_{i}\cap
K_{j}\smallsetminus\\{0,1\\}$ is either an atom or a co-atom in both $K_{i}$
and $K_{j}$. We call $\mathscr{K}$ a _pasted family_ of Kleene lattices, and
any $\mathbf{K}_{i}\in\mathscr{K}$ an _initial block_. Note that
$\mathbf{K}_{i}\cap\mathbf{K}_{j}\not\cong\mathbf{K}_{3}$, where
$\mathbf{K}_{3}$ is the three-elements Kleene lattice, since otherwise one
would have $\mathbf{K}_{i}\cong\mathbf{K}_{3}\cong\mathbf{K}_{j}$. In fact,
assume by way of contradiction that
$\mathbf{K}_{i}\cap\mathbf{K}_{j}\cong\mathbf{K}_{3}$; let $a\in
K_{i}\smallsetminus K_{3}$. Clearly one must have $a\parallel b$, where $b$ is
the fixed point of $\mathbf{K}_{3}$. By regularity, one must have $0=a\land
a^{\prime}<b$, and so $\mathbf{K}_{i}$ contains a sublattice isomorphic to
$\mathbf{N}_{5}$, which is impossible. Therefore,
$\mathbf{K}_{i}\cap\mathbf{K}_{j}$ must be orthoisomorphic either to the two-
element Boolean algebra $\mathbf{B}_{2}$, or to the four-element Boolean
algebra $\mathbf{B}_{4}$, or to the four-element Kleene lattice:

$0$$x$$x^{\prime}$$1=0^{\prime}$Fig. 3

Let $\mathscr{K}=\\{\mathbf{K}_{i}\\}_{i\in I}$ be a pasted family of Kleene
lattices. Set ${K}=\bigcup_{i\in I}K_{i}$, and let $\leq\ \subseteq K^{2}$ be
such that, for any $x,y\in K$,

$x\leq y\text{ if and only if there exists }i\in I\text{ such that
}x\leq^{\mathbf{K}_{i}}y.$

Customary arguments yield the following

######  Lemma 4.1.

Let $\mathscr{K}=\\{\mathbf{K}_{i}\\}_{i\in I}$ be a pasted family of Kleene
lattices. Then $(K,\leq,0,1)$ is a bounded partially ordered set.

The following lemma is obvious

######  Lemma 4.2.

Let $\mathscr{K}=\\{\mathbf{K}_{i}\\}_{i\in I}$ be a pasted family of Kleene
lattices. Upon defining, for any $x\in K$,
$x^{\prime}=x^{\prime\mathbf{K}_{i}}$, for some $i\in I$ such that $x\in
K_{i}$, one has that $(K,\leq,{}^{\prime},0,1)$ is a bounded poset with
antitone involution.

Given a pasted family of Kleene lattices
$\mathscr{K}=\\{\mathbf{K}_{i}\\}_{i\in I}$, we will call the bounded poset
with antitone involution $(K,\leq,{}^{\prime},0,1)$ obtained from a pasted
family of Kleene lattices $\mathscr{K}=\\{\mathbf{K}_{i}\\}_{i\in I}$ as by
Lemma 4.1 and Lemma 4.2 an _atomic amalgam_ of Kleene lattices. Note that, if
any $\mathbf{K}_{i}\in\mathscr{K}$ is a Boolean algebra (for any $i\in I$),
then our concept coincides with the one developed by R. Greechie (cf. [2, p.
144]). The following result shows that, indeed, any atomic amalgam of Kleene
lattices yields a paraorthomodular poset.

######  Lemma 4.3.

Let $\mathscr{K}$ be a pasted family of Kleene lattices, then
$\mathbf{K}=(K,\leq,{}^{\prime},0,1)$ is a paraorthomodular poset.

###### Proof.

Of course, $(K,\leq,{}^{\prime},0,1)$ is a bounded poset with antitone
involution by Lemma 4.1 and Lemma 4.3. Now, suppose towards a contradiction
that $\mathbf{K}$ is not paraorthomodular. Then, by [5, Theorem 2.5],
$\mathbf{K}$ contains a strong sub-poset orthoisomorphic to the Benzene ring
$\mathbf{B}_{6}$.

$0$$y^{\prime}$$x$$x^{\prime}$$y$$1=0^{\prime}$Fig. 4

Since $x\leq^{\mathbf{K}_{i}}y$, for some $i\in I$, one has that
$\mathbf{K}_{i}$ contains a sublattice isomorphic to
${\color[rgb]{0,0,1}\mathbf{N}_{5}}$, which is impossible. ∎

######  Remark 4.4.

Let $\mathscr{K}=\\{\mathbf{K}_{i}\\}_{i\in I}$ be a pasted family of Boolean
algebras and $J\subseteq I$. Then $(\bigcup_{j\in J}K_{j},\leq,^{\prime},0,1)$
is a sub-poset with antitone involution of $\mathbf{K}$.

######  Lemma 4.5.

Let $K=\bigcup_{i\in I}K_{i}$ be an atomic amalgam of a pasted family of
Kleene lattices $\\{\mathbf{K}_{i}\\}_{i\in I}$. Then, for any $i,j\in I$ such
that $i\neq j$, $K_{i}\cup K_{j}$ is the base set of a paraorthomodular
lattice which is isomorphic to the amalgam pasting together the lattices
$(K_{i},\land,\lor)$ and $(K_{j},\land,\lor)$ along the pasted sublattice
$(K_{i}\cap K_{j},\land,\lor)$.

###### Proof.

Note that, since $K_{i}\cap K_{j}$ is finite, it is a closed plexus in the
sense of [2, p.139]. Moreover, $K_{i}\cup K_{j}$ is a regularly pasted amalgam
of lattices which, by the first part of [2, Theorem 3.4] is the universe of a
lattice upon defining greatest lower bounds and least upper bounds as in the
proof of [2, Lemma 3.3]. Moreover, by Lemma 4.3, $(K_{i}\cup
K_{j},\land,\lor,{}^{\prime},0,1)$ is a paraorthomodular lattice. ∎

Let $\mathbf{A}=(A,\leq)$ be a poset. For any $x,y\in A$, we say that $x$
_covers_ $y$, written $x\lessdot y$, provided that, for any $z\in A$ one has
that $x\leq z\leq y$ implies $x=z$ or $y=z$.

######  Lemma 4.6.

Let $\mathscr{K}=\\{\mathbf{K}_{i}\\}_{i\in I}$ be a pasted family of Kleene
lattices and let $x,y\in K$ be such that $x\leq^{\mathbf{K}_{i}}y$ for some
$i\in I$. Then the following holds:

  1. (i)

If $x\lessdot y$ in $\mathbf{K}$ then $x\lessdot^{\mathbf{K}_{i}}y$ in
$\mathbf{K}_{i}$.

  2. (ii)

If $y\neq x^{\prime}$ then $x\lessdot y$ in $\mathbf{K}$ if and only if
$x\lessdot^{\mathbf{K}_{i}}y$ in $\mathbf{K}_{i}$.

###### Proof.

  1. (i)

If $x\lessdot y$ in $\mathbf{K}$ and $x<^{\mathbf{K}_{i}}z<^{\mathbf{K}_{i}}y$
then $x<z<y$ would hold also in $\mathbf{K}$, a contradiction.

  2. (ii)

Let us suppose that $x\lessdot^{\mathbf{K}_{i}}y$ but $x<z<y$. Therefore, for
some $j,k\in I$, one has $x<^{\mathbf{K}_{j}}z<^{\mathbf{K}_{k}}y$. We have
$x\in K_{i}\cap K_{j}$, $z\in K_{j}\cap K_{k}$, and $y\in K_{i}\cap K_{k}$.
Observe also that $i\neq j$ and $i\neq k$. Let us distinguish several cases
depending on the nature of $x$ and $y$. If $x=0$, then $y$ is an atom in
$\mathbf{K}_{i}$ and hence also in $\mathbf{K}_{k}$. Similarly, if $y=1$, then
$x$ is a co-atom in $\mathbf{K}_{i}$ and in $\mathbf{K}_{j}$. This means that
$z=x$ or $z=y$, a contradiction. Therefore, let us suppose without loss of
generality that $x\neq 0$ and $y\neq 0$. Hence $x$ is an atom in both
$\mathbf{K}_{i}$ and $\mathbf{K}_{j}$ and $y$ is a co-atom in both
$\mathbf{K}_{i}$ and $\mathbf{K}_{k}$. Now, if $j\neq k$, then $z$ must be
either an atom in $\mathbf{K}_{k}$ and hence in $\mathbf{K}_{j}$, and in this
case $z=x$, or a co-atom in $\mathbf{K}_{j}$ and hence in $\mathbf{K}_{k}$,
and so $z=y$, which is impossible. So we must have $j=k$. But this means that
$\mathbf{K}_{i}\cap\mathbf{K}_{j}=\\{0,x,y=x^{\prime},1\\}$, contradicting our
assumptions.

∎

######  Remark 4.7.

Observe that, when one deals with the general case of Kleene lattices, the
assumption of Lemma 4.6(ii) is necessary. Indeed, let us consider the amalgam
of Kleene lattices $\mathbf{K}$

$0$$b$$b^{\prime}$$a$$c$$c^{\prime}$$a^{\prime}$$1$Fig. 5

Note that $a\lessdot a^{\prime}$ in the isomorphic copy of
$\mathbf{K}_{3}\times\mathbf{B}_{2}$ $\\{0,1,a,a^{\prime},b,b^{\prime}\\}$,
but $a<c<a^{\prime}$ in $\mathbf{K}$.

Borrowing the analogous notion for Boolean algebras, given a natural number
$n\geq 3$, we say that initial blocks
$\mathbf{K}_{i_{1}},\dots,\mathbf{K}_{i_{n}}$ of an atomic amalgam
$\mathbf{K}$ form an _atomic loop_ of order $n$ for $\mathbf{K}$ provided that
the following conditions are satisfied:

  1. 1.

For any $1\leq j<n$, $|K_{i_{j}}\cap K_{i_{j+1}}|=4$. Moreover,
$|K_{i_{n}}\cap K_{i_{1}}|=4$.

  2. 2.

For any other pair of indexes $1\leq j,k\leq n$ not mentioned in the above
item, $K_{i_{j}}\cap K_{i_{k}}=\\{0,1\\}$.

  3. 3.

For any $1\leq j<k<l<n$, $K_{i_{j}}\cap K_{i_{k}}\cap K_{i_{l}}=\\{0,1\\}$.

Note that item $(3)$ ensures that ,for any $1\leq j<n$,
$\\{0,1\\}=K_{i_{j}}\cap K_{i_{j+1}}\cap
K_{i_{j+2}}=\\{0,1,a_{j}{\color[rgb]{0,0,1},}a^{\prime}_{j}\\}\cap\\{0,1,a_{j+1}{\color[rgb]{0,0,1},}a^{\prime}_{j+1}\\}$.
Reasoning in a similar way one has that the atoms $a_{1},\dots,a_{n}$ must be
pairwise distinct.

######  Lemma 4.8.

Let $\mathscr{K}=\\{\mathbf{K}_{i}\\}_{i\in I}$ be a pasted family of Kleene
lattices which does not contain an atomic loop of order $3$. If $x,y\in
K_{i}$, then $x\lor^{\mathbf{K}_{i}}y$ is the least upper bound of $x$ and $y$
in $\mathbf{K}$.

###### Proof.

Let us assume without loss of generality that $x\parallel y$. Let $z\in K$ be
such that $x,y\leq z$. If $z=1$ then $x\lor^{\mathbf{K}_{i}}y\leq z$. Suppose
now that $z\neq 1$. Then there exists $j,k\in I$ such that
$x<^{\mathbf{K}_{j}}z$ and $y<^{\mathbf{K}_{k}}z$, $x$ is an atom in both
$\mathbf{K}_{j}$ and $\mathbf{K}_{i}$, $y$ is an atom in both $\mathbf{K}_{k}$
and $\mathbf{K}_{i}$, and $z$ is a co-atom in both $\mathbf{K}_{j}$ and
$\mathbf{K}_{k}$. If $j,i,k$ are pairwise distinct, then
$(\mathbf{K}_{i},\mathbf{K}_{j},\mathbf{K}_{k})$ forms an atomic loop of order
$3$. Therefore, one must have $i=j$ or $j=k$. Therefore, by Lemma 4.5,
$K_{i}\cup K_{j}\cup K_{k}$ forms a paraorthomodular lattice and so we have
again $x\lor^{\mathbf{K}_{i}}y\leq z$. ∎

######  Remark 4.9.

Recall that in any Kleene lattice $\mathbf{K}$ it holds that

$x\land y=0\text{ implies }x\leq y^{\prime}.$

######  Theorem 4.10.

An atomic amalgam of Kleene lattices is a sharply paraorthomodular poset if
and only if it does not contain any loop of order $3$.

###### Proof.

Suppose that the atomic amalgam $\mathbf{K}$ of Kleene lattices does not
contain an atomic loop of order $3$. For any $x,y\in K$, if $x\leq
y^{\prime}$, then there exists $i\in I$ such that $x,y\in K_{i}$. But then
$x\lor y$ exists and equals $x\lor^{\mathbf{K}_{i}}y$, by Lemma 4.8. Moreover,
$\mathbf{K}$ is a paraorthomodular poset by Lemma 4.3 and so it is sharply
paraorthomodular. Conversely, let us assume that $\mathbf{K}$ contains an
atomic loop of order $3$, say
$(\mathbf{K}_{1},\mathbf{K}_{2},\mathbf{K}_{3})$. Assume that $K_{1}\cap
K_{2}=\\{0,1,a_{1},a^{\prime}_{1}\\}$, $K_{2}\cap
K_{3}=\\{0,1,a_{2},a^{\prime}_{2}\\}$, $K_{3}\cap
K_{1}=\\{0,1,a_{3},a^{\prime}_{3}\\}$, where $a_{1},a_{2},a_{3}$ are pairwise
distinct atoms. We show that $a_{1}\lor a_{3}$ does not exist. Let us assume
by way of contradiction that this supremum exists. Since $a_{1}\land
a_{2}=0=a_{2}\land a_{3}$, by Remark 4.9, one must have $a_{1}\lor a_{3}\leq
a^{\prime}_{2}$.
Now, let us suppose that $a_{1}\lor^{\mathbf{K}_{1}}a_{3}$ equals
$a^{\prime}_{1}$ or $a^{\prime}_{3}$. Let us assume without loss of generality
that $a_{1}\lor^{\mathbf{K}_{1}}a_{3}=a^{\prime}_{1}$. Then one has
$a_{1}\leq^{\mathbf{K}_{i}}a_{1}\lor
a_{3}\leq^{\mathbf{K}_{j}}a^{\prime}_{1}$. If $i=1$ or $j=1$, then one must
have $a_{1}\lor a_{3}=a^{\prime}_{1}$ and so $a^{\prime}_{1}\leq
a^{\prime}_{2}$, i.e. $a_{1}=a_{2}$, which is absurd. So we must have $i\neq
1$ and $j\neq 1$. But then $a_{1}\lor a_{3}$ must be either an atom, or a co-
atom, which is equally impossible. Therefore, we can assume without loss of
generality that $a_{1}\lor^{\mathbf{K}_{1}}a_{3}$ is not equal to
$a^{\prime}_{1}$ or $a^{\prime}_{3}$. Note that
$a_{1}\lessdot^{\mathbf{K}_{1}}a_{1}\lor^{\mathbf{K}_{1}}a_{3}$. In fact, if
$a_{1}<x<a_{1}\lor^{\mathbf{K}_{1}}a_{3}$, then the following would be a sub-
lattice of $\mathbf{K}_{1}$:

$0$$a_{1}$$x$$a_{3}$$a_{1}\lor^{\mathbf{K}_{1}}a_{3}$Fig. 6

But this contradicts the distributivity of $\mathbf{K}_{1}$. Similarly,
$a_{3}\lessdot^{\mathbf{K}_{1}}a_{1}\lor^{\mathbf{K}_{1}}a_{3}$. So, by Lemma
4.6, we must have that $a_{1},a_{3}\lessdot a_{1}\lor^{\mathbf{K}_{1}}a_{3}$
in $\mathbf{K}$. Hence, we have $a_{1}\lor^{\mathbf{K}_{1}}a_{3}=a_{1}\lor
a_{3}\leq^{\mathbf{K}_{j}}a^{\prime}_{2}$, for some $j\in I$. If $j=1$, then
$a^{\prime}_{2}\in K_{1}\cap K_{2}\cap K_{3}=\\{0,1\\}$, but this is
impossible since $a^{\prime}_{2}$ is a co-atom. If $i\neq 1$, then
$a_{1}\lor^{\mathbf{K}_{1}}a_{3}\neq a^{\prime}_{2}$ and it must be a co-atom
but this is impossible, since one would have $a^{\prime}_{2}=1$, which is
absurd. ∎

######  Theorem 4.11.

An atomic amalgam of Kleene lattices is a paraorthomodular lattice if and only
if it does not contain an atomic loop of order $3$ or $4$.

###### Proof.

If $\mathbf{K}$ contains a loop of order $3$ then it is not a lattice by
Theorem 4.10. Therefore, we can assume without loss of generality that
$\mathbf{K}$ does not contain loops of order $3$. Suppose that $\mathbf{K}$
contains a loop
$(\mathbf{K}_{1},\mathbf{K}_{2},\mathbf{K}_{3},\mathbf{K}_{4})$ of order $4$.
Let $K_{i}\cap K_{i+1}=\\{0,1,a_{i},a^{\prime}_{i}\\}$ and $K_{4}\cap
K_{1}=\\{0,1,a_{4},a^{\prime}_{4}\\}$, $a_{1},a_{2},a_{3},a_{4}$ atoms. Let us
assume towards a contradiction that $a_{1}\lor a_{3}$ exists. Note that

  1. ($\ast$)

$a_{1}\lor a_{3}\leq a^{\prime}_{2},a^{\prime}_{4}$.

Therefore, there are $i,j\in I$ such that $a_{1}\lor
a_{3}\leq^{\mathbf{K}_{i}}a^{\prime}_{2}$ and $a_{1}\lor
a_{3}\leq^{\mathbf{K}_{j}}a^{\prime}_{4}$. If $i\neq j$, then $a_{1}\lor
a_{3}$ is either an atom, or a co-atom, or $1$. But this is impossible because
of ($\ast$). So we conclude that $i=j$. Now, we have
$a_{1}\leq^{\mathbf{K}_{k}}a_{1}\lor a_{3}$. Therefore, reasoning as above, we
conclude that $k=i$. But then $(\mathbf{K}_{1},\mathbf{K}_{i},\mathbf{K}_{4})$
is a loop of order $3$, which is impossible.
Conversely, suppose that $\mathbf{K}$ does not contain loops of order $3$ or
$4$. Assume towards a contradiction that $\\{x,y\\}\subseteq K$ does not have
a least upper bound _A fortiori_ this means that $x\parallel y$ (and, of
course, that $x,y\notin\\{0,1\\}$). Let us consider the following cases.

  1. 1.

There exists $i\in I$ such that $x,y\in K_{i}$. By hypothesis there is $z\in
K$ such that $x\lor^{\mathbf{K}_{i}}y\not\leq z$. Clearly, $z\neq 1$.
Moreover, we have also $x\leq^{\mathbf{K}_{j}}z$ and
$y\leq^{\mathbf{K}_{k}}z$. If $j=i=k$, then $x\lor^{\mathbf{K}_{i}}y\leq z$
which is impossible. If e.g. $i\neq j$ and $k=i$ then $y$ must be an atom. So
$x\land y=0$ and, by Remark 4.9, $x\leq y^{\prime}$. But then, by Theorem
4.10, $x\lor y$ exists, a contradiction. So one must have $i\neq j$ and $i\neq
k$ and so it is easily seen that
$(\mathbf{K}_{i},\mathbf{K}_{j},\mathbf{K}_{k})$ forms a loop of order $3$,
against our assumptions.

  2. 2.

If $x$ and $y$ are in distinct blocks, then let $x\leq^{\mathbf{K}_{i}}z$ and
$y\leq^{\mathbf{K}_{j}}z$ with $z\neq 1$ and consider $u\in K$ such that
$x\leq^{\mathbf{K}_{k}}u$, $y\leq^{\mathbf{K}_{l}}u$ but $z\not\leq u$ (and so
$u\neq 1$). By hypothesis, one has $i\neq j$. Therefore $z$ must be a co-atom.
Now, if $j\neq l$ and $i\neq k$, since necessarily $k\neq l$, one has that
$(\mathbf{K}_{i},\mathbf{K}_{j},\mathbf{K}_{l},\mathbf{K}_{k})$ is a loop of
order $4$, against our assumptions. Therefore, one must have $j=l$ or $i=k$.
In both cases, it can be seen that one must have $u=z$. Again a contradiction.

∎

##  5 Relatively paraorthomodular posets

In the sequel we introduce posets closely related to paraorthomodular ones.

A relatively paraorthomodular poset is an ordered quintuple
$\big{(}P,\leq,({}^{x};x\in P),0,1\big{)}$ such that $(P,\leq,0,1)$ is a
bounded poset and for every $x\in P$, $([x,1],\leq,{}^{x},x,1)$ is a
paraorthomodular poset. Hence, a bounded poset $(P,\leq,0,1)$ is relatively
paraorthomodular if in each principal filter $[x,1]$ there exists an antitone
involution x such that

  1. (RP)

$x\leq y\leq z$ and $y^{x}\wedge z=x$ imply $y=z$.

######  Example 5.1.

An example of a poset (even a lattice) where every proper interval $[x,1]$ can
be equipped with an antitone involution, but which is not relatively
paraorthomodular, is visualized in Fig. 7.

It is evident how to introduce the involution in every interval $[x,1]$; for example, in $[a,1]$ we have $a^{a}:=1$, $(b^{\prime})^{a}:=b^{\prime}$ and $1^{a}:=a$. However this poset is not paraorthomodular and hence also not relatively paraorthomodular since $a\leq b^{\prime}$ and $a^{\prime}\wedge b=0$, but $a\neq b^{\prime}$. If, however, the involution in $[0,1]$ is defined as in Fig. 2 then this poset is relatively paraorthomodular. |  |
$0$$a$$b$$b^{\prime}$$a^{\prime}$$1=0^{\prime}$Fig. 7non-paraorthomodular
lattice
---|---|---

######  Example 5.2.

Examples of relatively paraorthomodular posets which are not lattices are
visualized in Fig. 1.

It is a question whether also relatively paraorthomodular lattices and posets
can be converted into a certain logic, i.e. whether we can introduce a
connective implication $\rightarrow$ having some expected properties.

For mub-complete relatively paraorthomodular posets we can define

  1. (I3)

$x\rightarrow y:=\big{(}\operatorname{Min}U(x,y)\big{)}^{y}$

where $A^{y}:=\\{a^{y}\mid a\in A\\}$. Since $y\leq U(x,y)$ we have
$\operatorname{Min}U(x,y)\subseteq[y,1]$ and hence $x\rightarrow y$ is well-
defined.

Several interesting properties of the operator $\rightarrow$ defined by (I3)
in a relatively paraorthomodular poset are listed in the following result.

######  Theorem 5.3.

Let $\mathbf{P}=\big{(}P,\leq,({}^{z};z\in P),0,1\big{)}$ be a relatively
paraorthomodular mub-complete poset, $\rightarrow$ defined by (I3) and $x,y\in
L$. Then the following hold:

  1. (i)

$y\leq x\rightarrow y$,

  2. (ii)

$x\leq y$ if and only if $x\rightarrow y=1$,

  3. (iii)

$x\rightarrow y=\left\\{\begin{array}[]{ll}1&\text{ if }x\leq y,\\\ (x\vee
y)^{y}&\text{ if }x\vee y\text{ exists},\\\ x^{y}&\text{ if }y\leq
x,\end{array}\right.$

  4. (iv)

$(x\rightarrow y)\rightarrow y\approx\operatorname{Min}U(x,y)$,

  5. (v)

$\big{(}(x\rightarrow y)\rightarrow y\big{)}\rightarrow y\approx x\rightarrow
y$.

###### Proof.

  1. (i)

follows directly from (I3).

  2. (ii)

The following are equivalent: $x\leq y$, $\operatorname{Min}U(x,y)=y$,
$\big{(}\operatorname{Min}U(x,y)\big{)}^{y}=1$, $x\rightarrow y=1$.

  3. (iii)

$x\rightarrow
y=\big{(}\operatorname{Min}U(x,y)\big{)}^{y}=\left\\{\begin{array}[]{ll}y^{y}=1&\text{
if }x\leq y,\\\ (x\vee y)^{y}&\text{ if }x\vee y\text{ exists},\\\
x^{y}&\text{ if }y\leq x.\end{array}\right.$

  4. (iv)

$\displaystyle(x\rightarrow y)\rightarrow y$
$\displaystyle\approx\bigcup_{z\in x\rightarrow y}(z\rightarrow
y)\approx\bigcup_{z\in\big{(}\operatorname{Min}U(x,y)\big{)}^{y}}\big{(}\operatorname{Min}U(z,y)\big{)}^{y}\approx$
$\displaystyle\approx\bigcup_{w\in\operatorname{Min}U(x,y)}\big{(}\operatorname{Min}U(w^{y},y)\big{)}^{y}\approx\bigcup_{w\in\operatorname{Min}U(x,y)}w^{yy}\approx\operatorname{Min}U(x,y).$

  5. (v)

According to (iv) and (iii) we have

$\displaystyle\big{(}(x\rightarrow y)\rightarrow y\big{)}\rightarrow y$
$\displaystyle\approx\operatorname{Min}U(x,y)\rightarrow
y\approx\bigcup_{z\in\operatorname{Min}U(x,y)}(z\rightarrow
y)\approx\bigcup_{z\in\operatorname{Min}U(x,y)}z^{y}\approx$
$\displaystyle\approx\big{(}\operatorname{Min}U(x,y)\big{)}^{y}\approx
x\rightarrow y.$

∎

Surprisingly, the implication as defined by (I3) can be used for a
characterization of paraorthomodularity, see the next theorem.

######  Theorem 5.4.

Let $(P,\leq,0,1)$ be a bounded mub-complete poset, for all $x\in P$ let
$([x,1],{}^{x},x,1)$ be a bounded poset with an antitone involution and let
$\rightarrow$ be defined by (I3). Then the following are equivalent:

  1. (i)

$(P,\leq,{}^{0},0,1)$ is paraorthomodular.

  2. (ii)

$x\leq y^{0}$ and $x\rightarrow y=y$ together imply $x=y^{0}$.

###### Proof.

The following are equivalent: $x\rightarrow y=y$,
$\big{(}\operatorname{Min}U(x,y)\big{)}^{y}=y$, $\operatorname{Min}U(x,y)=1$,
$U(x,y)=1$, $x\vee y=1$, $x^{0}\wedge y^{0}=0$. Hence (ii) is equivalent to
(ii’):

  1. (ii’)

$x\leq y^{0}$ and $x^{0}\wedge y^{0}=0$ imply $x=y^{0}$.

But this is nothing else than (P) for $(P,\leq,{}^{0},0,1)$. ∎

Our results can be illuminated by the following example.

######  Example 5.5.

Consider the relatively paraorthomodular poset depicted in Fig. 1(a). In
sections define antitone involutions as follows:
$[a,1]$: $(a^{\prime})^{a}:=b^{\prime}$, $(b^{\prime})^{a}:=a^{\prime}$,
$a^{a}:=1$, $1^{a}:=a$,
$[b,1]$: $(a^{\prime})^{b}:=b^{\prime}$, $(b^{\prime})^{b}:=a^{\prime}$,
$b^{b}:=1$, $1^{b}:=b$
and trivially in the two-element intervals $[a^{\prime},1]$ and
$[b^{\prime},1]$. Then the table for the operator $\rightarrow$ defined by
$x\rightarrow y:=\big{(}\operatorname{Min}U(x,y)\big{)}^{y}$ is as follows:

$\begin{array}[]{c|cccccc}\rightarrow&0&a&b&a^{\prime}&b^{\prime}&1\\\
\hline\cr 0&1&1&1&1&1&1\\\ a&a^{\prime}&1&\\{a^{\prime},b^{\prime}\\}&1&1&1\\\
b&b^{\prime}&\\{a^{\prime},b^{\prime}\\}&1&1&1&1\\\
a^{\prime}&a&b^{\prime}&b^{\prime}&1&b^{\prime}&1\\\
b^{\prime}&b&a^{\prime}&a^{\prime}&a^{\prime}&1&1\\\
1&0&a&b&a^{\prime}&b^{\prime}&1\end{array}$

If we assume a certain compatibility condition, we can prove a stronger
result.

######  Theorem 5.6.

Let $(P,\leq,0,1)$ be a bounded mub-complete poset and for every $x\in P$ let
x be an antitone involution on $([x,1],\leq)$ such that the following
compatibility condition holds:

  1. (C)

$x\leq y\leq z$ implies $z^{y}=z^{x}\vee y$.

Moreover, let $\rightarrow$ be defined by (I3). Then the following are
equivalent:

  1. (i)

$\mathbf{P}=\big{(}P,\leq,(^{x};x\in P),0,1\big{)}$ is relatively
paraorthomodular.

  2. (ii)

$x\rightarrow y=1$ implies $x\leq y$.

###### Proof.

(i) $\Rightarrow$ (ii):
This follows from Theorem 5.3.
(ii) $\Rightarrow$ (i):
We want to prove that $([x,1],{}^{x},x,1)$ is paraorthomodular. If $x\leq
y\leq z$ and $y^{x}\wedge z=x$

$z\rightarrow y=\big{(}\operatorname{Min}U(z,y)\big{)}^{y}=z^{y}=z^{x}\vee
y=(z\wedge y^{x})^{x}=x^{x}=1$

according to (C) and hence $z\leq y$ according to (ii) which together with
$y\leq z$ yields $y=z$. ∎

##  6 Relatively paraorthomodular join-semilattices

More interesting results can be obtained if we assume our paraorthomodular
poset to be a join-semilattice. Then definition (I3) of the implication
reduces to the following one:

  1. (I4)

$x\rightarrow y:=(x\vee y)^{y}$

where $(x\vee y)^{y}$ denotes the result of applying the involution y in the
interval $[y,1]$ to $x\vee y$. Since $x\vee y\in[y,1]$, $(x\vee y)^{y}$ is
well-defined.

######  Example 6.1.

Consider the lattice $\mathbf{L}=(L,\leq,{}^{\prime},0,1)$ depicted in Fig. 8:

$0$$a$$b$$c$$d$$d^{\prime}$$c^{\prime}$$b^{\prime}$$a^{\prime}$$1=0^{\prime}$Fig.
8relatively paraorthomodular lattice

Evidently, this poset is paraorthomodular. Consider e.g. the interval $[a,1]$.
Then we define an involution a as follows: $a^{a}:=1$,
$(d^{\prime})^{a}:=d^{\prime}$, $1^{a}:=a$; similarly for the intervals
$[b,1]$, $[c,1]$ and trivially for two-element intervals $[d^{\prime},1]$,
$[c^{\prime},1]$, $[b^{\prime},1]$ and $[a^{\prime},1]$. For $[d,1]$ we define

$d^{d}:=1,(a^{\prime})^{d}:=d^{\prime},(b^{\prime})^{d}:=c^{\prime},(c^{\prime})^{d}:=b^{\prime},(d^{\prime})^{d}:=a^{\prime}\text{
and, of course, }1^{d}:=d.$

One can easily check that every interval of the form $[x,1]$ is a
paraorthomodular poset and hence $\mathbf{L}$ is a relatively paraorthomodular
lattice.

######  Remark 6.2.

Assume that $\mathbf{L}=(L,\vee,\wedge,{}^{\prime},0,1)$ is an orthomodular
lattice and $a,b\in L$ with $a\leq b$. It is well-known (cf. e.g. [18]) that
then $\big{(}[a,b],\vee,\wedge,x\mapsto(x^{\prime}\vee a)\wedge b,a,b\big{)}$
is an orthomodular lattice, too. Hence $\mathbf{L}$ is relatively
paraorthomodular and satisfies (C) since

$x\leq y\leq z\text{ implies }z^{y}=z^{\prime}\vee y=z^{\prime}\vee x\vee
y=z^{x}\vee y$

and (I4) has the form

$x\rightarrow y=(x\vee y)^{y}=\big{(}y\vee(x\vee y)^{\prime}\big{)}\wedge
1=y\vee(x^{\prime}\wedge y^{\prime})$

since $x\vee y\in[y,1]$. We see that this definition of $x\rightarrow y$
coincides with (I2).

######  Lemma 6.3.

Let $\big{(}L,\vee,(^{x};x\in L),0,1\big{)}$ be a relatively paraorthomodular
join-semilattice and $\rightarrow$ be defined by (I4). Then $x\leq y$ implies
$y\rightarrow z\leq x\rightarrow z$.

###### Proof.

$x\leq y$ implies $x\vee z\leq y\vee z$ and hence $y\rightarrow z=(y\vee
z)^{z}\leq(x\vee z)^{z}=x\rightarrow z$. ∎

##  7 Adjointness

Having defined the connective implication $\rightarrow$, e.g., as shown in the
previous sections, the question arises if there exists a binary connective
$\odot$ such that they form a so-called adjoint pair, i.e.

$x\odot y\leq z\text{ if and only if }x\leq y\rightarrow z.$

It is worth noticing that in case that $\odot$ and $\rightarrow$ are binary
operations, any of the two operations $\odot$ and $\rightarrow$ uniquely
determines the other one. Namely, for given $x$ and $y$, the element $x\odot
y$ is the smallest element $z$ satisfying $x\leq y\rightarrow z$, and for
given $y$ and $z$, the element $y\rightarrow z$ is the greatest element $x$
satisfying $x\odot y\leq z$.

Let us note that we do not require $\odot$ to be associative or commutative.
We will show that the requirement that $\odot$ and $\rightarrow$ form an
adjoint pair is not realistic for paraorthomodular lattices or posets and
hence we consider only the following two pairs of weaker conditions (A) and
(B), and (A)21 and (B)12 respectively:

  1. (A)

$x\odot y\leq z$ implies $x\leq y\rightarrow z$,

  2. (B)

$x\leq y\rightarrow z$ implies $x\odot y\leq z$.

  1. (A)21

$x\odot y\leq_{2}z$ implies $x\leq_{1}y\rightarrow z$,

  2. (B)12

$x\leq_{1}y\rightarrow z$ implies $x\odot y\leq_{2}z$.

When we will work with posets instead of lattices then the expressions $x\odot
y$ and $y\rightarrow z$ can be subsets of the poset in question and hence we
sometimes use the relations $\leq_{1}$ and $\leq_{2}$ instead of the partial
order relation $\leq$.

In what follows, we will consider also another $\rightarrow$, the so-called
Sasaki implication.

To simplify our reasonings, we will use for an orthogonal mlb-complete poset
$\mathbf{P}$ or a lattice $\mathbf{L}$ with an involution ′ the following
notation:

$\displaystyle
x\rightarrow_{I}y=y\vee\operatorname{Max}L(x^{\prime},y^{\prime}),x\rightarrow_{S}y=x^{\prime}\vee\operatorname{Max}L(x,y),x\odot_{S}y=y\wedge\operatorname{Min}U(x,y^{\prime}).$

Note that in the case when $\mathbf{L}$ is a lattice with an antitone
involution, $x\odot_{S}y$ is the so-called Sasaki projection and
$x\rightarrow_{S}y$ is the so-called Sasaki implication. Moreover,
$x\rightarrow_{I}y=y^{\prime}\rightarrow_{S}x^{\prime}$.

In fact we have the following lemma.

######  Lemma 7.1.

Let $\mathbf{L}=(L,\vee,\wedge,{}^{\prime})$ be a lattice with an antitone
involution ′ on $\mathbf{L}$. Then (A) and (B) for the connectives $\odot_{S}$
and $\rightarrow_{S}$ are equivalent, respectively.

###### Proof.

Assume that (A) holds. Let $a,b,c\in L$ and assume $a\leq b\rightarrow_{S}c$.
Then $a\leq b^{\prime}\vee(c\wedge b)$. Put $x=c^{\prime}$, $y=b$, and
$z=a^{\prime}$. Hence $z^{\prime}\leq y^{\prime}\vee(x^{\prime}\wedge y)$. We
obtain $x\odot_{S}y=y\wedge(x\vee y^{\prime})\leq z$. By (A) we have $x\leq
y\rightarrow_{S}z=y^{\prime}\vee(z\wedge y)$, i.e., $c^{\prime}\leq
b^{\prime}\vee(a^{\prime}\wedge b)$. We conclude that
$a\odot_{S}b=b\wedge(a\vee b^{\prime})\leq c$ and (B) holds. The reverse
implication follows analogously. ∎

Moreover, for the Sasaki implication $\rightarrow_{S}$, we obtain an adjoint
pair provided the lattice in question satisfies even weaker conditions, see
the following result.

######  Theorem 7.2.

Let $\mathbf{L}=(L,\vee,\wedge,{}^{\prime})$ be a lattice with an involution
′. Then the following conditions are equivalent .

  1. (i)

The orthomodular identities (OI∨) and (OI∧) hold in $\mathbf{L}$.

  2. (ii)

$x\odot_{S}y\leq z$ if and only if $x\leq y\rightarrow_{S}z$.

###### Proof.

(i) $\Rightarrow$ (ii):
$a\odot_{S}b\leq c$ implies

$\displaystyle a$ $\displaystyle\leq a\vee
b^{\prime}\stackrel{{\scriptstyle\mbox{\rm(OI${}_{\vee}$)}}}{{=}}b^{\prime}\vee\big{(}(a\vee
b^{\prime})\wedge b\big{)}=b^{\prime}\vee\big{(}b\wedge(a\vee
b^{\prime})\wedge b\big{)}=b^{\prime}\vee\big{(}(a\odot_{S}b)\wedge
b\big{)}\leq$ $\displaystyle\leq b^{\prime}\vee(c\wedge b)=b\rightarrow_{S}c.$

Similarly, $a\leq b\rightarrow_{S}c$ implies

$\displaystyle a\odot_{S}b$ $\displaystyle=b\wedge(a\vee b^{\prime})\leq
b\wedge\big{(}(b\rightarrow_{S}c)\vee
b^{\prime}\big{)}=b\wedge\big{(}b^{\prime}\vee(c\wedge b)\vee
b^{\prime}\big{)}$ $\displaystyle=b\wedge\big{(}(c\wedge b)\vee
b^{\prime}\big{)}\stackrel{{\scriptstyle\mbox{\rm(OI${}_{\wedge}$)}}}{{=}}c\wedge
b\leq c.$

(ii) $\Rightarrow$ (i):
Let $a,b\in L$. Since $a\odot_{S}b^{\prime}\leq a\odot_{S}b^{\prime}$ we
obtain

$\displaystyle a\leq b^{\prime}\rightarrow(a\odot
b^{\prime})=b\vee\big{(}b^{\prime}\wedge[b^{\prime}\wedge(a\vee
b)]\big{)}=b\vee\big{(}b^{\prime}\wedge(a\vee b)\big{)}.$

Hence $a\vee b\leq b\vee\big{(}b^{\prime}\wedge(a\vee b)\big{)}$. Because the
converse inequality is evident we have

$a\vee b=b\vee\big{(}b^{\prime}\wedge(a\vee b)\big{)}.$

Similarly, since $a\rightarrow_{S}b\leq a\rightarrow_{S}b$ we obtain

$\displaystyle
b\geq(a\rightarrow_{S}b)\odot_{S}a=a\wedge\big{(}a^{\prime}\vee[a^{\prime}\vee(a\wedge
b)]\big{)}=a\wedge\big{(}a^{\prime}\vee(a\wedge b)\big{)}.$

Clearly, $a\wedge\big{(}a^{\prime}\vee(a\wedge b)\big{)}\leq a\wedge b$ and
the converse inequality is evident. We conclude that

$a\wedge\big{(}a^{\prime}\vee(a\wedge b)\big{)}=a\wedge b.$

∎

For the case of lattices, we can state and prove the following result showing
that condition (A) for the connectives $\odot_{S}$ and $\rightarrow_{I}$
already yields orthomodularity.

######  Theorem 7.3.

Let $\mathbf{L}=(L,\vee,\wedge,{}^{\prime},0,1)$ be a bounded lattice with an
antitone involution and assume (A) for $\odot_{S}$ and $\rightarrow_{I}$. Then
$\mathbf{L}$ is orthomodular.

###### Proof.

For $a,b\in L$ every of the following statements implies the next one:

$\displaystyle a\odot_{S}b^{\prime}$ $\displaystyle\leq a\odot_{S}b^{\prime},$
$\displaystyle a$ $\displaystyle\leq
b^{\prime}\rightarrow_{I}(a\odot_{S}b^{\prime}),$ $\displaystyle a$
$\displaystyle\leq\big{(}b^{\prime}\wedge(a\vee
b)\big{)}\vee\Big{(}b\wedge\big{(}b\vee(a^{\prime}\wedge
b^{\prime})\big{)}\Big{)},$ $\displaystyle a$
$\displaystyle\leq\big{(}b^{\prime}\wedge(a\vee b)\big{)}\vee b,$
$\displaystyle a\vee b$ $\displaystyle\leq\big{(}b^{\prime}\wedge(a\vee
b)\big{)}\vee b.$

The converse inequality is evident. Thus $a\vee
b=\big{(}b^{\prime}\wedge(a\vee b)\big{)}\vee b$. ∎

Our next goal is to obtain a result analogous to Theorem 7.2 but formulated
for posets. Here we work with subsets of a given poset $P$ instead of its
elements, thus we should use the equality sign “$\approx_{2}$” instead of
“$=$”.

######  Theorem 7.4.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthogonal mub-complete poset.
Then the following conditions are equivalent:

  1. (i)

$\mathbf{P}$ is an orthomodular poset.

  2. (ii)

$\mathbf{P}$ satisfies the identity

    1. (OMU)

$x\vee\big{(}\operatorname{Min}U(x,y)\wedge
x^{\prime}\big{)}\approx\operatorname{Min}U(x,y)$.

  3. (iii)

$\mathbf{P}$ satisfies the condition

    1. (OMUE)

$x\vee\big{(}\operatorname{Min}U(x,y)\wedge
x^{\prime}\big{)}\approx_{2}\operatorname{Min}U(x,y)$.

###### Proof.

(i) $\Rightarrow$ (ii):
This follows immediately from $x\vee(z\wedge x^{\prime})=z$ for all
$z\in\operatorname{Min}U(x,y)$.
(ii) $\Rightarrow$ (iii):
It is transparent.
(iii) $\Rightarrow$ (i):
Assume that $x\leq y$. Then $\operatorname{Min}U(x,y)=\\{y\\}$ and
$x\vee\big{(}\operatorname{Min}U(x,y)\wedge x^{\prime}\big{)}=\\{x\vee(y\wedge
x^{\prime})\\}$. Hence $x\vee(y\wedge x^{\prime})\leq y\leq x\vee(y\wedge
x^{\prime})$. We conclude that $y=x\vee(y\wedge x^{\prime})$. ∎

Now we are ready to formulate our result for posets. Let us note that the
operation $\odot$ defined for lattices is now replaced by a binary operator.

######  Lemma 7.5.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthogonal mlb-complete poset.
Then (A) and (B), and (A)21 and (B)12 for the connectives $\odot_{S}$ and
$\rightarrow_{S}$ are equivalent, respectively.

###### Proof.

We only show that (A) and (B) are equivalent. The remaining equivalence
follows the same proof steps.

Assume (A). Let $a,b,c\in P$. Suppose that $a\leq b\rightarrow_{S}c$. Then
$a\leq b^{\prime}\vee\operatorname{Max}L(c,b)$. Put $x=c^{\prime}$, $y=b$, and
$z=a^{\prime}$. Hence $z^{\prime}\leq
y^{\prime}\vee\operatorname{Max}L(x^{\prime},y)$. We obtain
$x\odot_{S}y=y\wedge\operatorname{Min}U(x,y^{\prime})\leq z$. By (A) we have
$x\leq y\rightarrow_{S}z=y^{\prime}\vee\operatorname{Max}L(z,y)$, i.e.,
$c^{\prime}\leq b^{\prime}\vee\operatorname{Max}L(a^{\prime},b)$. We conclude
that $a\odot_{S}b=b\wedge\operatorname{Min}U(a,b^{\prime})\leq c$ and (B)
holds.

Following the same reasoning we obtain the converse, i.e., (B) implies (A). ∎

As for lattices in Theorem 7.2, we obtain, for orthogonal mlb-complete posets
the following theorem:

######  Theorem 7.6.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthogonal mlb-complete poset.
Then the following are equivalent:

  1. (i)

$\mathbf{P}$ is orthomodular.

  2. (ii)

$x\odot_{S}y\leq_{2}z$ if and only if $x\leq_{1}y\rightarrow_{S}z$.

###### Proof.

(i) $\Rightarrow$ (ii):
Let $a,b,c\in P$. Suppose that $a\leq_{1}b\rightarrow_{S}c$. Then
$a\leq_{1}b^{\prime}\vee\operatorname{Max}L(c,b)$. There is
$z\in\operatorname{Max}L(c,b)$ such that $a\leq b^{\prime}\vee z\in
U(a,b^{\prime})$. We compute:

$a\odot_{S}b=b\wedge\operatorname{Min}U(a,b^{\prime})\leq_{2}b\wedge(b^{\prime}\vee
z)=z\leq c.$

We conclude that (B)12 and by Lemma 7.5 also (A)21 hold for $\odot_{S}$ and
$\rightarrow_{S}$. Hence we conclude (ii).
(ii) $\Rightarrow$ (i):
Assume that $a\leq b$, $a,b\in P$. Let us compute:

$b\odot_{S}a^{\prime}=a^{\prime}\wedge\operatorname{Min}U(a,b)=\\{a^{\prime}\wedge
b\\}\leq_{2}b.$

Hence by (A)21 we obtain
$b\leq_{1}a^{\prime}\rightarrow_{S}b=\\{a\vee(a^{\prime}\wedge b)\\}\leq b$,
i.e., $b=a\vee(a^{\prime}\wedge b)$. ∎

The next result is analogous to Theorem 7.3 but formulated for posets.

######  Theorem 7.7.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthogonal poset and assume
(A)21 for $\odot_{S}$ and $\rightarrow_{I}$. Then $\mathbf{P}$ is
orthomodular.

###### Proof.

Let $a,b\in L$ and $a\leq b$. Then
$b\odot_{S}a^{\prime}=a^{\prime}\wedge(b\vee a)=a^{\prime}\wedge b$ and
$\\{b\odot_{S}a^{\prime}\\}\leq_{2}\\{b\odot_{S}a^{\prime}\\}$. Therefore
$b\leq_{1}a^{\prime}\rightarrow_{I}(a^{\prime}\wedge b)=(a^{\prime}\wedge
b)\vee\operatorname{Max}L(a,a\vee b^{\prime})=\\{a\vee(a^{\prime}\wedge
b)\\}\leq b$. We conclude that $a\vee(a^{\prime}\wedge b)=b$. ∎

######  Corollary 7.8.

  1. (i)

If $\mathbf{L}=(L,\vee,\wedge,{}^{\prime},0,1)$ is a paraorthomodular lattice
and if $\odot_{S}$ and $\rightarrow_{I}$ satisfy (A) then $\mathbf{L}$ is
orthomodular.

  2. (ii)

If $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ is a sharply paraorthomodular poset
and if $\odot_{S}$ and $\rightarrow_{I}$ satisfy (A)21 then $\mathbf{P}$ is
orthomodular.

######  Proposition 7.9.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthogonal mlb-complete poset
and assume that there exists a binary operator $\odot_{I}$ on $P$ such that
$x\odot_{I}y\leq z$ if and only if $x\leq y\rightarrow_{I}z$ for all $x,y,z\in
P$. Then

  1. (i)

$x\odot_{I}x^{\prime}=\\{0\\}$ for all $x\in P$,

  2. (ii)

$\mathbf{P}$ is an orthomodular poset,

  3. (iii)

$x\odot_{I}y\leq_{1}\operatorname{Max}L(x,y)\leq x,y$ for all $x,y\in P$,

  4. (iv)

$\mathbf{P}$ is weakly Boolean.

###### Proof.

  1. (i)

Let $x\in P$. Then
$x\leq\\{x\\}=0\vee\operatorname{Max}L(x,1)=x^{\prime}\rightarrow_{I}0$. Hence
also $x\odot_{I}x^{\prime}\leq\\{0\\}$, i.e., $x\odot_{I}x^{\prime}=\\{0\\}$.

  2. (ii)

Let $x,y\in P$, $x\leq y$. From (i) we know that
$y\odot_{I}y^{\prime}=\\{0\\}\leq x$. Therefore $y\leq
y^{\prime}\rightarrow_{I}x=x\vee\operatorname{Max}L(y,x^{\prime})=\\{x\vee(y\wedge
x^{\prime})\\}$. Hence $y\leq x\vee(y\wedge x^{\prime})\leq y$, i.e.,
$y=x\vee(y\wedge x^{\prime})$.

  3. (iii)

Let $x,y\in P$. By (ii), Theorem 3.2, (i) and (iii), and the fact that every
orthomodular poset is complemented we know that $x\leq y\rightarrow_{I}x$ and
$x\leq y\rightarrow_{I}y=\\{y\vee y^{\prime}\\}=\\{1\\}$, respectively. Hence
$x\odot_{I}y\leq x$ and $x\odot_{I}y\leq y$. We conclude
$x\odot_{I}y\leq_{1}\operatorname{Max}L(x,y)$.

  4. (iv)

Let $x,y\in P$ such that $x\wedge y=0$ and $x\wedge y^{\prime}=0$. From (iii)
we conclude $x\odot_{I}y=\\{0\\}$ and $x\odot_{I}y^{\prime}=\\{0\\}$. By
adjointness and Theorem 3.2 (iii) we have $x\leq
y\rightarrow_{I}0=\\{y^{\prime}\\}$ and $x\leq
y^{\prime}\rightarrow_{I}0=\\{y\\}$. Since $\mathbf{P}$ is complemented we
obtain $x\leq y\wedge y^{\prime}=0$.

∎

######  Proposition 7.10.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthogonal Boolean poset with
maximality property. Then $\mathbf{P}$ is a Boolean algebra.

###### Proof.

Recall that $\mathbf{P}$ is orthomodular. Namely, let $x,y\in P$, $x\leq y$.
Since $\mathbf{P}$ is a distributive poset we can compute:

$\displaystyle LU\big{(}x,L(x^{\prime},y)\big{)}$
$\displaystyle=L\big{(}U(x,x^{\prime}),U(x,y)\big{)}=L\big{(}\\{1\\},U(y)\big{)}=LU(y).$

Let us show that $\mathbf{P}$ is a weakly Boolean poset. Suppose $x,y\in P$
such that $x\wedge y=0$ and $x\wedge y^{\prime}=0$. Again by distributivity we
obtain

$\displaystyle L(x)=L\big{(}x,U(y,y^{\prime})\big{)}$
$\displaystyle=LU\big{(}L(x,y),L(x,y^{\prime})\big{)}=LU\big{(}\\{0\\},\\{0\\}\big{)}=L(0).$

From [22, Theorem 4.2] we conclude that $\mathbf{P}$ is a Boolean algebra. ∎

Finally, we are going to show that $\odot_{I}$ and $\rightarrow_{I}$ can form
an adjoint pair in orthogonal mlb-complete posets only in a very specific
case, i.e., if the poset in question is a Boolean algebra.

######  Theorem 7.11.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthogonal mlb-complete poset.
Then the following conditions are equivalent:

  1. (i)

There exists a binary operator $\odot_{I}$ on $\mathbf{P}$ such that

$x\odot_{I}y\leq z$ if and only if $x\leq y\rightarrow_{I}z$

for all $x,y,z\in P$.

  2. (ii)

$\mathbf{P}$ is a Boolean algebra.

###### Proof.

(i) $\Rightarrow$ (ii):
From Proposition 7.9 we know that $\mathbf{P}$ is a weakly Boolean poset which
has the maximality property. From [22, Theorem 4.2] we obtain that
$\mathbf{P}$ is a Boolean algebra.
(ii) $\Rightarrow$ (i):
Clearly, $x\rightarrow_{I}y=y\vee(x^{\prime}\wedge y^{\prime})=(y\vee
x^{\prime})\wedge(y\vee y^{\prime})=y\vee x^{\prime}=(x^{\prime}\vee
x)\wedge(y\vee x^{\prime})=x^{\prime}\vee(x\wedge y)=x\rightarrow_{S}y$ and
$x\odot_{S}y=y\wedge(x\vee y^{\prime})=(y\wedge x)\vee(y\wedge
y^{\prime})=x\wedge y$.

Since $\mathbf{P}$ is an orthomodular lattice from Theorem 7.2 we immediately
obtain that

$x\odot_{S}y\leq z$ if and only if $x\leq y\rightarrow_{S}z$ if and only if
$x\leq y\rightarrow_{I}z$

for all $x,y,z\in P$. Hence $\odot_{I}$ exists, and
$\odot_{I}=\odot_{S}=\wedge$ and $x\rightarrow_{I}y=x\rightarrow_{S}y=y\vee
x^{\prime}$ realize the required adjointness. ∎

######  Corollary 7.12.

Let $\mathbf{L}=(L,\vee,\wedge,{}^{\prime},0,1)$ be a bounded lattice with an
antitone involution. Then the following conditions are equivalent:

  1. (i)

There exists a binary operation $\odot_{I}$ such that $x\odot_{I}y\leq z$ if
and only if $x\leq y\rightarrow_{I}z$ for all $x,y,z\in L$.

  2. (ii)

$\mathbf{L}$ is a Boolean algebra.

### Acknowledgements

Ivan Chajda and Helmut Länger gratefully acknowledge the Austrian Science Fund
(FWF), project I 4579-N, and the Czech Science Foundation (GAČR), project
20-09869L, entitled “The many facets of orthomodularity”. Antonio Ledda
gratefully acknowledges the support of MIUR within the project PRIN 2017:
“Logic and cognition. Theory, experiments, and applications”, CUP: 2013YP4N3.

## References

  * [1] S. Abramsky and A. Jung, Domain theory. In Handbook of Logic in Computer Science, Vol. III, Oxford Univ. Press, 1994, 1–168.
  * [2] L. Beran, Orthomodular Lattices. Algebraic Approach. Reidel, Dordrecht 1985. ISBN 90-277-1715-X.
  * [3] G. Birkhoff and J. von Neumann, The logic of quantum mechanics. Annals of Mathematics, 37 (4), (1936), 823–843.
  * [4] I. Chajda and D. Fazio, On residuation in paraorthomodular lattices. Soft Computing 24 (2020), 10295–10304.
  * [5] I. Chajda, D. Fazio, H. Länger, A. Ledda and J. Paseka, Algebraic properties of paraorthomodular posets. Logic J. IGPL 30 (2022), 840–869.
  * [6] I. Chajda and H. Länger, How to introduce the connective implication in orthomodular posets. Asian-Eur. J. Math. 14 (2021), 2150066-1 – 2150066-8.
  * [7] I. Chajda and H. Länger, Residuation in finite posets. Math. Slovaca 71 (2021), 807–820.
  * [8] I. Chajda and H. Länger, The logic of orthomodular posets of finite height. Log. J. IGPL 30 (2022), 143–154.
  * [9] F. Chovanec and M. Jurečková, MV-algebra pasting. _International Journal of Theoretical Physics_. 42 (2003), 1913–1926.
  * [10] J. Czelakowski, Logics based on partial Boolean $\sigma$-algebras. I. Studia Logica, 33 (4), (1974), 371–396.
  * [11] J. Czelakowski, Logics based on partial Boolean $\sigma$-algebras. II. Studia Logica, 34 (1), (1975), 69–86.
  * [12] J. Czelakowski, Partial Boolean algebras in a broader sense. Studia Logica, 38 (1), (1979), 1–16.
  * [13] D. Fazio, A. Ledda and F. Paoli, On Finch’s conditions for the completion of orthomodular posets. Found. Sci. (2020).
  * [14] P. D. Finch, On orthomodular posets. J. Austral. Math. Soc. 11 (1970), 57–62.
  * [15] R. Giuntini, A. Ledda and F. Paoli, A new view of effects in a Hilbert space. Studia Logica 104 (2016), 1145–1177.
  * [16] R. Giuntini, A. Ledda and F. Paoli, On some properties of PBZ*-lattices. Internat. J. Theoret. Phys. 56 (2017), 3895–3911.
  * [17] R. J. Greeehie, Orthomodular lattices admitting no states, Journ. Comb. Theory 10 (1971), 119–132.
  * [18] G. Kalmbach, Orthomodular Lattices, Academic Press, London 1983. ISBN 0-12-394580-1.
  * [19] S. Kochen, and E.P. Specker, Logical structures arising in quantum theory, in J.W. Addison, L. Henkin and A. Tarski (eds.), _The Theory of Models_ , in Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam 1965, 177–189.
  * [20] S, Pulmannová, Compatibility and partial compatibility in quantum logics, Annales de l’I. H. P., section A, 34 (1981), 391–403.
  * [21] P. Suppes, Logics appropriate to empirical theories, in _Symposium on the Theory of Models_ , Proceedings of the 1963 International Symposium at Berkeley, North-Holland, Amsterdam, 1965, 364–375
  * [22] J. Tkadlec, Conditions that force an orthomodular poset to be a Boolean algebra. Tatra Mountains Mathematical Publications 10 (1997), 55–62.
  * [23] Y. Xie, Y. Li and A. Yang, The pasting constructions of lattice ordered effect algebras, _Information Sciences,_ 180(12) (2010), 2476–2486.

Authors’ addresses:

Ivan Chajda
Palacký University Olomouc
Faculty of Science
Department of Algebra and Geometry
17\. listopadu 12
771 46 Olomouc
Czech Republic
<EMAIL_ADDRESS>
Davide Fazio
Università degli Studi di Teramo
Facoltà di Science della Comunicazione
Campus “Aurelio Saliceti”
Via R. Balzarini 1
64100 Teramo
Italy
<EMAIL_ADDRESS>
Helmut Länger
TU Wien
Faculty of Mathematics and Geoinformation
Institute of Discrete Mathematics and Geometry
Wiedner Hauptstraße 8-10
1040 Vienna
Austria, and
Palacký University Olomouc
Faculty of Science
Department of Algebra and Geometry
17\. listopadu 12
771 46 Olomouc
Czech Republic
<EMAIL_ADDRESS>
Antonio Ledda
University of Cagliari
A. Lo. P. Hi. S Research Group
Via Is Mirrionis 1
09123 Cagliari
Italy
<EMAIL_ADDRESS>
Jan Paseka
Masaryk University
Faculty of Science
Department of Mathematics and Statistics
Kotlářská 2
611 37 Brno
Czech Republic
<EMAIL_ADDRESS>


# Music Era Recognition Using Supervised Contrastive Learning and Artist
Information

###### Abstract

Does popular music from the 60s sound different than that of the 90s? Prior
study has shown that there would exist some variations of patterns and
regularities related to instrumentation changes and growing loudness across
multi-decadal trends. This indicates that perceiving the era of a song from
musical features such as audio and artist information is possible. Music era
information can be an important feature for playlist generation and
recommendation. However, the release year of a song can be inaccessible in
many circumstances. This paper addresses a novel task of music era
recognition. We formulate the task as a music classification problem and
propose solutions based on supervised contrastive learning. An audio-based
model is developed to predict the era from audio. For the case where the
artist information is available, we extend the audio-based model to take
multimodal inputs and develop a framework, called MultiModal Contrastive (MMC)
learning, to enhance the training. Experimental result on Million Song Dataset
demonstrates that the audio-based model achieves 54% in accuracy with a
tolerance of 3-years range; incorporating the artist information with the MMC
framework for training leads to 9% improvement further.

##  1 Introduction

As a central goal of the popular music industry over the past 60 years, music
is created to be attractive to listeners [1, 2]. It is believed that music
must incorporate variations of patterns and regularities that build upon
people’s expectations and memories across years [2, 3]. Although scientific
evidence still remains less specific, prior study has shown that these
variations could be related to instrumentation changes and growing loudness
levels [4]. Timbral and mood variations were also found across multi-decadal
trends in [5]. Inspired by these prior findings, developing a predictive model
to recognize the music era from audio can be a potentially feasible task.

In music categorization, people typically break down music era by decades
(e.g., 80s, 90s, and 2000s), which are commonly used as tags in major music
streaming services such as Spotify and Pandora. A song’s release year can
carry a meaningful context of culture, mood, a time in life, or a peek into
history, offering a straightforward way to organize songs for use such as
playlist generation and recommendation [6, 7]. Although the music era can be
inferred via the release year, estimating the era for a song from audio is
practically useful in various scenarios. As the Internet has become
ubiquitous, users’ content sharing and derivative work have grown rapidly on
media platforms such as TikTok and YouTube. These activities may involve music
reuse or edits (e.g., excerpt and remix) which can cause loss of the original
metadata. Moreover, the cover version of an old tune recorded recently would
also yield confusion.

In this paper, we introduce the music era recognition task, which can be
formulated as an application-specific music classification problem that aims
to classify songs into different _year ranges_ (e.g., decades), depending on
the granularity. A major challenge is to differentiate songs from nearby year
ranges, because the variation between two songs can be less notable if they
were released within a range of years. In addition, we also observe an
imbalanced distribution of songs across years in data. Therefore, our design
principle requires the model to learn representations near each other for two
song inputs if both are in the same year range, and otherwise far apart. For
this purpose, we adopt the supervised contrastive learning framework [8],
which has shown state-of-the-art performance in image classification. To
incorporate artist information, we introduce the MultiModal Contrastive (MMC)
learning framework, which includes a novel architecture for multimodal inputs
and an unsupervised contrastive loss for learning a robust combination of the
audio and artist embeddings. The MMC loss can force clustering the embeddings
of the songs from the same artist [9].

To our knowledge, this work represents the first attempt to develop automatic
methods for music era recognition. From a technical perspective, our work is
related to music auto-tagging. Early approaches to music tagging include
methods using the context information of music to predict the user preference
tags [10]. Recent advances in deep learning accelerated the development of
content-based music tagging technology [11]. Numerous systems based on
convolutional neural networks (CNN) were proposed [12, 13, 14]. However,
little attention was paid to music era recognition, possibly because the song
release year is typically accessible through metadata in common use cases such
as streaming. Nevertheless, as aforementioned, era information can be missing
in many circumstances.

As the sounds of an era could be better defined by the popular songs of the
era, we consider this work is also related to the task of _hit song
prediction_ [15], where the goal is to predict if a song would be
successful/popular within a period of (future) time based on its musical
features. Two types of features were explored: _internal_ features extracted
from audio [16, 17] and _external_ ecosystem-related features such as social
media and market data [5]. It is also found that adding an artist factor
(i.e., if the artist had succeeded in the near past) can significantly improve
the prediction accuracy [5].

To summarize our technical contributions, we propose three variants of
methods, including: (1) _Audio-CNN_ : a CNN model that predicts music era from
audio (Section 2.1); (2) _Audio-SUC_ : an enhanced model that incorporates
supervised contrastive learning to improve era recognition from audio (Section
2.2); (3) _AudioArt-MMC_ : an augmented model of _Audio-SUC_ that joints the
audio and artist embeddings, and is trained with an additional MMC loss
(Section 2.4). We evaluate the proposed methods on two datasets (Section 3).
Our results show that _Audio-SUC_ significantly outperforms _Audio-CNN_ , and
incorporating the artist’s information with the MMC loss (_AudioArt-MMC_)
further improves the performance.

##  2 Proposed Method

###  2.1 Convolutional Neural Network (CNN)

Due to insufficient research on music era recognition, we develop _Audio-CNN_
, a modified CNN proposed in [18] as our baseline model, which contains a
stack of CNN layers as a filter to extract local features, followed by a
linear layer as to output classification logits. The feature map of $l$-th CNN
layer is computed as follows:

$\displaystyle{\bf H}_{\ell}=\text{ELU}(\text{BN}(\text{CNN}({\bf
H}_{\ell-1}))),$ (1)

where BN is the batch normalization operation, and ELU is the exponential
linear unit. Each convolution is operated based on a kernel size of $3\times
3$, followed by an average pooling layer. The model is trained with cross-
entropy loss:

$\displaystyle\mathcal{L}_{\mathrm{MLE}}=-\frac{1}{N}\sum_{i=1}^{N}y_{i}\cdot\log(f(\operatorname{\mathbf{x}}_{i})),$
(2)

where $N$ is the number of songs, $\operatorname{\mathbf{x}}_{i}$ is the mel-
spectrogram extracted from the audio of song $i$, and $y_{i}$ is the
corresponding one-hot encoded label out of $C$ possible era classes.

(a) without EC loss.

(b) with EC loss.

(c) without MMC loss.

(d) with MMC loss.

Fig. 1:  Illustrating the effects of contrastive learning. From (a) to (b), EC
loss help aggregating songs based on the era classes. From (c) to (d), MMC
loss improves clustering songs based on artists.

###  2.2 Supervised Contrastive (SUC) Learning

Despite the large scale CNNs can be a powerful model, we consider its
architecture and training scheme using cross-entropy loss are still
insufficient to tackle the discrimination of nearby eras. Therefore, we
develop a novel architecture, called _Audio-SUC_ , to incorporate the
contrastive learning [8, 19, 20, 21, 22], which aims to learn robust
representations by discriminating positive and negative embedding pairs based
on the era labels.

Specifically, we follow [8] to design the _era contrastive_ (EC) loss, which
forces clusters of audio embedding belonging to the same era class to be
pulled together in the embedding space, and meanwhile pushes apart the audio
embeddings from different era classes. To this end, an learnable projection
head is defined as $g_{{\bm{\theta}}}:\\{{\bm{h}}_{a}\\}\rightarrow{\bm{z}}$,
where ${\bm{h}}_{a}$ is the encoded embedding from mel-spectrogram
$\operatorname{\mathbf{x}}$, and ${\bm{z}}$ is the latent representation.

For training, we first organize the training songs by their era labels. Let
$I$ be the training set, ${\bm{z}}_{i}$ be the latent representation for song
$i$, and $P(i)$ be the set with the same era label as song $i$’s, while $N(i)$
be the set with different labels from song $i$’s. The learning process is
illustrated in Fig. 1(a) and Fig. 1(b). Then, the EC loss can be defined as
follows:

$\displaystyle\mathcal{L}_{EC}=-\sum_{i\in I}\sum_{j\in
P(i)}\text{log}\frac{\sigma({\bm{z}}_{i}\cdot{\bm{z}}_{j}/\tau)}{\sum_{k\in
N(i)}\sigma({\bm{z}}_{i},{\bm{z}}_{k}/\tau)},$ (3)

where $\sigma(\cdot)$ is the non-linear transform which adopts an exponential
function, and $\tau$ is the temperature parameter. Note that
$\mathcal{L}_{EC}$ brings two beneficial properties [8]: $(\textup{\it i})$
ability to align all available positive samples; $(\textup{\it ii})$ ability
to mine hard positive and negative pairs. These properties can lead to a more
robust representation of learning in our supervised scenario.

###  2.3 Multi-Modal Contrastive (MMC) Learning

Fig. 2 presents the era distribution of an In-House dataset across years,
indicating a high imbalance between pre- and post-2000. In this ection, we
propose the Multi-Modal Contrastive Learning to tackle this issue.

####  2.3.1 Multi-Modal Fusion Module

Fig. 2: Era distribution of our In-House dataset. Upper sub-figure covers the
years of from 1960 to 2020. Lower sub-figure enlarges the years of pre-1990.

Multi-modal approaches have increasingly attracted researchers’ attention
owing to their promising results on various deep learning tasks [23, 24]. To
learn a better music representation, text about the music, such as lyrics, can
be helpful augmented information and has been widely used in MIR tasks [25,
26, 27]. However, given a song, its lyrics are not always available for
reasons such as copyright issues. Alternatively, the text of the artist’s
biography contains rich information about the music style during the artist’s
active years [28]. We choose the artist’s biography considering its
availability as opposed to lyrics.

We follow the previous work [29] to make full use of contextual and
correspondence information between music and text via attention mechanism, as
illustrated in Fig. 3(a). Given an input song $i$, we use two embedding
vectors, $\operatorname{\mathbf{a}}_{i}\in\mathbb{R}^{d_{h}}$ and
$\operatorname{\mathbf{t}}_{i}\in\mathbb{R}^{d_{h}}$, to represent its audio
and artist biography information, respectively, and $d_{h}=64$ is adopted in
the experiments. Specifically, $\operatorname{\mathbf{a}}_{i}$ is derived from
the mel-spectrogram $\operatorname{\mathbf{x}}_{i}$ using the audio encoder.
For $\operatorname{\mathbf{t}}_{i}$, we utilize Sentences-BERT [30] to encode
the corresponding artist biography text. Then, we concatenate
$\operatorname{\mathbf{a}}_{i}$ and $\operatorname{\mathbf{t}}_{i}$ to obtain
${\bm{s}}_{i}\in\mathbb{R}^{2d_{h}}$ as the input of the self-attention layer.

The fusion module is based on Transformer blocks [31]. Specifically, the
output of $l$-th Transformer block $\mathbf{\tilde{H}}_{\ell}$ is the
concatenation of multiple attention heads:
$\mathbf{\tilde{H}}_{\ell}=[\mathbf{A}_{{\ell},\text{1}},\cdots,\mathbf{A}_{\ell,h}]$
where $h$ is the number of heads. Each head $\bf A$ contains trainable
parameters
$\mathbf{W^{Q}}_{\ell},\mathbf{W}^{K}_{\ell},\mathbf{W}^{V}_{\ell}\in\mathbb{R}^{2d_{h}\times
d_{k}}$, where $d_{k}$ is the projection dimension of the attention layer.

(a) _Audio-Artist Encoder_ (b) Contrastive Learning Framework

Fig. 3: The overall illustration of _AudioArt-MMC_. (a) is the _Audio-Artist
Encoder_ mainly contains audio encoder, artist biography text encoder and a
multi-modal fusion module. (b) shows the encoded latent representation
${\bm{z}}$ that goes through corresponding projection heads for MMC loss and
EC loss, respectively.

####  2.3.2 Multi-Modal Contrastive (MMC) Loss

The EC loss $\mathcal{L}_{EC}$ conditioning on the era labels directly helps
improve the era classification task. However, it may excessively blur the
differences of songs in the multi-modal embedding space within the same era
class. Therefore, to avoid the representation collapse, we develop the multi-
modal contrastive (MMC) loss, an unsupervised contrastive learning objective.
The critical part for the effectiveness of MMC learning is a reasonable design
of different views of the input. For example, in computer vision, data
augmentation techniques such as cropping and rotation are applied to one
anchor image and the other to create positive and negative views of the anchor
image. Similarly, we propose text-shuffle, a technique to create multiple
views for the combination of audio and artist embeddings input. To be
specific, we consider the concatenation of matched audio and artist embeddings
${\bm{s}}_{i,i}=[\operatorname{\mathbf{a}}_{i},\operatorname{\mathbf{t}}_{i}]$
as a reference, so the concatenation of mismatched audio and artist creates
negative views, e.g.,
${\bm{s}}_{i,k}=[\operatorname{\mathbf{a}}_{i},\operatorname{\mathbf{t}}_{k}]$,
where $\operatorname{\mathbf{t}}_{k}$ is a randomly sampled biography
embedding from a different artist. To apply the MMC loss, two components are
designed as follows:

  * •

_Audio-Artist Encoder_ : $f_{E}:\\{{\bm{s}}\\}\rightarrow{\bm{h}}_{m}$, where
${\bm{h}}_{m}$ is the multi-modal embedding learned from the audio-artist
concatenation embedding ${\bm{s}}$.

  * •

_Projection Head_ :
$f_{T}:\\{\operatorname{\mathbf{t}}\\}\rightarrow{\bm{h}}_{m}$, which projects
the artist biography $\operatorname{\mathbf{t}}$ into the multi-modal
embedding space.

For simplicity, we take the artist biography as the anchor. Let an artist
embedding $\operatorname{\mathbf{t}}_{i}$ be the anchor, its positive and
negative examples in the audio-artist concatenation embedding space are
${\bm{s}}_{i,i}$ and ${\bm{s}}_{i,k}$, respectively. The MMC loss is then
defined as:

$\displaystyle\mathcal{L}_{MMC}=-\sum_{c\in\mathrm{C}}\sum_{i\in
c(i)}\text{log}\frac{\sigma\left(f_{T}(\operatorname{\mathbf{t}}_{i})\cdot
f_{E}({\bm{s}}_{i,i})/\tau\right)}{\sum_{k\in
c(i)}\sigma\left(f_{T}(\operatorname{\mathbf{t}}_{i})\cdot
f_{E}({\bm{s}}_{i,k})/\tau\right)},$ (4)

where $\sigma(\cdot)$ is the exponential function, $\mathrm{C}$ is the all
possible era labels, $c$ is the era label of song $i$, and $c(i)$ is the set
having the era label $c$. The learning process is illustrated in Fig. 1(c) and
Fig. 1(d). As illustrated in Fig. 3(b), we obtain the _AudioArt-MMC_ model,
which is trained with a sum of the cross-entropy (MLE) loss, era contrastive
(EC) loss, and multi-modal contrastive (MMC) loss:

$\displaystyle\mathcal{L}=\mathcal{L}_{\mathrm{MLE}}+\alpha\mathcal{L}_{MMC}+\beta\mathcal{L}_{EC},$
(5)

where $\alpha$ and $\beta$ are hyperparameters weighting the loss terms. Note
that, _AudioArt-MMC_ uses the multi-modal embedding for $g_{\theta}$ to
calculate $\mathcal{L}_{EC}$ (see Eq. 3), while _Audio-SUC_ uses the audio
embedding encoded from mel-spectrogram for $\mathcal{L}_{EC}$.

Fig. 4: The t-SNE visualization of the latent representation of different
methods. The active years of an artist is marked in the parenthesis.

##  3 Experiments

###  3.1 Experiment Setup

Datasets & Metrics. To evaluate our proposed methods, we used a public
dataset, Million Song Dataset (MSD) [32], and an internal music collection
(In-House). MSD, which has been widely used in music auto-tagging evaluations
[11], contains the metadata and pre-computed audio features of one million
contemporary songs. Our In-House dataset includes about 800K songs, where the
era distribution is presented in Fig. 2. For artist biography, we adopted the
AllMusic dataset [33], which covers the biography text data of about 200K
artists. We created the audio-text input pairs by matching the artist names.
We selected 85,475 tracks from MSD for the experiments, considering the
availability of audio features, release years, and artist biographies.
Whereas, the full set of In-House was used, since it covers all the required
information.

The difference in instrumentation and genre of songs released in close years
can be subtle. Therefore, we follow similar works [34, 35] to propose a metric
that allows false prediction tolerance. We define the accuracy with $\pm x$
years of tolerance (ACCx) as follows:

$\text{ACC}_{x}=\frac{1}{N}\sum_{i=1}^{N}\frac{x-\text{min}[\text{abs}(y_{i}-\hat{y_{i}}),x]}{x},$
(6)

where $y_{i}$ and $\hat{y_{i}}$ represent the ground truth and predicted years
of song $i$, and $N$ is the total number of test songs. For instance, ACC0
counts only the cases when the prediction exactly matches the ground truth. We
define two evaluation scenarios of granularity for year ranges: $(\textup{\it
i})$ _one year per class_ : each year represents a class. For example, songs
released in 1981 are assigned with a different class than songs released in
1982. $(\textup{\it ii})$ ten years per class: each class represents a decade
(e.g., 60s, 80s, and 00s). Therefore, songs released in 1981 and 1982 are
within the ‘1980s’ class, while the song in 2011 is labeled with ‘2010s’
class.

Training Details. To extract the input mel-spectrograms, we first resample the
original audio waveform to 22,050 Hz. Then, we use a window size of 2,048 with
a hop size of 512 for a frame, and transform each frame into a 224-band mel-
scaled magnitude spectrum. In the training stage, we set a batch size of 64.
Each audio example, which includes 1,024 frames (roughly 6-seconds long), is
randomly sampled from an arbitrary song in the training data. We use the Adam
optimizer [36] with a learning rate of 1e-4.

| MSD | In-House
---|---|---
Method | ACC0 | ACC1 | ACC2 | ACC3 | ACC0 | ACC1 | ACC2 | ACC3
64 classes (one year per class)
_AudioArt-MMC_ | .238 | .451 | .588 | .628 | .188 | .350 | .476 | .650
_Audio-SUC_ | .213 | .313 | .426 | .544 | .140 | .196 | .436 | .579
_Audio-CNN_ | .138 | .169 | .271 | .302 | .097 | .157 | .334 | .486
8 classes (ten years per class)
_AudioArt-MMC_ | .788 | .975 | .988 | - | .750 | .978 | .986 | -
_Audio-SUC_ | .675 | .917 | .958 | - | .563 | .920 | .973 | -
_Audio-CNN_ | .519 | .902 | .964 | - | .433 | .865 | .954 | -
Table 1: Results of the one year/class and ten year/class are reported. Best
results of each dataset and metric are marked bold.

###  3.2 Results and Discussion

We performed 10-fold cross-validation to obtain the results, and this process
was done on MSD and In-House independently. We split 10% of the training set
as the validation set, and selected the snapshot with best performance on
validation set as the testing model.

Era Prediction Comparison. Table 1 presents the overall comparison of the two
datasets. It is clear that incorporating the supervised contrastive learning
(_Audio-SUC_) can significantly outperform the baseline model (_Audio-CNN_)
for most of the cases (except the ACC2 in the 8 classes scenario on MSD), and
that jointing the audio and artist information for input with the multi-modal
contrastive learning (i.e., _AudioArt-MMC_) can consistently further the
performance. Such a result is in line with our expectations when designing the
models. As suggested in prior work [5], artist information is very helpful in
hit song prediction of a period of time, so it also offers strong clues about
the musical sounds representing an era. Comparing among ACC values with
different tolerances for the 64 classes scenario, the superiority of the
contrastive learning-based models over CNN seems to become more apparent when
the tolerance is getting larger. This indicates that the learned features
indeed possess abilities for era discrimination. Lastly, for the 8 classes
scenario (i.e., decade tagging), ACC1 can achieve 90%+ accuracy solely based
on audio, demonstrating promising results for a usable system in relevant MIR
applications.

Robustness to Data Imbalance. As discussed, our dataset is imbalanced in era
distribution. We are interested in the performance of the pre-2000 era, so we
present the result of the corresponding subset in In-House. Table 2 shows the
results on the pre-2000 subset and the difference $\Delta$ from their
counterparts in Table 1. Note that smaller $\Delta$ is better. Comparing the
$\Delta$ values among the three variants, _AudioArt-MMC_ shows significantly
smaller performance gaps between the full-set and subset results, indicating
its stronger ability to handle the imbalanced data.

Visualization of Embeddings. Finally, we perform a qualitative examination of
the effect of the proposed MMC loss by visualizing the embeddings of the
penultimate layer from the three proposed models using t-SNE plots. We
randomly select 400 songs associated with 13 artists and plot the
corresponding song embeddings in the 2D space. Each artist is assigned a
unique color, with brighter colors for recenter eras. As displayed in Fig. 4,
EC loss can facilitate the embeddings of same era class to be closer while
maintaining the diversity of different songs (see Fig. 4(b)), and MMC loss
further forces aggregating songs of the same artist (see Fig. 4(c)).

Method | ACC0 | ACC1 | ACC2 | ACC3 | $\Delta_{0}$ | $\Delta_{1}$ | $\Delta_{2}$ | $\Delta_{3}$
---|---|---|---|---|---|---|---|---
_AudioArt-MMC_ | .175 | .287 | .393 | .488 | .013 | .063 | .083 | .162
_Audio-SUC_ | .094 | .122 | .237 | .379 | .046 | .074 | .097 | .200
_Audio-CNN_ | .055 | .093 | .162 | .295 | .042 | .064 | .172 | .191
Table 2: Results for the pre-2000 subset of In-House in the 64 classes (one
year/class) scenario. $\Delta_{i}$ corresponds to the ACCi difference between
the pre-2000 subset and the whole dataset.

##  4 Conclusion

In this paper, we have presented a novel multi-modal contrastive learning
framework to recognize music era based on audio and artist information. The
design of the EC and MMC losses enables a satisfactory clustering behaviors,
showing convincing results for the task. For future work, we will try to
improve the performance with more sophisticated solutions. We also plan to
study the possibility of including additional metadata, such as
instrumentation, genre, and mood, into a unified framework.

##  5 Acknowledgement

This work was supported by National Key R&D Program of China (2019YFC1711800)
and NSFC (62171138).

## References

  * [1] P. Ball,  The music instinct: how music works and why we can’t do without it,  Random House, 2010.
  * [2] D. Huron,  Sweet anticipation: Music and the psychology of expectation,  MIT press, 2008.
  * [3] H. Honing,  Musical cognition: a science of listening,  Routledge, 2017.
  * [4] J. Serrà, Á. Corral, M. Boguñá, M. Haro Berois, and J. L. Arcos,  “Measuring the evolution of contemporary western popular music,”  Scientific reports. 2012;(2): 521, 2012.
  * [5] M. Interiano, K. Kazemi, L. Wang, J. Yang, Z. Yu, and N. L. Komarova,  “Musical trends and predictability of success in contemporary songs in and out of the top charts,”  Royal Society open science, vol. 5, no. 5, pp. 171274, 2018.
  * [6] T. Pohle, P. Knees, M. Schedl, E. Pampalk, and G. Widmer,  ““reinventing the wheel”: a novel approach to music player interfaces,”  IEEE Transactions on Multimedia, vol. 9, no. 3, pp. 567–575, 2007\.
  * [7] M. Schedl, D. Hauger, and D. Schnitzer,  “A model for serendipitous music retrieval,”  in Proc. Workshop on Context-awareness in Retrieval and Recommendation, 2012, pp. 10–13.
  * [8] P. Khosla, P. Teterwak, C. Wang, A. Sarna, Y. Tian, P. Isola, A. Maschinot, C. Liu, and D. Krishnan,  “Supervised contrastive learning,”  Proc. NeurIPS, vol. 33, pp. 18661–18673, 2020.
  * [9] T. Chen, S. Kornblith, M. Norouzi, and G. Hinton,  “A simple framework for contrastive learning of visual representations,”  in Proc. ICML, 2020, pp. 1597–1607.
  * [10] K. Hoashi, K. Matsumoto, and N. Inoue,  “Personalization of user profiles for content-based music retrieval based on relevance feedback,”  in Proc. ACM Multimedia, 2003, p. 110–119.
  * [11] J. Nam, K. Choi, J. Lee, S.-Y. Chou, and Y.-H. Yang,  “Deep learning for audio-based music classification and tagging: Teaching computers to distinguish rock from bach,”  IEEE signal processing magazine, vol. 36, no. 1, pp. 41–51, 2018\.
  * [12] K. Choi, G. Fazekas, and M. Sandler,  “Automatic tagging using deep convolutional neural networks,”  arXiv preprint arXiv:1606.00298, 2016.
  * [13] J. Pons, O. Nieto, M. Prockup, E. Schmidt, A. Ehmann, and X. Serra,  “End-to-end learning for music audio tagging at scale,”  arXiv preprint arXiv:1711.02520, 2017.
  * [14] M. Won, S. Chun, O. Nieto, and X. Serrc,  “Data-driven harmonic filters for audio representation learning,”  in Proc. ICASSP, 2020, pp. 536–540.
  * [15] E. Zangerle, M. Vötter, R. Huber, and Y.-H. Yang,  “Hit song prediction: Leveraging low-and high-level audio features.,”  in ISMIR, 2019, pp. 319–326.
  * [16] L.-C. Yang, S.-Y. Chou, J.-Y. Liu, Y.-H. Yang, and Y.-A. Chen,  “Revisiting the problem of audio-based hit song prediction using convolutional neural networks,”  in Proc. ICASSP, 2017, pp. 621–625.
  * [17] L.-C. Yu, Y.-H. Yang, Y.-N. Hung, and Y.-A. Chen,  “Hit song prediction for pop music by siamese cnn with ranking loss,”  arXiv preprint arXiv:1710.10814, 2017.
  * [18] K. Ibrahim, E. Epure, G. Peeters, and G. Richard,  “Should we consider the users in contextual music auto-tagging models?,”  in Proc. ISMIR, 2020.
  * [19] J. Yu, H. Yin, X. Xia, T. Chen, L. Cui, and Q. V. H. Nguyen,  “Are graph augmentations necessary? simple graph contrastive learning for recommendation,”  in Proc. ACM SIGIR, 2022, pp. 1294–1303.
  * [20] Y. Marrakchi, O. Makansi, and T. Brox,  “Fighting class imbalance with contrastive learning,”  in International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, 2021, pp. 466–476.
  * [21] Z. Jiang, T. Chen, B. J. Mortazavi, and Z. Wang,  “Self-damaging contrastive learning,”  in Proc. ICML, 2021, pp. 4927–4939.
  * [22] P. Wang, K. Han, X.-S. Wei, L. Zhang, and L. Wang,  “Contrastive learning based hybrid networks for long-tailed image classification,”  in Proc. CVPR, 2021, pp. 943–952.
  * [23] W. Ding, M. Xu, D. Huang, W. Lin, M. Dong, X. Yu, and H. Li,  “Audio and face video emotion recognition in the wild using deep neural networks and small datasets,”  in Proc. ACM Multimodal Interaction, 2016, pp. 506–513.
  * [24] S. Pini, O. B. Ahmed, M. Cornia, L. Baraldi, R. Cucchiara, and B. Huet,  “Modeling multimodal cues in a deep learning-based framework for emotion recognition in the wild,”  in Proc. ACM Multimodal Interaction, 2017, pp. 536–543.
  * [25] P. Lamere,  “Social tagging and music information retrieval,”  Journal of new music research, vol. 37, no. 2, pp. 101–114, 2008\.
  * [26] S. Oramas, O. Nieto, F. Barbieri, and X. Serra,  “Multi-label music genre classification from audio, text, and images using deep features,”  arXiv preprint arXiv:1707.04916, 2017.
  * [27] S. Oramas, F. Barbieri, O. Nieto Caballero, and X. Serra,  “Multimodal deep learning for music genre classification,”  Trans. ISMIR, vol. 1, no. 1, pp. 4–21, 2018.
  * [28] T. Li and M. Ogihara,  “Music artist style identification by semi-supervised learning from both lyrics and content,”  in Proc. ACM Multimedia, 2004, pp. 364–367.
  * [29] F. Simonetta, S. Ntalampiras, and F. Avanzini,  “Multimodal music information processing and retrieval: Survey and future challenges,”  in International Workshop on Multilayer Music Representation and Processing, 2019, pp. 10–18.
  * [30] N. Reimers and I. Gurevych,  “Sentence-bert: Sentence embeddings using siamese bert-networks,”  arXiv preprint arXiv:1908.10084, 2019.
  * [31] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin,  “Attention is all you need,”  Proc. NeurIPS, vol. 30, 2017.
  * [32] T. Bertin-Mahieux, D. P. Ellis, B. Whitman, and P. Lamere,  “The million song dataset,”  in Proc. ISMIR, 2011.
  * [33] R. Malheiro, R. Panda, P. Gomes, and R. P. Paiva,  “Emotionally-relevant features for classification and regression of music lyrics,”  IEEE Transactions on Affective Computing, vol. 9, no. 2, pp. 240–254, 2016.
  * [34] R. Angulu, J. R. Tapamo, and A. O. Adewumi,  “Age estimation via face images: a survey,”  EURASIP Journal on Image and Video Processing, vol. 2018, no. 1, pp. 1–35, 2018.
  * [35] A. Saraf and E. Khoury,  “Distribution learning for age estimation from speech,”  in Proc. ICASSP, 2022, pp. 8552–8556.
  * [36] D. P. Kingma and J. Ba,  “Adam: A method for stochastic optimization,”  arXiv preprint arXiv:1412.6980, 2014.