00footnotetext: Support of the research by the Austrian Science Fund (FWF),
project I 4579-N, and the Czech Science Foundation (GAČR), project 20-09869L,
entitled “The many facets of orthomodularity”, as well as by ÖAD, project CZ
02/2019, entitled “Function algebras and ordered structures related to logic
and data fusion”, and, concerning the first author, by IGA, project PřF 2021
030, is gratefully acknowledged.

# Algebras describing pseudocomplemented, relatively pseudocomplemented and
sectionally pseudocomplemented posets

Ivan Chajda and Helmut Länger

###### Abstract

In order to be able to use methods of Universal Algebra for investigating
posets, we assign to every pseudocomplemented poset, to every relatively
pseudocomplemented poset and to every sectionally pseudocomplemented poset a
certain algebra (based on a commutative directoid or on a $\lambda$-lattice)
which satisfies certain identities and implications. We show that the assigned
algebras fully characterize the given corresponding posets. It turns out that
the assigned algebras satisfy strong congruence properties which can be
transferred back to the posets. We also mention applications of such posets in
certain non-classical logics.

AMS Subject Classification: 06A11, 06D15, 08A62, 08B05

Keywords: Pseudocomplemented poset, relatively pseudocomplemented poset,
sectionally pseudocomplemented poset, Stone poset, commutative directoid,
$\lambda$-lattice, congruence permutability, congruence distributivity, weak
regularity.

##  1 Introduction

For investigating algebras researchers usually apply well-known algebraic
methods and results. Unfortunately, this does not work in the case of
partially ordered sets (posets, shortly). The reason for this is that posets
need not have operations and hence basic theorems of General Algebra cannot be
applied. In 1990, J. Ježek and R. Quackenbush ([10]) showed that if a poset
$\mathbf{P}$ is up-directed or down-directed then a certain algebra with one
binary operation, a so-called directoid, can be assigned to $\mathbf{P}$. This
assignment is in general not unique, but, conversely, from every such assigned
directoid, $\mathbf{P}$ can be reconstructed in a unique way. This fact allows
to convert directed posets into algebras which bear all the information on the
given poset.

In a similar way to a given poset that is both up- and down-directed one can
assign an algebra with two binary operations $\sqcup$ and $\sqcap$ as shown by
V. Snášel ([14]). Such an algebra is called a $\lambda$-lattice. An overview
concerning results on directoids and $\lambda$-lattices can be found in our
monograph [5].

The aforementioned approach was used also for bounded complemented posets by
the authors and M. Kolařík in [4]. In fact, the class of directoids assigned
to such posets forms a variety of algebras. This is of great advantage since
there exist many methods and results for studying varieties in General
Algebra. The same machinery was used in [3] where so-called orthoposets and
orthomodular posets (used in the formalization of the logic of quantum
mechanics) were converted into algebras forming a variety.

The natural question arises if such a method can be applied also to
pseudocomplemented, relatively pseudocomplemented and sectionally
pseudocomplemented posets (the last were introduced recently by the authors
and J. Paseka in [8]). We solve this question by using commutative directoids
and $\lambda$-lattices. We characterize the assigned algebras by means of
relatively simple conditions. Unfortunately, not all of these conditions can
be expressed in the form of identities or quasi-identities. Thus the
corresponding classes of algebras do not form varieties or quasivarieties. On
the other hand, these algebras still share nice congruence properties as we
will show.

We believe that our approach can bring new insight into the study of
pseudocomplemented, relatively pseudocomplemented and sectionally
pseudocomplemented posets since we provide a purely algebraic description of
them, thus enabling the application of algebraic tools for their
investigation.

##  2 Preliminaries

Let $\mathbf{P}:=(P,\leq)$ be a poset, $A,B\subseteq P$ and $a,b\in P$. Then
$A\leq B$ means that $x\leq y$ for all $x\in A$ and $y\in B$. Instead of
$A\leq\\{b\\}$ and $\\{a\\}\leq B$ we simply write $A\leq b$ and $a\leq B$.
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 and upper cone of $A$, respectively. Instead of
$L(\\{a\\})$ and $L(\\{a,b\\})$ we simply write $L(a)$ and $L(a,b)$ ,
respectively. In a similar way we proceed for $U$. It is easy to see that
$A\subseteq L(b)$ if and only if $A\leq b$. If the infimum $\inf(a,b)$ of $a$
and $b$ exists in $\mathbf{P}$ then we will denote it by $a\wedge b$. The
poset $\mathbf{P}$ is called

  * •

down-directed if $L(x,y)\neq\emptyset$ for all $x,y\in P$,

  * •

up-directed if $U(x,y)\neq\emptyset$ for all $x,y\in P$,

  * •

directed if it is both down- and up-directed.

Of course, if $\mathbf{P}$ has a top element $1$ then it is up-direceted, and
if it has a bottom element $0$ then it is down-directed.

The concept of a commutative directoid was introduced by J. Ježek and R.
Quackenbush ([10]), see also [5] for details and elementary theory. A
commutative meet-directoid is a groupoid $(D,\sqcap)$ satisfying the following
identities:

  * •

$x\sqcap x\approx x$,

  * •

$x\sqcap\big{(}(x\sqcap y)\sqcap z\big{)}\approx(x\sqcap y)\sqcap z$,

  * •

$x\sqcap y\approx y\sqcap x$.

If $\mathbf{D}=(D,\sqcap)$ is a commutative meet-directoid and one defines a
binary relation $\leq$ on $D$ by $x\leq y$ if $x\sqcap y=x$ then
$\mathbb{P}(\mathbf{D}):=(D,\leq)$ is a down-directed poset, called the poset
induced by $\mathbf{D}$.

Dually, one can define a commutative join-directoid $(D,\sqcup)$. If
$\mathbf{D}=(D,\sqcup)$ is a commutative join-directoid and one defines a
binary relation $\leq$ on $D$ by $x\leq y$ if $x\sqcup y=y$ then
$\mathbb{Q}(\mathbf{D}):=(D,\leq)$ is an up-directed poset, called the poset
induced by $\mathbf{D}$.

Assume $\mathbf{P}=(P,\leq)$ to be down-directed and define a binary operation
$\sqcap$ on $P$ by the following prescription: $x\sqcap y:=\min(x,y)$ if $x$
and $y$ are comparable with each other, and $x\sqcap y=y\sqcap x$ should be an
arbitrary element of $L(x,y)$ otherwise. Thus $x\sqcap y\in L(x,y)$ in any
case. Then $\mathbf{D}:=(P,\sqcap)$ is a commutative meet-directoid, called a
commutative meet-directoid assigned to $\mathbf{P}$, and
$\mathbb{P}(\mathbf{D})=\mathbf{P}$. Hence, the down-directed poset
$\mathbf{P}$ is uniquely determined by an assigned commutative meet-directoid
$\mathbf{D}$ in contrast to the fact that $\mathbf{D}$ is in general not
uniquely determined by $\mathbf{P}$.

Dually, we define a commutative join-directoid $\mathbf{D}=(P,\sqcup)$
assigned to an up-directed poset $\mathbf{P}=(P,\leq)$. We then have
$\mathbb{Q}(\mathbf{D})=\mathbf{P}$.

Using an assigned commutative meet- and join-directoid, respectively, we can
describe lower and upper cones of a given poset as follows.

######  Lemma 2.1.

Let $\mathbf{P}=(P,\leq)$ be a poset and $a,b,c\in P$. If $\mathbf{P}$ is
down-directed and $(P,\sqcap)$ an assigned commutative meet-directoid then

  1. (a)

$L(a,b)=\\{(a\sqcap x)\sqcap(b\sqcap x)\mid x\in P\\}$,

  2. (b)

$c\in L(a,b)$ if and only if $(a\sqcap c)\sqcap(b\sqcap c)=c$.

If $\mathbf{P}$ is up-directed and $(P,\sqcup)$ an assigned commutative join-
directoid then

  1. (c)

$U(a,b)=\\{(a\sqcup x)\sqcup(b\sqcup x)\mid x\in P\\}$,

  2. (d)

$c\in U(a,b)$ if and only if $(a\sqcup c)\sqcup(b\sqcup c)=c$.

###### Proof.

First assume $\mathbf{P}$ to be down-directed and $(P,\sqcap)$ to be an
assigned commutative meet-directoid.

  1. (a)

If $c\in L(a,b)$ then $c=c\sqcap c=(a\sqcap c)\sqcap(b\sqcap c)$. Conversely,
$(a\sqcap c)\sqcap(b\sqcap c)\leq a\sqcap c\leq a$ and, analogously, $(a\sqcap
c)\sqcap(b\sqcap c)\leq b$, i.e. $(a\sqcap c)\sqcap(b\sqcap c)\in L(a,b)$.

  2. (b)

If $c\in L(a,b)$ then $(a\sqcap c)\sqcap(b\sqcap c)=c\sqcap c=c$. The converse
direction follows from (i).

The rest of the lemma follows by duality. ∎

##  3 Pseudocomplemented posets

Pseudocomplemented posets were introduced and studied by O. Frink ([9]), see
also [12] and [15] for further development. Let us recall the definition.

######  Definition 3.1.

A pseudocomplemented poset is an ordered quadruple $(P,\leq,{}^{*},0)$ such
that $(P,\leq,0)$ is a poset with bottom element $0$ and ∗ is a unary
operation on $P$ such that for all $x\in P$, $x^{*}$ is the greatest element
$y$ of $P$ satisfying $L(x,y)=\\{0\\}$. The element $x^{*}$ is called the
pseudocomplement of $x$. A Stone poset is a pseudocomplemented poset
$(P,\leq,{}^{*},0)$ satisfying $U(x^{*},x^{**})=\\{0^{*}\\}$ for all $x\in P$.

It is worth noticing that $L(x,y)=\\{0\\}$ means the same as $x\wedge y=0$.
Hence, for every $x\in P$ we have $x\wedge x^{*}=0$. Moreover, every
pseudocomplemented poset $(P,\leq,{}^{*},0)$ has a top element, namely
$0^{*}=1$.

######  Example 3.2.

If $(P,\leq)$ denotes the poset visualized in Figure 1:

$a$$b$$c$$d$$1$$0$Fig. 1

and the unary operation ∗ on $P$ is defined by

$\begin{array}[]{c|cccccc}x&0&a&b&c&d&1\\\ \hline\cr
x^{*}&1&b&a&0&0&0\end{array}$

then $(P,\leq,{}^{*},0)$ is a pseudocomplemented poset which is neither a
lattice nor a Stone poset since

$\begin{array}[]{c|cccccc}x&0&a&b&c&d&1\\\ \hline\cr x^{*}&1&b&a&0&0&0\\\
x^{**}&0&a&b&1&1&1\end{array}$

and $U(a^{*},a^{**})=\\{c,d,1\\}\neq\\{1\\}$.

######  Example 3.3.

If $(P,\leq)$ denotes the poset visualized in Figure 2:

$0$$a$$b$$d$$e$$1$$c$$f$Fig. 2

and the unary operation ∗ on $P$ is defined by

$\begin{array}[]{c|cccccccc}x&0&a&b&c&d&e&f&1\\\ \hline\cr
x^{*}&1&f&c&f&0&0&c&0\end{array}$

then $(P,\leq,{}^{*},0)$ is not a lattice, but a Stone poset since

$\begin{array}[]{c|cccccccc}x&0&a&b&c&d&e&f&1\\\ \hline\cr
x^{*}&1&f&c&f&0&0&c&0\\\ x^{**}&0&c&f&c&1&1&f&1\end{array}$

and $U(x^{*},x^{**})=\\{1\\}$ for all $x\in P$.

Now we show how a pseudocomplemented poset can be characterized by an assigned
commutative meet-directoid equipped with a unary operation ∗ and a nullary
operation $0$.

######  Theorem 3.4.

Let $(P,\leq)$ be a down-directed poset, ∗ a unary operation on $P$, $0\in P$
and $(P,\sqcap)$ a commutative meet-directoid assigned to $(P,\leq)$. Then
$\mathbf{P}=(P,\leq,{}^{*},0)$ is a pseudocomplemented poset if and only if
$\mathbf{A}=(P,\sqcap,{}^{*},0)$ satisfies the following conditions:

  1. (i)

$0\sqcap x\approx 0$,

  2. (ii)

$(x\sqcap y)\sqcap(x^{*}\sqcap y)\approx 0$,

  3. (iii)

$(x\sqcap z)\sqcap(y\sqcap z)=0\,\forall z\in P\Rightarrow y\sqcap x^{*}=y$.

In this case we call $\mathbf{A}$ an algebra assigned to $\mathbf{P}$.

###### Proof.

$\mathbf{P}$ is a pseudocomplemented poset if and only if the following hold:

  1. (i’)

$0\leq x$,

  2. (ii’)

$L(x,x^{*})=\\{0\\}$,

  3. (iii’)

$L(x,y)=\\{0\\}\Rightarrow y\leq x^{*}$.

Now $\mathbf{P}$ satisfies (i’) if and only if $\mathbf{A}$ satisfies (i).
Because of Lemma 2.1 (a), $\mathbf{P}$ satisfies (ii’) if and only if
$\mathbf{A}$ satisfies (ii), and $\mathbf{P}$ satisfies (iii’) if and only if
$\mathbf{A}$ satisfies (iii). ∎

The concept of a $\lambda$-lattice was introduced by V. Snášel ([14]), see
also [5]. A $\lambda$-lattice is an algebra $(L,\sqcup,\sqcap)$ of type
$(2,2)$ satisfying the following identities:

  * •

$x\sqcup y\approx y\sqcup x$, $x\sqcap y\approx y\sqcap x$,

  * •

$x\sqcup\big{(}(x\sqcup y)\sqcup z\big{)}\approx(x\sqcup y)\sqcup z$,
$x\sqcap\big{(}(x\sqcap y)\sqcap z\big{)}\approx(x\sqcap y)\sqcap z$,

  * •

$(x\sqcup y)\sqcap x\approx x$, $(x\sqcap y)\sqcup x\approx x$.

Let $\mathbf{L}=(L,\sqcup,\sqcap)$ be a $\lambda$-lattice. Then $(L,\sqcup)$
and $(L,\sqcap)$ are commutative join- and meet-directoids, respectively, and
$x\sqcap y=x$ if and only if $x\sqcup y=y$. If one defines a binary relation
$\leq$ on $L$ by $x\leq y$ if $x\sqcap y=x$ (or, equivalently, $x\sqcup y=y$)
then $\mathbb{R}(\mathbf{L}):=(L,\leq)$ is a directed poset, called the poset
induced by $\mathbf{L}$.

Let $\mathbf{P}=(P,\leq)$ be a directed poset and define binary operations
$\sqcup$ and $\sqcap$ on $P$ by the following prescription: $x\sqcup
y:=\max(x,y)$ and $x\sqcap y:=\min(x,y)$ if $x$ and $y$ are comparable with
each other, and $x\sqcup y=y\sqcup x$ and $x\sqcap y=y\sqcap x$ should be
arbitrary elements of $U(x,y)$ and $L(x,y)$, respectively, otherwise. Then
$\mathbf{L}:=(P,\sqcup,\sqcap)$ is a $\lambda$-lattice, called a
$\lambda$-lattice assigned to $\mathbf{P}$, and
$\mathbb{R}(\mathbf{L})=\mathbf{P}$. Hence, a given directed poset
$\mathbf{P}$ is uniquely determined by an assigned $\lambda$-lattice
$\mathbf{L}$ contrary to the fact that $\mathbf{L}$ may be assigned to
$\mathbf{P}$ in a non-unique way. It is easy to see that $(P,\sqcup)$ and
$(P,\sqcap)$ are commutative join- and meet-directoids assigned to
$\mathbf{P}$, respectively.

As remarked above, every Stone poset $\mathbf{P}$ has the top element $0^{*}$,
thus it is also up-directed and hence directed. Due to this, we can assign to
$\mathbf{P}$ a $\lambda$-lattice.

######  Theorem 3.5.

Let $(P,\leq)$ be a directed poset, ∗ a unary operation on $P$, $0\in P$ and
$(P,\sqcup,\sqcap)$ a $\lambda$-lattice assigned to $(P,\leq)$. Then
$\mathbf{P}=(P,\leq,{}^{*},0)$ is a Stone poset if and only if
$\mathbf{A}=(P,\sqcup,\sqcap,{}^{*},0)$ satisfies (i) – (iii) of Theorem 3.4
as well as

  1. (iv)

$(x^{*}\sqcup y)\sqcup(x^{**}\sqcup y)\approx 0^{*}$.

In this case we call $\mathbf{A}$ an algebra assigned to $\mathbf{P}$.

###### Proof.

$\mathbf{P}$ is a Stone poset if and only if it is a pseudocomplemented poset
satisfying

  1. (iv’)

$U(x^{*},x^{**})=\\{0^{*}\\}$.

Because of Lemma 2.1 (c), $\mathbf{P}$ satisfies (iv’) if and only if
$\mathbf{A}$ satisfies (iv). The rest follows from Theorem 3.4. ∎

Let us recall that a poset $(P,\leq)$ is called distributive if it satisfies
the equality

$U\big{(}L(x,y),z\big{)}=UL\big{(}U(x,z),U(y,z)\big{)}$ (1)

or, equivalently,

$L\big{(}U(x,y),z\big{)}=LU\big{(}L(x,z),L(y,z)\big{)}$

for all $x,y,z\in P$. We are going to show that distributive
pseudocomplemented posets satisfying $U(x,x^{*})=\\{0^{*}\\}$ can be
characterized by means of equalities only.

######  Theorem 3.6.

Let $(P,\leq,{}^{*},0,1)$ be a bounded distributive poset with a unary
operation ∗ satisfying $U(x,x^{*})=\\{1\\}$ for all $x\in P$. Then
$\mathbf{P}=(P,\leq,{}^{*},0)$ is pseudocomplemented if and only if it
satisfies the following equalities for all $x,y\in P$:

  1. (i)

$L(x,x^{*})=\\{0\\}$,

  2. (ii)

$U\big{(}x^{*},L(x,y)\big{)}=U(x^{*},y)$.

###### Proof.

Let $a,b\in P$. If $\mathbf{P}$ is pseudocomplemented then (i) follows from
Definition 3.1 and (ii) follows from

$U\big{(}a^{*},L(a,b)\big{)}=UL\big{(}U(a^{*},a),U(a^{*},b)\big{)}=UL\big{(}1,U(a^{*},b)\big{)}=ULU(a^{*},b)=U(a^{*},b)$

by using distributivity of $(P,\leq)$ and $U(a,a^{*})=\\{1\\}$. If,
conversely, $\mathbf{P}$ satisfies (i) and (ii) Then $L(a,a^{*})=\\{0\\}$ by
(i) and if $L(a,b)=\\{0\\}$ then

$a^{*}\in U(a^{*})=U(a^{*},0)=U\big{(}a^{*},L(a,b)\big{)}=U(a^{*},b)\subseteq
U(b)$

by (ii) whence $b\leq a^{*}$. ∎

An example of a poset satisfying the assumptions of Theorem 3.6 is the
following.

######  Example 3.7.

If $(P,\leq)$ denotes the poset visualized in Figure 3:

$0$$a$$b$$c$$d$$e$$f$$g$$h$$i$$j$$1$Fig. 3

and the unary operation ∗ on $P$ is defined by

$\begin{array}[]{c|cccccccccccc}x&0&a&b&c&d&e&f&g&h&i&j&1\\\ \hline\cr
x^{*}&1&j&i&h&g&f&e&d&c&b&a&0\end{array}$

then $(P,\leq,{}^{*},0)$ is a distributive Stone poset (which is not a
lattice) satisfying $U(x,x^{*})=\\{1\\}$ for all $x\in P$ since

$\begin{array}[]{c|cccccccccccc}x&0&a&b&c&d&e&f&g&h&i&j&1\\\ \hline\cr
x^{*}&1&j&i&h&g&f&e&d&c&b&a&0\\\ x^{**}&0&a&b&c&d&e&f&g&h&i&j&1\end{array}$

##  4 Relatively pseudocomplemented posets

Relatively pseudocomplemented posets were studied by numerous authors from
different points of view, e.g. as a base of some non-classical logics where
the relative pseudocomplementation is considered as the connective implication
(see e.g. [11]) or as an approach to Hilbert algebras (see e.g. [7] and [13]),
or for purely algebraic reasons, see e.g. [6]. Recall the following
definition.

######  Definition 4.1.

A relatively pseudocomplemented poset is an ordered quadruple $(P,\leq,*,1)$
such that $(P,\leq)$ is a poset and $*$ is a binary operation on $P$ such that
for all $x,y\in P$, $x*y$ is the greatest element $z$ of $P$ satisfying
$L(x,z)\subseteq L(y)$ and $1$ denotes the top element of $(P,\leq)$. (The
existence of such an element follows from the fact that $x*x$ is the greatest
element $y$ of $(P,\leq)$ satisfying $L(x,y)\subseteq L(x)$.) The element
$x*y$ is called the relative pseudocomplement of $x$ with respect to $y$.

It is clear that a relatively pseudocomplemented poset having a bottom element
$0$ is pseudocomplemented since $x^{*}=x*0$. As it was shown by the authors in
[6], every relatively pseudocomplemented poset is distributive.

It is well-known that relatively pseudocomplemented semilattices or lattices
play important roles in the axiomatization of intuitionistic logics. They are
known under the names Brouwerian (semi-)lattices or Heyting algebras, see e.g.
[10]. However, also relatively pseudocomplemented posets can be recognized as
a formalization of certain logics of this sort, where conjunction is unsharp
in the following sense. Directly by Definition 4.1 we have

$L(a,c)\leq b\text{ if and only if }c\leq a*b.$

The binary operation $*$ can be considered as the logical connective of
implication. From above we have

$L(a,a*b)\leq b.$

This can be transferred to the language of propositional calculus as follows:
If we know truth values of $a$ and $a*b$ then the truth value of $b$ cannot be
less than that of $a$ and $a*b$. This expresses an unsharp version of the
derivation rule Modus Ponens.

A typical example of a relatively pseudocomplemented poset is the following.

######  Example 4.2.

If $(P,\leq)$ denotes the poset from Example 3.2 and the binary operation $*$
on $P$ is defined by

$\begin{array}[]{c|cccccc}*&0&a&b&c&d&1\\\ \hline\cr 0&1&1&1&1&1&1\\\
a&b&1&b&1&1&1\\\ b&a&a&1&1&1&1\\\ c&0&a&b&1&d&1\\\ d&0&a&b&c&1&1\\\
1&0&a&b&c&d&1\end{array}$

then $(P,\leq,*,1)$ is a relatively pseudocomplemented poset which is not a
lattice.

Also down-directed relatively pseudocomplemented posets can be characterized
by means of assigned directoids equipped with a binary operation.

######  Theorem 4.3.

Let $(P,\leq)$ be a down-directed poset, $*$ a binary operation on $P$, $1\in
P$ and $(P,\sqcap)$ a commutative meet-directoid corresponding to $(P,\leq)$.
Then $\mathbf{P}=(P,\leq,*,1)$ is a relatively pseudocomplemented poset if and
only if $\mathbf{A}=(P,\sqcap,*,1)$ satisfies the following conditions:

  1. (i)

$x\sqcap 1\approx x$,

  2. (ii)

$\Big{(}(x\sqcap z)\sqcap\big{(}(x*y)\sqcap z\big{)}\Big{)}\sqcap
y\approx(x\sqcap z)\sqcap\big{(}(x*y)\sqcap z\big{)}$,

  3. (iii)

$\big{(}(x\sqcap t)\sqcap(z\sqcap t)\big{)}\sqcap y=(x\sqcap t)\sqcap(z\sqcap
t)\,\forall t\in P\Rightarrow z\sqcap(x*y)=z$.

In this case we call $\mathbf{A}$ an algebra assigned to $\mathbf{P}$.

###### Proof.

$\mathbf{P}$ is a relatively pseudocomplemented poset if and only if the
following hold:

  1. (i’)

$x\leq 1$,

  2. (ii’)

$L(x,x*y)\leq y$,

  3. (iii’)

$L(x,z)\leq y\Rightarrow z\leq x*y$.

Obviously, $\mathbf{P}$ satisfies (i’) if and only if $\mathbf{A}$ satisfies
(i). Because of Lemma 2.1 (a), $\mathbf{P}$ satisfies (ii’) if and only if
$\mathbf{A}$ satisfies (ii), and $\mathbf{P}$ satisfies (iii’) if and only if
$\mathbf{A}$ satisfies (iii). ∎

As shown above, a relatively pseudocomplemented poset $\mathbf{P}$ is
determined by its assigned algebra $\mathbf{A}$ which is a commutative meet-
directoid with constant $1$ and equipped with a binary oparation $*$. Hence,
this algebra shares all the properties of $\mathbf{P}$, but expressed in the
language of $\mathbf{A}$. In the following we show how some properties of
$\mathbf{A}$ can be derived directly from conditions (i) – (iii) of Theorem
4.3.

######  Theorem 4.4.

Let $\mathbf{A}=(P,\sqcap,*,1)$ be an algebra assigned to a relatively
pseudocomplemented poset $(P,\leq,*,1)$. Then it satisfies the following
identities:

  1. (a)

$x*x\approx 1$,

  2. (b)

$1*x\approx x$,

  3. (c)

$x\sqcap\big{(}(x*y)*y\big{)}\approx x$.

###### Proof.

  1. (a)

We have

$(x\sqcap t)\sqcap(1\sqcap t)\leq x\sqcap t\leq x\forall t\in P$

and hence

$\big{(}(x\sqcap t)\sqcap(1\sqcap t)\big{)}\sqcap x=(x\sqcap t)\sqcap(1\sqcap
t)\forall t\in P$

which implies

$x*x=1\sqcap(x*x)=1$

by (i) and by (iii) when replacing $y$ and $z$ by $x$ and $1$, respectively.

  2. (b)

We have

$1*x=1\sqcap(1*x)=(1\sqcap 1)\sqcap\big{(}(1*x)\sqcap 1\big{)}=\Big{(}(1\sqcap
1)\sqcap\big{(}(1*x)\sqcap 1\big{)}\Big{)}\sqcap x\leq x$

by (i) and by (ii) when replacing $x$, $y$ and $z$ by $1$, $x$ and $1$,
respectively. Conversely, we have

$\displaystyle\big{(}(1\sqcap t)\sqcap(x\sqcap t)\big{)}\sqcap x$
$\displaystyle=\big{(}t\sqcap(x\sqcap t)\big{)}\sqcap x=(x\sqcap t)\sqcap
x=x\sqcap t=t\sqcap(x\sqcap t)=$ $\displaystyle=(1\sqcap t)\sqcap(x\sqcap
t)\forall t\in P$

and hence

$x=x\sqcap(1*x)\leq 1*x$

by (iii) when replacing $x$, $y$ and $z$ by $1$, $x$ and $x$, respectively.
Altogether, we obtain $1*x\approx x$.

  3. (c)

We have

$\Big{(}(x\sqcap t)\sqcap\big{(}(x*y)\sqcap t\big{)}\Big{)}\sqcap y=(x\sqcap
t)\sqcap\big{(}(x*y)\sqcap t\big{)}\forall t\in P$

by (ii) when replacing $z$ by $t$ and hence

$\Big{(}\big{(}(x*y)\sqcap t\big{)}\sqcap(x\sqcap t)\Big{)}\sqcap
y=\big{(}(x*y)\sqcap t\big{)}\sqcap(x\sqcap t)\forall t\in P$

which implies

$x\sqcap\big{(}(x*y)*y\big{)}=x$

by (iii) when replacing $x$ and $z$ by $x*y$ and $x$, respectively,

∎

##  5 Sectionally pseudocomplemented posets

Sectionally pseudocomplemented posets were recently introduced by the authors
and J. Paseka in [8].

######  Definition 5.1.

A sectionally pseudocomplemented poset is an ordered triple $(P,\leq,\circ)$
such that $(P,\leq)$ is a poset and $\circ$ is a binary operation on $P$ such
that for all $x,y\in P$, $x\circ y$ is the greatest element $z$ of $P$
satisfying $L(U(x,y),z)=L(y)$. The element $x\circ y$ is called the sectional
pseudocomplement of $x$ with respect to $y$. A sectionally pseudocomplemented
poset with $1$ is an ordered quadruple $(P,\leq,\circ,1)$ such that
$(P,\leq,\circ)$ is a sectionally pseudocomplemented poset and $1$ is the top
element of $(P,\leq)$. A strongly sectionally pseudocomplemented poset (see
[8]) is a sectionally pseudocomplemented poset $(P,\leq,\circ,1)$ with $1$
satisfying $x\leq(x\circ y)\circ y$ for all $x,y\in P$.

If a sectionally pseudocomplemented poset $(P,\leq,\circ)$ has a top element
$1$ then $1=x\circ x$ for all $x\in P$, but not every sectionally
pseudocomplemented poset has a top element, see the following example.

######  Example 5.2.

If $(P,\leq)$ denotes the poset visualized in Figure 4:

$a$$b$$c$$d$Fig. 4

and the binary operation $\circ$ on $P$ is defined by $x\circ y:=y$ for all
$x,y\in P$ then $(P,\leq,\circ)$ is a sectionally pseudocomplemented poset
which is neither a lattice nor relatively pseudocomplemented nor strongly
sectionally pseudocomplemented since it has no top element.

######  Example 5.3.

If $\mathbf{P}=(P,\leq)$ denotes the poset visualized in Figure 5:

$b$$a$$c$$d$$e$$1$$0$Fig. 5

and the binary operation $\circ$ on $P$ is defined by

$\begin{array}[]{c|ccccccc}\circ&0&a&b&c&d&e&1\\\ \hline\cr 0&1&1&1&1&1&1&1\\\
a&c&1&1&c&1&1&1\\\ b&c&a&1&c&1&1&1\\\ c&b&a&b&1&1&1&1\\\ d&0&a&b&c&1&e&1\\\
e&0&a&b&c&d&1&1\\\ 1&0&a&b&c&d&e&1\end{array}$

then $(P,\leq,\circ,1)$ is a strongly sectionally pseudocomplemented poset
which is neither a lattice nor relatively pseudocomplemented since the
relative pseudocomplement of $b$ with respect to $a$ does not exist. Since

$L\big{(}U(a,c),b\big{)}=LU(b)=L(b)\neq
L(a)=LUL(a)=LU\big{(}L(a,b),L(c,b)\big{)},$

$\mathbf{P}$ is not distributive.

An example of a sectionally pseudocomplemented poset (having a top element
$1$) which is not strongly sectionally pseudocomplemented is given in [8].

It is well-known that every relatively pseudocomplemented lattice is
distributive. Sectional pseudocomplementation was introduced by the first
author ([1]) in order to extend the concept of relative pseudocomplementation
to non-distributive lattices. It was shown that sectionally pseudocomplemented
lattices form a variety and, considered as posets, they are strongly
sectionally pseudocomplemented. It is a natural question if also these posets
can be considered as a formalization of certain unsharp propositional logic.
We can consider the binary operation $\circ$ of sectional
pseudocomplementation as the logical connective of implication. Having two
propositions with truth values $a$ and $b$ such that $b\leq a$, the formula

$L(U(a,b),c)=L(b)$

can be rewritten as

$a\wedge c=b.$

Hence, by Definition 5.1, for $b\leq a$,

$a\wedge c=b\text{ if and only if }c\leq a\circ b.$

Therefore, for $b\leq a$ we have a form of adjointness of operations $\wedge$
and $\circ$. Moreover, we have

$a\wedge(a\circ b)=b.$

Thus the truth value of $b$ is the same as truth value of conjunction
$a\wedge(a\circ b)$ provided $b\leq a$. This is again a version of Modus
Ponens restricted to elements $a,b$ with $b\leq a$.

Also directed sectionally pseudocomplemented posets can be characterized by
assigned $\lambda$-lattices with a binary operation as follows.

######  Theorem 5.4.

Let $(P,\leq)$ be a directed poset, $\circ$ a binary operation on $P$ and
$(P,\sqcup,\sqcap)$ a $\lambda$-lattice corresponding to $(P,\leq)$. Then the
following hold:

  1. (a)

$\mathbf{P}=(P,\leq,\circ)$ is a sectionally pseudocomplemented poset if and
only if $\mathbf{A}=(P,\sqcup,\sqcap,\circ)$ satisfies the following
conditions:

    1. (i)

$y\sqcap(x\circ y)\approx y$,

    2. (ii)

$\Big{(}\big{(}(x\sqcup t)\sqcup(y\sqcup t)\big{)}\sqcap
z\Big{)}\sqcap\big{(}(x\circ y)\sqcap z\big{)}=z\,\forall t\in P\Rightarrow
z\sqcap y=z$,

    3. (iii)

$\bigg{(}\Big{(}\big{(}(x\sqcup t)\sqcup(y\sqcup t)\big{)}\sqcap
s\Big{)}\sqcap(z\sqcap s)=s\,\forall t\in P\Leftrightarrow s\sqcap
y=s\bigg{)}\,\forall s\in P\Rightarrow z\sqcap(x\circ y)=z$.

  2. (b)

If $1\in P$ then $\mathbf{P}=(P,\leq,\circ,1)$ is a sectionally
pseudocomplemented poset with $1$ if and only if
$\mathbf{A}=(P,\sqcup,\sqcap,\circ,1)$ satisfies (i) – (iii) as well as

    1. (iv)

$x\sqcap 1\approx x$.

In both cases we call $\mathbf{A}$ an algebra assigned to $\mathbf{P}$.

###### Proof.

  1. (a)

$\mathbf{P}$ is a sectionally pseudocomplemented poset if and only if the
following hold:

    1. ($\alpha$)

$L\big{(}U(x,y),x\circ y\big{)}=L(y)$,

    2. (iii’)

$L\big{(}U(x,y),z\big{)}=L(y)\Rightarrow z\leq x\circ y$.

Now $\mathbf{P}$ satisfies ($\alpha$) and (iii’) if and only if it satisfies
(i’) – (iii’) where (i’) and (ii’) denote the following conditions:

    1. (i’)

$y\leq x\circ y$,

    2. (ii’)

$L\big{(}U(x,y),x\circ y\big{)}\leq y$.

This can be seen as follows: If $\mathbf{P}$ satisfies ($\alpha$) and (iii’)
then it satisfies (ii’) and because of $L\big{(}U(x,y),y\big{)}=L(y)$ and
(iii’) it satisfies (i’). If, conversely, $\mathbf{P}$ satisfies (i’) – (iii’)
then because of

$L(y)=L\big{(}U(x,y),y\big{)}\subseteq L\big{(}U(x,y),x\circ y\big{)}\subseteq
L(y)$

it satisfies ($\alpha$). This shows that $\mathbf{P}$ is a sectionally
pseudocomplemented poset if and only it satisfies (i’) – (iii’). Now
$\mathbf{P}$ satisfies (i’) if and only if $\mathbf{A}$ satisfies (i). Because
of Lemma 2.1 (c) and (b), $\mathbf{P}$ satisfies (ii’) if and only if
$\mathbf{A}$ satisfies (ii), and $\mathbf{P}$ satisfies (iii’) if and only if
$\mathbf{A}$ satisfies (iii).

  2. (b)

$\mathbf{P}$ is a sectionally pseudocomplemented poset with $1$ if and only if
$(P,\leq,\circ)$ is a sectionally pseudocomplemented poset satisfying

    1. (iv’)

$x\leq 1$.

Now $\mathbf{P}$ satisfies (iv’) if and only if $\mathbf{A}$ satisfies (iv).
The rest follows from (a).

∎

In the next theorem we show how some properties of algebras assigned to
sectionally pseudocomplemented posets with $1$ can be derived from conditions
(i) – (iv) of Theorem 5.4. As we remarked above, in a $\lambda$-lattice
$(A,\sqcup,\sqcap)$ the identities $x\sqcap 1\approx x$ and $x\sqcup 1\approx
1$ are equivalent.

######  Theorem 5.5.

Let $\mathbf{A}=(P,\sqcup,\sqcap,\circ,1)$ be an algebra assigned to a
sectionally pseudocomplemented poset $(P,\leq,\circ,1)$ with $1$. Then it
satisfies the following identities:

  1. (a)

$x\circ x\approx 1$,

  2. (b)

$1\circ x\approx x$.

###### Proof.

  1. (a)

According to (iv) we have

$\Big{(}\big{(}(x\sqcup t)\sqcup(x\sqcup t)\big{)}\sqcap
s\Big{)}\sqcap(1\sqcap s)=\big{(}(x\sqcup t)\sqcap s\big{)}\sqcap s=(x\sqcup
t)\sqcap s$

and hence the following are equivalent:

$\displaystyle\Big{(}\big{(}(x\sqcup t)\sqcup(x\sqcup t)\big{)}\sqcap
s\Big{)}\sqcap(1\sqcap s)$ $\displaystyle=s\forall t\in P,$
$\displaystyle(x\sqcup t)\sqcap s$ $\displaystyle=s\forall t\in P,$
$\displaystyle s$ $\displaystyle\leq x\sqcup t\forall t\in P,$ $\displaystyle
s$ $\displaystyle\leq x,$ $\displaystyle s\sqcap x$ $\displaystyle=s$

which implies

$x\circ x=1\sqcap(x\circ x)=1$

by (iv) and by (iii) when replacing $y$ and $z$ by $x$ and $1$, respectively.

  2. (b)

According to (iv) we have

$\displaystyle\Big{(}\big{(}(1\sqcup t)\sqcup(x\sqcup t)\big{)}\sqcap(1\circ
x)\Big{)}\sqcap\big{(}(1\circ x)\sqcap(1\circ x)\big{)}=$
$\displaystyle=\Big{(}\big{(}1\sqcup(x\sqcup t)\big{)}\sqcap(1\circ
x)\Big{)}\sqcap(1\circ x)=\big{(}1\sqcap(1\circ x)\big{)}\sqcap(1\circ
x)=(1\circ x)\sqcap(1\circ x)=$ $\displaystyle=1\circ x\forall t\in P$

and hence

$(1\circ x)\sqcap x=1\circ x$ (2)

by (ii) when replacing $x$, $y$ and $z$ by $1$, $x$ and $1\circ x$,
respectively. Moreover, according to (iv) we have

$\displaystyle\Big{(}\big{(}(1\sqcup t)\sqcup(x\sqcup t)\big{)}\sqcap
s\Big{)}\sqcap(x\sqcap s)$ $\displaystyle=\Big{(}\big{(}1\sqcup(x\sqcup
t)\big{)}\sqcap s\Big{)}\sqcap(x\sqcap s)=$ $\displaystyle=(1\sqcap
s)\sqcap(x\sqcap s)=s\sqcap(x\sqcap s)=x\sqcap s=s\sqcap x$

and hence the following are equivalent:

$\displaystyle\Big{(}\big{(}(1\sqcup t)\sqcup(x\sqcup t)\big{)}\sqcap
s\Big{)}\sqcap(x\sqcap s)$ $\displaystyle=s\forall t\in P,$ $\displaystyle
s\sqcap x$ $\displaystyle=s.$

This implies

$x\sqcap(1\circ x)=x$ (3)

by (iii) when replacing $x$, $y$ and $z$ by $1$, $x$ and $x$, respectively.
From (2) and (3) we obtain (b).

∎

Analogously to the characterizations in Theorem 5.4 we can characterize also
strongly sectionally pseudocomplemented posets.

######  Theorem 5.6.

Let $(P,\leq)$ be a directed poset, $\circ$ a binary operation on $P$, $1\in
P$ and $(P,\sqcup,\sqcap)$ a $\lambda$-lattice corresponding to $(P,\leq)$.
Then $\mathbf{P}=(P,\leq,\circ,1)$ is a strongly sectionally
pseudocomplemented poset if and only if $\mathbf{A}=(P,\sqcup,\sqcap,\circ,1)$
satisfies (i) – (iv) of Theorem 5.4 as well as

  1. (v)

$x\sqcap\big{(}(x\circ y)\circ y\big{)}\approx x$.

In this case we call $\mathbf{A}$ an algebra assigned to $\mathbf{P}$.

###### Proof.

$\mathbf{P}$ is a strongly sectionally pseudocomplemented poset if and only if
$(P,\leq,\circ)$ is a sectionally pseudocomplemented poset with $1$ satisfying

  1. (v’)

$x\leq(x\circ y)\circ y$.

Now $\mathbf{P}$ satisfies (v’) if and only if $\mathbf{A}$ satisfies (v). The
rest follows from Theorem 5.4. ∎

##  6 Congruence properties

Next we consider congruence properties of algebras. For the convenience of the
reader we recall the corresponding definitions.

An algebra $\mathbf{A}$ is called

  * •

congruence permutable if $\Theta\circ\Phi=\Phi\circ\Theta$ for all
$\Theta,\Phi\in\operatorname{Con}\mathbf{A}$,

  * •

congruence distributive if the congruence lattice of $\mathbf{A}$ is
distributive,

  * •

arithmetical if it is both congruence permutable and congruence distributive,

  * •

weakly regular (with respect to an equationally definable constant $1$) if
$\Theta,\Phi\in\operatorname{Con}\mathbf{A}$ and $[1]\Theta=[1]\Phi$ imply
$\Theta=\Phi$.

The following results are well-known (see e.g. [2]) :

Let $\mathbf{A}$ be an algebra.

  * •

If there exists a ternary term $p$ satisfying the identities $p(x,x,y)\approx
p(y,x,x)\approx y$ then $\mathbf{A}$ is congruence permutable.

  * •

If there exists a ternary term $m$ satisfying the identities $m(x,x,y)\approx
m(x,y,x)\approx m(y,x,x)\approx x$ then $\mathbf{A}$ is congruence
distributive.

  * •

If $\mathbf{A}$ has an equationally definable constant $1$ and there exists a
positive integer $n$, binary terms $t_{1},\ldots,t_{n}$ and quaternary terms
$s_{1},\ldots,s_{n}$ satisfying the identities

$\displaystyle t_{1}(x,x)\approx\cdots\approx t_{n}(x,x)\approx 1,$
$\displaystyle s_{1}\big{(}t_{1}(x,y),1,x,y\big{)}\approx x,$ $\displaystyle
s_{i}\big{(}1,t_{i}(x,y),x,y\big{)}\approx
s_{i+1}\big{(}t_{i+1}(x,y),1,x,y\big{)}\text{ for }i=1,\ldots,n-1,$
$\displaystyle s_{n}\big{(}1,t_{n}(x,y),x,y\big{)}\approx y$

then $\mathbf{A}$ is weakly regular.

The terms $p$ and $m$ are called Maltsev term and majority term, respectively.

Now we can prove

######  Theorem 6.1.

  1. (i)

Let $(P,\leq,{}^{*},0)$ be a Stone poset and
$\mathbf{A}=(P,\sqcup,\sqcap,{}^{*},0)$ be an assigned algebra. Then
$\mathbf{A}$ is congruence distributive.

  2. (ii)

Let $(P,\leq,*,1)$ be a down-directed relatively pseudocomplemented poset and
$\mathbf{A}=(P,\sqcap,*,1)$ an assigned algebra. Then $\mathbf{A}$ is
congruence permutable and weakly regular.

  3. (iii)

Let $(P,\leq,\circ)$ be a directed sectionally pseudocomplemented poset and
$\mathbf{A}=(P,\sqcup,\sqcap,\circ)$ an assigned algebra. Then $\mathbf{A}$ is
congruence distributive.

  4. (iv)

Let $(P,\leq,\circ,1)$ be a down-directed sectionally pseudocomplemented poset
with $1$ and $\mathbf{A}=(P,\sqcup,\sqcap,\circ,1)$ an assigned algebra. Then
$\mathbf{A}$ is congruence distributive and weakly regular.

  5. (v)

Let $(P,\leq,\circ,1)$ be a down-directed strongly sectionally
pseudocomplemented poset and $\mathbf{A}=(P,\sqcup,\sqcap,\circ,1)$ an
assigned algebra. Then $\mathbf{A}$ is arithmetical and weakly regular.

###### Proof.

  1. (i)

If

$m(x,y,z):=\big{(}(x\sqcup y)\sqcap(y\sqcup z)\big{)}\sqcap(z\sqcup x)$

then

$\displaystyle m(x,x,y)$ $\displaystyle\approx\big{(}(x\sqcup x)\sqcap(x\sqcup
y)\big{)}\sqcap(y\sqcup x)\approx\big{(}x\sqcap(x\sqcup
y)\big{)}\sqcap(y\sqcup x)\approx x\sqcap(y\sqcup x)\approx$
$\displaystyle\approx x,$ $\displaystyle m(x,y,x)$
$\displaystyle\approx\big{(}(x\sqcup y)\sqcap(y\sqcup x)\big{)}\sqcap(x\sqcup
x)\approx(x\sqcup y)\sqcap x\approx x,$ $\displaystyle m(y,x,x)$
$\displaystyle\approx\big{(}(y\sqcup x)\sqcap(x\sqcup x)\big{)}\sqcap(x\sqcup
y)\approx\big{(}(y\sqcup x)\sqcap x\big{)}\sqcap(x\sqcup y)\approx
x\sqcap(x\sqcup y)\approx$ $\displaystyle\approx x.$

  2. (ii)

If

$\displaystyle p(x,y,z)$
$\displaystyle:=\big{(}(x*y)*z\big{)}\sqcap\big{(}(z*y)*x\big{)},$
$\displaystyle t_{1}(x,y)$ $\displaystyle:=x*y,$ $\displaystyle t_{2}(x,y)$
$\displaystyle:=y*x,$ $\displaystyle s_{1}(x,y,z,u)$
$\displaystyle:=(x*u)\sqcap z,$ $\displaystyle s_{2}(x,y,z,u)$
$\displaystyle:=(y*z)\sqcap u$

then by Theorem 4.4 we obtain

$\displaystyle p(x,x,y)$
$\displaystyle\approx\big{(}(x*x)*y\big{)}\sqcap\big{(}(y*x)*x\big{)}\approx(1*y)\sqcap\big{(}(y*x)*x\big{)}\approx$
$\displaystyle\approx y\sqcap\big{(}(y*x)*x\big{)}\approx y,$ $\displaystyle
p(y,x,x)$
$\displaystyle\approx\big{(}(y*x)*x\big{)}\sqcap\big{(}(x*x)*y\big{)}\approx\big{(}(y*x)*x\big{)}\sqcap(1*y)\approx$
$\displaystyle\approx\big{(}(y*x)*x\big{)}\sqcap y\approx y,$ $\displaystyle
t_{1}(x,x)$ $\displaystyle\approx x*x\approx 1,$ $\displaystyle t_{2}(x,x)$
$\displaystyle\approx x*x\approx 1,$ $\displaystyle
s_{1}\big{(}t_{1}(x,y),1,x,y\big{)}$
$\displaystyle\approx\big{(}(x*y)*y\big{)}\sqcap x\approx x,$ $\displaystyle
s_{1}\big{(}1,t_{1}(x,y),x,y\big{)}$ $\displaystyle\approx(1*y)\sqcap x\approx
y\sqcap x\approx x\sqcap y\approx(1*x)\sqcap y\approx
s_{2}\big{(}t_{2}(x,y),1,x,y\big{)},$ $\displaystyle
s_{2}\big{(}1,t_{2}(x,y),x,y\big{)}$
$\displaystyle\approx\big{(}(y*x)*x\big{)}\sqcap y\approx y.$

  3. (iii)

This follows like in (i).

  4. (iv)

This follows like in (i) and (ii) using Theorem 5.5 instead of Theorem 4.4.

  5. (v)

This follows like in (i) and (ii) using Theorem 5.5 instead of Theorem 4.4.

∎

##  7 Conclusions

The advantage of our approach of introducing algebras $\mathbf{A}$ closely
related to given posets $\mathbf{P}$ is that we can introduce congruences in
$\mathbf{P}$ by means of $\mathbf{A}$. Namely, we may introduce congruences on
$\mathbf{P}$ as congruences on a fixed algebra $\mathbf{A}$ assigned to
$\mathbf{P}$. Of course, the assignment of $\mathbf{A}$ to $\mathbf{P}$ is in
general not unique as mentioned above and hence for different algebras
assigned to $\mathbf{P}$ we could obtain different congruences on
$\mathbf{P}$. However, regardless which algebra $\mathbf{A}$ is assigned to
$\mathbf{P}$, the lattice of congruences on $\mathbf{P}$ will be distributive
if $\mathbf{A}$ is a $\lambda$-lattice and congruences on $\mathbf{P}$ will
permute and will be fully determined by their $1$-class if $\mathbf{P}$ is
down-directed and either relatively pseudocomplemented or strongly sectionally
pseudocomplemented. This may shed new light on the concept of congruences on
posets.

Another application of our approach may be the following. Given a directed
relatively or sectionally pseudocomplemented poset $\mathbf{P}$, we can ask if
there are posets $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ of the same kind as
$\mathbf{P}$ such that $\mathbf{P}_{1}\times\mathbf{P}_{2}\cong\mathbf{P}$.
For posets, it is not easy to decide if such a decomposition is possible.
However, we can assign to $\mathbf{P}$ an algebra $\mathbf{A}$ as constructed
above. Now we have a simple criterion for deciding if there exist algebras
$\mathbf{A}_{1}$ and $\mathbf{A}_{2}$ with
$\mathbf{A}_{1}\times\mathbf{A}_{2}\cong\mathbf{A}$. Namely, such a
decomposition is possible if and only if there exist
$\Theta,\Phi\in\operatorname{Con}\mathbf{A}$ such that

$\displaystyle\Theta\vee\Phi$ $\displaystyle=\nabla_{A},$
$\displaystyle\Theta\cap\Phi$ $\displaystyle=\Delta_{A},$
$\displaystyle\Theta\circ\Phi$ $\displaystyle=\Phi\circ\Theta.$

Hence, if such congruences exist, we have
$\mathbf{A}\cong\mathbf{A}_{1}\times\mathbf{A}_{2}$ with
$\mathbf{A}_{1}=\mathbf{A}/\Theta$ and $\mathbf{A}_{2}=\mathbf{A}/\Phi$. Now
we can conversely assign posets $\mathbf{P}_{i}$ to $\mathbf{A}_{i}$ for
$i=1,2$ and we can write $\mathbf{P}_{1}\times\mathbf{P}_{2}\cong\mathbf{P}$.

Moreover, if $\mathbf{P}=\mathbf{P}_{1}\times\mathbf{P}_{2}$ then we can ask
if a given congruence $\Theta$ on $\mathbf{P}$ is directly decomposable, i.e.
if there are $\Theta_{i}\in\operatorname{Con}\mathbf{P}_{i}$ for $i=1,2$ such
that $\Theta_{1}\times\Theta_{2}=\Theta$. If for assigned algebras we have
$\mathbf{A}=\mathbf{A}_{1}\times\mathbf{A}_{2}$ then, provided $\mathbf{A}$ is
congruence distributive, there exist
$\Theta_{i}\in\operatorname{Con}\mathbf{A}_{i}$ for $i=1,2$ such that
$\Theta_{1}\times\Theta_{2}=\Theta$ and hence $\mathbf{P}$ has directly
decomposable congruences.

## References

  * [1] I. Chajda, An extension of relative pseudocomplementation to non-distributive lattices. Acta Sci. Math. (Szeged) 69 (2003), 491–496.
  * [2] I. Chajda, G. Eigenthaler and H. Länger, Congruence Classes in Universal Algebra. Heldermann, Lemgo 2012. ISBN 3-88538-226-1.
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  * [4] I. Chajda, M. Kolařík and H. Länger, Varieties corresponding to classes of complemented posets. Miskolc Math. Notes (to appear). http://arxiv.org/abs/1911.05138.
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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>
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>


# ITA-ELECTION-2022: A multi-platform dataset of social media conversations
around the 2022 Italian general election

Francesco Pierri, Geng Liu, Stefano Ceri

###### Abstract

Online social media play a major role in shaping public discourse and opinion,
especially during political events. We present the first public multi-platform
dataset of Italian-language political conversations, focused on the 2022
Italian general election taking place on September 25th. Leveraging public
APIs and a keyword-based search, we collected millions of posts published by
users, pages and groups on Facebook, Instagram and Twitter, along with
metadata of TikTok and YouTube videos shared on these platforms, over a period
of four months. We augmented the dataset with a collection of political ads
sponsored on Meta platforms, and a list of social media handles associated
with political representatives. Our data resource will allow researchers and
academics to further our understanding of the role of social media in the
democratic process.

## Introduction

Online social media provide researchers and academics with unprecedented
opportunities to observe a wide range of political and societal phenomena
(Rossi, Righetti, and Marino 2021). They also play a critical role in shaping
public opinion during political events (Vitak et al. 2011), and represent a
rich source of data to study the interplay between political actors’ campaigns
(Sahly, Shao, and Kwon 2019), media outlets’ agenda settings (Kim et al.
2016), and users’ news consumption (Allcott and Gentzkow 2017).

In Italy, as of 2022111www.statista.com/statistics/1311549/top-social-
platforms-italy/, YouTube is the platform used by the largest amount of
internet users (88%), followed by Meta platforms (64%) and TikTok (54%),
whereas Twitter only accounts for approximately
7%222datareportal.com/reports/digital-2022-italy. However, previous studies of
online social media during Italian elections and referendum mostly focused on
Twitter (Rossi, Righetti, and Marino 2021), due to the large availability of
data via its APIs. In this work, instead, we present a public data resource of
political conversations and user-generated content shared around the 2022
Italian general election, which allows researchers and academics to study
multiple social platforms simultaneously.

The 2022 Italian general election was the first ever to take place in autumn,
as a consequence of the fall of the government of national unity led by Mario
Draghi in July333en.wikipedia.org/wiki/2022˙Italian˙general˙election. The
election had a record-low voter turnout and it was won by the right-wing
coalition of Giorgia Meloni with over 43% of the vote share. Among the
opponents, the Centre-left coalition led by Enrico Letta obtained
approximately 25% of the voters, the populist Movimento 5 Stelle led by former
PM Giuseppe Conte reached less than 16%, and the liberal and centrist Third
Pole, which included former PM Matteo Renzi, obtained almost 8% of the vote
share.

We present ITA-ELECTION-2022, the first public multi-platform dataset of
Italian-language political conversations taking place on online social media,
with a focus on the 2022 Italian general election. We collected millions of
social media posts from Facebook, Instagram and Twitter, as well as
advertisements sponsored on Meta platforms and metadata for TikTok and YouTube
videos shared on the aforementioned platforms. We finally augment the dataset
with a collection of social media handles associated with Italian political
representatives. To collect the data, we employed a snowball sampling
procedure and curated a list of relevant terms to accordingly perform a
keyword-based search during a period of four months (July 2022 - October
2022). We provide public access to the data via GitHub and DataVerse
repositories, as detailed next.

The outline of this paper is the following: in the next section we review
existing public data resources related to the present work; then, we describe
the data collection procedure(s) carried out to build the dataset; next, we
describe a few potential applications of the collected data; finally, we
discuss limitations, draw conclusions and provide some ethical remarks.

## Related Work

There are several public datasets that allow to study social media
conversations around political issues. We focus our literature review on the
Italian context, and then describe a few datasets related to other countries.
We also refer the interested reader to (Rossi, Righetti, and Marino 2021) for
an overview of studies that describe the interplay between social media and
Italian politics.

(Basile, Lai, and Sanguinetti 2018) collect tweets in the Italian language
continuously from 2012 to 2018, extracting a number of smaller datasets
enriched with different kinds of annotations for linguistic purposes. They
provide access to tweet IDs and annotations in a public repository.

(Pierri, Artoni, and Ceri 2020) analyze the prevalence of Italian
disinformation spreading on Twitter in the five months preceding the 2019
European Parliament election. They collect over 300 k tweets sharing thousands
of news articles originating from websites flagged as unreliable by
journalists and fact-checkers, providing public access to tweet IDs and lists
of websites. The same authors provide a similar dataset collected in a
different period of 2019, and that contains tweets sharing links to mainstream
and traditional news websites, both in the Italian and French language (Pierri
2020).

(Di Giovanni and Brambilla 2021) study the polarization around the 2020
Italian constitutional referendum. They collect a dataset of 1.2 M tweets
discussing the event – and provide access to their IDs –, with the goal of
designing a hashtag-based semi-automatic approach to label Twitter users’
stance towards the referendum.

Following the COVID-19 pandemic, several researchers collected social media
data to study conversations around the crisis, with a particular focus on the
impact of vaccine misinformation. (Crupi et al. 2022) study the evolution of
Italian Twitter conversations around vaccines during the period 2019-2021,
whereas (Di Giovanni et al. 2022) collect tweets in multiple languages
(French, German and Italian) during the first year of world vaccination
programs. Both contributions give public access to tweet IDs, with the latter
providing also a set of labeled pro/anti-vaccines tweets and hashtags that can
be used for training machine learning classifiers.

(Calisir and Brambilla 2020) provide a dataset of tweets discussing Brexit for
a period of 45 months, from January 2016 until September 2019. The data, which
comprises 50.8 million tweets and 3.97 million users, is enriched with
metadata such as the bot score of users, sentiment score of tweets, and
political stance labels predicted by a classifier developed by the authors.

There is a large number of datasets that focus on the U.S. elections (both
presidential and midterms), and we provide here a non-exhaustive list of
available resources. (Hanna et al. 2011) mapped candidates from the 2010 U.S.
Midterm election with their Twitter accounts and a random sample of their
followers. (Bovet and Makse 2019) collected over 171 M tweets in the English
language, mentioning Donald Trump and Hillary Clinton during the 2016 U.S.
Presidential election. (Deb et al. 2019) and (Yang, Hui, and Menczer 2022)
collected tweets discussing the 2018 U.S. Midterm election, both using a
hashtag-based search (e.g. tweets sharing the hashtag ”#ivoted” on election
day) and querying Twitter APIs with general keywords related to the midterm
election. (Chen, Deb, and Ferrara 2022) provide a longitudinal dataset of over
1.2 billion U.S. politics- and election-related tweets shared around the
period of the 2020 U.S. Presidential election. Related to the same election,
(Abilov et al. 2021) released a multi-modal dataset of 7.6 M tweets and 25.6 M
retweets from 2.6 M users related to voter fraud claims. They augmented the
data with cluster labels, users’ suspension status, and perceptual hashes of
tweeted images as well as aggregate data from external links and YouTube
videos shared on Twitter.

## Data Collection

This section describes the data collection procedure(s) carried out to gather
data from different social media platforms. We remark that we employed the
same list of keywords related to the Italian election, which we obtained
through a snowball sampling approach using Twitter data only, to query
different APIs. Our dataset conforms with FAIR principles: it is Findable,
Accessible and Reusable as it is publicly accessible in an online
Github444github.com/frapierri/ita-election-2022 and DataVerse
repository555doi.org/10.7910/DVN/EALXH2, where we provide the means to
recreate it almost completely (see limitations discussed next). It is also
Interoperable as the data files are released in “.csv” and “.txt” formats. We
summarize some statistics of the dataset in Table 1.

Figure 1: Example of an ad run on Meta platforms along with the information
provided by Meta Ad Library API. Figure 2: Daily number of social media posts
and ads collected in our dataset, for different platforms. Solid lines show
3-day moving averages.

### Twitter

We collected all tweets in the Italian language related to the election by
using tweepy Python library to query Twitter v1.1 Filter streaming API
endpoint666developer.twitter.com/en/docs/twitter-api/v1/tweets/filter-
realtime/overview in the period September 2nd, 2022 - October 20th, 2022. We
also leveraged Twitter’s historical Search API v2
endpoint777developer.twitter.com/en/docs/twitter-
api/tweets/search/introduction to collect tweets retrospectively in the period
July 1st, 2022 - September 2nd, 2022. To query Twitter’s APIs we employed a
snowball sampling approach, following existing work (Di Giovanni et al. 2022;
DeVerna et al. 2021), and generated a list of relevant keywords starting with
seed terms such as “elezioni2022” and “elezioni”888In the Italian language
“elezioni” means elections.; the final list contains 62 keywords and it is
available in the repository associated with this paper. A sample is provided
in Table 2. The total collection of tweets contains 19,087,594 tweets shared
by 618,089 unique users. We remark that to abide by Twitter’s terms of service
we only share tweet IDs publicly. These can be “re-hydrated” to retrieve tweet
objects, with the exception of removed or protected tweets, by querying
Twitter API directly or using tools like Hydrator999github.com/DocNow/hydrator
or twarc.101010github.com/DocNow/twarc

Twitter | 19,087,594 tweets
---|---
618,089 unique accounts
Facebook | 1,142,812 posts
445,461 unique accounts
Instagram | 68,078 posts
5,274 unique accounts
Meta | 29,211 ads
3,750 unique sponsors
YouTube |  | 22,754 unique videos (Twitter)
---
| 17,401 unique videos (FB)
---
TikTok |  | 1,903 unique videos (Twitter)
---
| 1,744 unique videos (FB)
---
Table 1: Statistics of the dataset. elezioni | partito democratico | berlusconi
---|---|---
renzi | movimento 5 stelle | salvini
calenda | di maio | politiche2022
meloni | elezioni2022 | conte
Table 2: A sample of Italian language keywords related to the 2022 election
that were used to retrieve social media posts in our dataset.

### Facebook and Instagram posts

We collected Facebook and Instagram data by employing CrowdTangle, a public
tool owned and operated by Meta (CrowdTangle Team 2022) that allows retrieving
posts shared by public pages and groups with a certain amount of followers or
that were manually added by other researchers on the platform.111111More
details are available in the official documentation:
help.crowdtangle.com/en/articles/1140930-what-data-is-crowdtangle-tracking We
queried the /posts/search endpoint121212github.com/CrowdTangle/API/wiki/Search
using the same list of keywords employed for collecting Twitter data. For each
post, the API returns several attributes related to the post and the account
(page or group) that shared it; the full list of attributes is available in
the official documentation131313github.com/CrowdTangle/API/wiki. We retained
only posts in the Italian language by filtering on the languageCode parameter:
the final dataset contains 1,142,812 Facebook posts, shared by 445,461 unique
pages and groups and generating over 233 M interactions (shares, comments,
reactions), and 68,078 Instagram posts, shared by 5,274 unique pages and
generating over 97 M interactions (likes and comments). We provide access to
the URLs and IDs141414For each post we provide both platform and Crowdtangle
ID that can be given as input to the GET Post ID endpoint accessible here:
github.com/CrowdTangle/API/wiki/Posts#get-postid of these posts, which can be
used to access and retrieve those that are not removed or deleted, in the
repository associated with this paper.

### TikTok and YouTube videos

We augmented our dataset of social media posts by extracting metadata for
TikTok and YouTube videos shared in Facebook151515There were no links shared
on Instagram. and Twitter messages present in our dataset. For what concerns
YouTube, we identified all external links to the platform and employed the
official YouTube API161616developers.google.com/youtube/v3 to extract video
information such as the author, channel id, video title, description, Top 10
popular comments, etc. The resulting collection yields metadata for 22,754
unique YouTube videos shared on Twitter and 17,401 unique YouTube videos
shared on Facebook. For what concerns TikTok, given the lack of an official
API, we employed pyktok Python library171717github.com/dfreelon/pyktok to
collect metadata about TikTok videos such as the title, description, length as
well as information about the author of the video. The resulting collection
yields metadata for 1,903 unique TikTok videos shared on Twitter and 1,744
unique TikTok videos shared on Facebook.

### Facebook and Instagram ads

We leveraged Meta Ad Library API181818www.facebook.com/ads/library/api to
collect all ads about “social issues, elections or politics” that were active
on Meta platforms191919These are: Facebook, Instagram, Messenger, and the
Audience Network. Notice that only a dozen ads were placed on platforms other
than Facebook and Instagram. in the period July 1st, 2022 - October 20th,
2022. We provide an example of a sponsored ad in Figure 1. We queried the API
with the same set of keywords mentioned beforehand; the API allows to search
ads using one keyword at a time, and we queried the endpoint multiple times
eventually discarding duplicated ads. The resulting collection contains 29,211
unique ads paid by 3,750 unique sponsors. For each ad, the API provides
several different attributes: date of creation, period when the ad is active,
name of the sponsor, message, platform on which the ad is active, lower and
upper bound for the amount spent and the number of impressions generated, etc.
In the repository associated with this dataset we provide access to the ID of
ads, which can be then used to retrieve ads through Meta Ad Library
interactive search console or API. In particular, to abide by Meta’s terms of
use, an identification procedure is required to access the API endpoint,
whereas the interactive search console only requires a Meta account to access
it.

### Social media handles of political representatives

We compiled a list of Facebook, Instagram and Twitter handles of elected
members in the Senate and Chamber of deputies based on the official list
released by the Italian Ministry of Interior202020 github.com/ondata/elezioni-
politiche-2022. Specifically, for each representative, we manually checked
whether their official account was present on the three platforms. Insofar, we
were able to match around 500 Twitter accounts, and approximately 100-150
Facebook and Instagram accounts. The full list is available in the repository
associated with this paper. We refer the interested reader to a similar useful
resource presented by (Haman and Školník 2021), who provide an online running
database of politicians’ activity on social media (currently only Twitter is
supported) spanning multiple countries.

Figure 3: Distributions of the number of tweets/posts and
retweets/interactions for each Twitter, Facebook and Instagram account. Dashed
lines indicate median values: 2 tweets and 3 retweets for Twitter accounts; 2
posts and 8 interactions for Facebook accounts; 1 post and 174 interactions
for Instagram accounts. Figure 4: Distributions of the mean amount spent and
the mean number of impressions generated at the ad and sponsor level for Meta
ads. Dashed lines indicate median values: 148.5 EUR spent for ads and by
sponsors; 7498.5 impressions generated by ads and 12498.5 impressions by
sponsors.

## Data Characterization

In this section, we provide a few basic descriptive statistics of the data
presented in this work and leave more detailed analyses for future research.

In Figure 2, we show the daily number of social media posts and ads collected
in our dataset, for each platform. We can observe a significant increasing
trend (Mann-Kendall $P<0.001$) toward election day in all cases, with a sharp
drop in the weeks afterward. We also notice that Twitter activity in our
dataset is much more represented than other platforms, followed by Facebook
and Instagram.

In Figure 3, we show the distribution of account-wise metrics for Twitter,
Facebook and Instagram. Specifically, we show the Cumulative Distribution
Function (CDF) for the number of tweets/posts created and
retweets/interactions received by accounts on each platform. All distributions
show an exponential-like behavior, with most of the accounts being very rarely
active and receiving little engagement, and only a minority of them exhibiting
a large number of posts created and engagement received. Median values are
shown by dashed lines and are available in the caption of the figure.

In Figure 4, we show distributions of metrics for Meta ads. Specifically, we
show the CDF of the mean amount spent and the mean number of impressions
generated at both the ad and sponsor level; the mean value is computed by
taking the average between the lower and upper bound estimates of the
amount/impressions provided by Meta Ad Library API for each ad. Median values
are shown by dashed lines and are available in the caption of the figure. We
refer the reader to (Pierri 2022) for a more detailed analysis of political
advertising on Meta platforms during the 2022 Italian general election.

In Figure 5, we show the top 10 accounts on Twitter, Facebook and Instagram
ranked by the number of tweets/posts created and the number of
retweets/interactions received. We can observe notable politicians from the
entire political spectrum as well as journalists and news outlets, but also
supporting pages and groups. We also notice that some of the most active
accounts do not appear among the most engaged ones. Finally, we can see that
Italian PM Giorgia Meloni is the most engaged account on the three platforms.

Figure 5: Top 10 accounts ranked by the number of tweets/posts created and
retweets/interactions received on Twitter, Facebook and Instagram. Due to
space limitations, some account names are truncated.

## Potential Applications

There are several potential applications for our dataset, which can consider a
single platform or multiple ones at the same time.

Interested researchers could further the current understanding of polarization
processes taking place during election seasons by analyzing content shared on
multiple social platforms at once. They could study whether “echo-chamber”
effects take place on different platforms, highlighting similarities and
differences in their formation process.

Other researchers might leverage the data in order to study how political
candidates interacted with potential voters on social media platforms, thus
analyzing in detail the political communication strategies put in place by
different candidates. They could also investigate the presence of
correlational effects between online signals and electoral outcomes, or detect
the presence of toxic and hateful speech originating in communities of
political supporters.

Some researchers could investigate the presence of mis/disinformation and
astroturfing campaigns taking place in the run-up to the election, studying
patterns of similarities and differences among different platforms. They could
also analyze how fringe and harmful content spreads across communities present
on different platforms, and whether influential accounts play a role in
amplifying certain malicious narratives.

## Discussion

We released ITA-ELECTION-2022, a large-scale dataset of social media posts in
the Italian language discussing the 2022 Italian General election, which took
place on 25th September 2022, spanning multiple online platforms and covering
a period of four months. In addition to gathering posts shared on Twitter,
Facebook and Instagram, we collected ads sponsored on Meta platforms, we
extracted metadata for YouTube and TikTok videos shared on different platforms
during the collection period, and we compiled a list of social media handles
associated to political representatives that can be used to gather further
data. We described in detail the collection procedures carried out to build
the dataset, and provided a few basic statistics about the collected data. We
also suggested promising directions for future research.

Our work is not without limitations. First, our keyword-based search might
entail results that are not completely accurate, e.g., one of the terms
employed for the query is “conte”, which might refer both to former PM
Giuseppe Conte and football manager Antonio Conte. From another perspective,
election-related terms might have been employed for marketing campaigns and
promoting content that is not pertinent to the election. However, while we are
unable to address these issues, which would require non-trivial efforts,
researchers can further refine our data collection to meet their needs.
Moreover, we performed a backward search to retrieve Twitter, Facebook and
Instagram posts shared from July to September 2022, and we missed those that
were deleted or removed during the same period. Similarly, by providing access
only to IDs and URLs of collected posts, posts that have been removed or made
private by users cannot be retrieved, thus limiting reproducibility analyses.
Furthermore, we did not filter out the activity of automated and inauthentic
accounts that might have polluted organic conversations around the election.
Another limitation concerns Meta which, as highlighted in (Le Pochat et al.
2022), might not accurately label all political ads as such and our collection
might be missing some data. Finally, the user base of different platforms
analyzed in this work might not be fully representative of the actual Italian
population, and this should be taken into consideration by future research.

Despite these limitations, we believe that our dataset provides fertile ground
for a number of intriguing and interesting research applications, and we hope
that this resource can advance our understanding of the interplay between
online social media and democratic processes.

## Ethical considerations

We performed our data collection and public release in complete agreement with
the platforms’ terms of service. We acknowledge that TikTok metadata was
scraped from the platform, thus potentially violating the platform’s terms of
service, but this was due to the lack of an official API(Freelon 2018). We do
not directly share the content of social media posts, but rather provide
access to IDs and URLs that can be used to retrieve the original data, with
the exception of posts that have been deleted by platforms, and removed or
made private by their author. We did not cause any harm nor expose information
about individual users in the process of collecting and releasing the data,
with the only exception of political representatives and a handful of popular
accounts shown in the descriptive statistics. We understand that disclosing
their social media accounts might open up to potential abuse by malicious
actors, but at the same time, it enables researchers, journalists and other
stakeholders to put important public actors, such as the members of the
Italian Parliament and Senate, to scrutiny in order to better understand the
influence of social media platforms on the democratic process.

## Acknowledgments

We are thankful to M.Sc. students Valeria Panté and Ilaria Saini for helping
match social media accounts to political representatives. Work supported in
part by PRIN grant HOPE (FP6, Italian Ministry of Education).

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