∎

11institutetext: Ismael Calomino 22institutetext: CIC and Departamento de
Matemática, Facultad de Ciencias Exactas, Universidad Nacional del Centro,
Tandil, Argentina
22email<EMAIL_ADDRESS>33institutetext: Paula Menchón
44institutetext: CONICET and Departamento de Matemática, Facultad de Ciencias
Exactas, Universidad Nacional del Centro, Tandil, Argentina
44email<EMAIL_ADDRESS>55institutetext: William J. Zuluaga
Botero 66institutetext: Laboratoire J. A. Dieudonné, Université Côte d’Azur,
Nice, France and Departamento de Matemática, Facultad de Ciencias Exactas,
Universidad Nacional del Centro, Tandil, Argentina
66email<EMAIL_ADDRESS>
# A topological duality for monotone expansions of semilattices ††thanks: This
research was supported by ANPCyT under grant 2019-00882. The third author has
been funded by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program (grant agreement No. 670624).

Ismael Calomino  Paula Menchón  William J. Zuluaga Botero

(Received: date / Accepted: date)

###### Abstract

In this paper we provide a Stone style duality for monotone semilattices by
using the topological duality developed in Celani2020  for semilattices
together with a topological description of their canonical extension. As an
application of this duality we obtain a characterization of the congruences of
monotone semilattices by means of monotone lower-Vietoris-type topologies.

###### Keywords:

Semilattice Duality Monotone operator Canonical extension

###### pacs:

06A12 06E15

##  1 Introduction

In Birkhoff  Birkhoff proved that every finite distributive lattice is
represented by the lower sets of a finite partial order. In Stone2  Stone
generalized the results given by Birkhoff establishing a categorical duality
for bounded distributive lattices through spectral spaces with continuous
functions. Later, in Priestley I  Priestley proved another categorical duality
for bounded distributive lattices by means of certain ordered topological
spaces and continuous order-preserving maps. Along the years both Stone
duality and Priestley duality have served as powerful tools not only for the
study of bounded distributive lattices, but also as an inspiration for the
study of many ordered algebraic structures associated with non-classical
logics. Evidence of this is supplied by the several topological
representations obtained for diverse classes of algebras with distributive
semilattice reducts and semilattice reducts. This is the case of the Stone
style duality for distributive semilattices developed in CelaniTopological ;
CelaniCalomino , which extends Stone duality and completes the representation
given by Grätzer in Gr78 , the Priestley style duality developed by
Bezhanishvili and Jansana in Guram-Jansana I ; Guram-Jansana III  for the
class of distributive semilattices and implicative semilattices, and more
recently the Stone style duality for semilattices and lattices obtained by
Celani and González in Celani2020 .

A particular case of interest raises when noticing that any of these dualities
can be combined with some other tools coming from algebra. This is the case of
canonical extensions (Gehrke -Jonsson2000 ; GJ2 ; PalmigianoDunn ;
GPsemilattices ). The studies on this subject started with Jónsson and
Tarski’s works for Boolean algebras with operators (Jonsson y Tarski ; JT2 ).
The most remarkable contribution of these papers was that they provided a
procedure for transfer the benefits and the working methodology from the
duality for Boolean algebras to several classes of algebras with additional
operators. The crucial observation is that given a class of algebras with
lattice reduct $\bf{L}$ and a suitable topological duality for $\bf{L}$, it is
possible to build new topological dualities for monotone expansions of
$\bf{L}$ by using a topological description of the canonical extension of the
members of $\bf{L}$ in terms of the given duality. This approach was used by
Celani and Menchón in menchon2  to obtain a Stone style duality for monotone
distributive semilattices. In MoshierJipsen  Moshier and Jipsen employed a
duality different from the ones of Stone and Priestley to give a topological
description of the canonical extension of a (not necessarily distributive)
bounded lattice.

The aim of this paper is to establish a Stone style duality for monotone
semilattices. To do so, we provide a topological description of the canonical
extension of a semilattice by using the duality obtained in Celani2020 . We
conclude by applying our duality to characterize the congruences of
semilattices and monotone semilattices in terms of lower-Vietoris-type
topologies (Ivanova-dimova ). These results generalize the results obtained in
CelaniCalomino  for the class of distributive semilattices.

The paper is organized as follows. In Section 2 we give several notions and
results which are needed for the rest of the paper. In Section 3 we present a
topological description of the canonical extension of a semilattice in terms
of the Stone style duality given in Celani2020  and we explore the underlying
connection of our results with the description of canonical extensions of
semilattices given by Gouveia and Priestley in GPsemilattices . Further, we
extend our results to a topological description of canonical extensions of
monotone maps of semilattices. In Section 4 we establish our main result: a
topological duality for the category of monotone semilattices and
homomorphisms of monotone semilattices. Finally, in Section 5 as an
application of the duality, we prove that the congruences of a semilattice and
a monotone semilattice are in correspondence with the lower-Vietoris-type
families and monotone lower-Vietoris-type topologies of their respective dual
spaces. Moreover, we use such a characterization to obtain a version for
semilattices of the results of Fajtlowicz and Schmidt in Fajtlowicz
concerning the lattice of congruences.

##  2 Preliminaries

If $f\colon X\rightarrow Y$ is a function and $U\subseteq X$ we write $f[U]$
for the direct image of $U$ through $f$. If $V\subseteq Y$ we write
$f^{-1}[V]$ for the inverse image of $V$ through $f$. If $g\colon Y\rightarrow
Z$ is a function we write $gf$ for the composition of $g$ with $f$. If
$R\subseteq X\times Y$ is a binary relation and $x\in X$, then $R(x)=\\{y\in
Y\colon(x,y)\in R\\}$. If $T\subseteq Y\times Z$ is some other binary
relation, then the composition of $T$ with $R$ is the set $T\circ
R=\\{(x,z)\colon\exists y\in Y[(x,y)\in R\ \text{and}\ (y,z)\in T]\\}$.

Let $\langle X,\leq\rangle$ be a poset. For each $Y\subseteq X$, let
$[Y)=\\{x\in X\colon\exists y\in Y(y\leq x)\\}$ and $(Y]=\\{x\in
X\colon\exists y\in Y(x\leq y)\\}$. We will say that $Y$ is an _upset_ of $X$
(a _downset_ of $X$) if $Y=[Y)$ ($Y=(Y]$). We write $\mathcal{P}(X)$ for the
set of all subsets of $X$ and $\mathrm{Up}(X)$ for the set of all upsets of
$X$. The complement of $Y\subseteq X$ will be denoted by $Y^{c}$. A subset
$Z\subseteq X$ is called _directed_ if for every $x,y\in Z$, there exists
$z\in Z$ such that $x\leq z$ and $y\leq z$. Analogously, $Z\subseteq X$ is
called _dually directed_ if for every $x,y\in Z$, there exists $z\in Z$ such
that $z\leq x$ and $z\leq y$. Two posets $\langle X,\leq_{X}\rangle$ and
$\langle Y,\leq_{Y}\rangle$ are said to be _dually isomorphic_ if there exists
a onto function $f\colon X\rightarrow Y$ such that $x\leq_{X}y$ if and only if
$f(y)\leq_{Y}f(x)$. If the context is clear, the subscripts of the partial
orders shall be omitted.

A _meet-semilattice with greatest element_ , or _semilattice_ for short, is an
algebra $\mathbf{A}=\langle A,\wedge,1\rangle$ of type $(2,0)$ such that the
operation $\wedge$ is idempotent, commutative, associative, and $a\wedge 1=a$
for all $a\in A$. The partial order $\leq$ on $\mathbf{A}$ is given by $a\leq
b$ if and only if $a=a\wedge b$. For each poset $\langle X,\leq\rangle$, the
structure $\langle\mathrm{Up}(X),\cap,X\rangle$ is a semilattice. If
$\mathbf{A}$ is a semilattice, a subset $F\subseteq A$ is a _filter_ of
$\mathbf{A}$ if it is an upset, $1\in F$ and if $a,b\in F$, then $a\wedge b\in
F$. The set of all filters of $\mathbf{A}$ will be denoted by
$\operatorname{Fi}({\mathbf{A}})$. The filter generated by the subset
$X\subseteq A$ will be denoted by $F(X)$. If $X=\\{a\\}$, then
$F(\\{a\\})=[\\{a\\})$, or simply, $[a)$. We say that a proper filter
$F\in\operatorname{Fi}({\mathbf{A}})$ is _irreducible_ if for all
$F_{1},F_{2}\in\operatorname{Fi}(\mathbf{A})$, if $F=F_{1}\cap F_{2}$, then
$F=F_{1}$ or $F=F_{2}$. We write $X(\mathbf{A})$ the set of all irreducible
filters of $\mathbf{A}$.

###### Theorem 2.1

(CelaniTopological ) Let $\mathbf{A}$ be a semilattice. A proper filter $F$ is
irreducible if and only if for every $a,b\notin F$, there exist $c\notin F$
and $f\in F$ such that $a\wedge f\leq c$ and $b\wedge f\leq c$.

A nonempty subset $I\subseteq A$ is an _order-ideal_ of $\mathbf{A}$ if it is
a directed downset. We denote by $\operatorname{Id}(\mathbf{A})$ the set of
all order-ideals of $\mathbf{A}$.

###### Theorem 2.2

(CelaniTopological ) Let $\mathbf{A}$ be a semilattice. Let
$F\in\operatorname{Fi}(\mathbf{A})$ and let
$I\in\operatorname{Id}(\mathbf{A})$. If $F\cap I=\emptyset$, then there exists
$P\in X(\mathbf{A})$ such that $F\subseteq P$ and $P\cap I=\emptyset$.

Let $\mathbf{A}$ and $\mathbf{B}$ be two semilattices. A map $h\colon
A\rightarrow B$ is a _homomorphism_ if $h(1)=1$ and $h(a\wedge b)=h(a)\wedge
h(b)$ for all $a,b\in A$. If we consider the poset $\langle
X(\mathbf{A}),\subseteq\rangle$ and the map $\beta_{\mathbf{A}}\colon
A\rightarrow\mathrm{Up}(X(\mathbf{A}))$ given by
$\beta_{\mathbf{A}}(a)=\\{P\in X(\mathbf{A})\colon a\in P\\}$, then it is
proved that $\mathbf{A}$ is isomorphic to the subalgebra
$\beta_{\mathbf{A}}[A]=\\{\beta_{\mathbf{A}}(a)\colon a\in A\\}$ of
$\langle\mathrm{Up}(X({\mathbf{A}})),\cap,X(\mathbf{A})\rangle$. Throughout
the paper and to simplify notation, we will omit the subscript of
$\beta_{\mathbf{A}}$ where appropriate.

By last, if $\langle X,\tau\rangle$ is a topological space and $Y\subseteq X$,
we write $\mathrm{cl}(Y)$ for the topological closure of $Y$. In particular,
if $Y=\\{y\\}$, then $\mathrm{cl}(\\{y\\})=\mathrm{cl}(y)$. It is known that
every topological space can be endowed with a partial order $\sqsubseteq$
defined by $x\sqsubseteq y$ if and only if $x\in\mathrm{cl}(y)$. Such an order
is called the _specialization order_ of $X$.

###  2.1 S-spaces

The authors in Celani2020  developed topological dualities for semilattices
and lattices that are natural generalizations of the Stone duality for bounded
distributive lattices through spectral spaces. To do so, they proposed the
following definition.

###### Definition 1

Let $\mathbf{A}$ be a semilattice and let $F\in\operatorname{Fi}(\mathbf{A})$.
A subset $I\subseteq A$ is an _$F$ -ideal_ if it is a downset and for every
$a,b\in I$, there exist $c\in I$ and $f\in F$ such that $a\wedge f\leq c$ and
$b\wedge f\leq c$.

It is clear that a proper filter $F$ is irreducible if and only if $F^{c}$ is
an $F$-ideal. The following result generalizes Theorem 2.2.

###### Theorem 2.3

(Celani2020 ) Let $\mathbf{A}$ be a semilattice. Let
$F\in\operatorname{Fi}(\mathbf{A})$ and let $I$ be an $F$-ideal. If $F\cap
I=\emptyset$, then there exists $P\in X(\mathbf{A})$ such that $F\subseteq P$
and $P\cap I=\emptyset$.

We stress that along this paper, by _topological space_ we mean pairs $\langle
X,\mathcal{K}\rangle$ where $\langle X,\tau\rangle$ is a topological space and
$\mathcal{K}$ is a subbase for $\tau$. We consider the following family of
subsets of $X$:

$S(X)=\\{U^{c}\colon U\in\mathcal{K}\\}.$

Let $C_{\mathcal{K}}(X)$ be the closure system on $X$ generated by $S(X)$,
i.e., $C_{\mathcal{K}}(X)=\\{\bigcap A\colon A\subseteq S(X)\\}$. The elements
of $C_{\mathcal{K}}(X)$ are called subbasic closed subsets of $X$. Note that
$S(X)\subseteq C_{\mathcal{K}}(X)\subseteq C(X)$, where $C(X)$ is the
collection of all closed subsets of the topology. A subset $Y\subseteq X$ is
called _saturated_ if it is an intersection of open sets.

###### Definition 2

Let $\langle X,\mathcal{K}\rangle$ be a topological space. Let $Y\subseteq X$.
A family $\mathcal{Z}\subseteq S(X)$ is a _$Y$ -family_ if for every
$A,B\in\mathcal{Z}$, there exist $H,C\in S(X)$ such that $Y\subseteq H$,
$C\in\mathcal{Z}$, $A\cap H\subseteq C$ and $B\cap H\subseteq C$.

###### Definition 3

(Celani2020 ) An _$S$ -space_ is a topological space $\langle
X,\mathcal{K}\rangle$ satisfying the following conditions:

  1. (S1)

$\langle X,\mathcal{K}\rangle$ is a $T_{0}$-space and $X=\bigcup\mathcal{K}$.

  2. (S2)

$\mathcal{K}$ is a subbase of compact open subsets, it is closed under finite
unions and $\emptyset\in\mathcal{K}$.

  3. (S3)

For every $U,V\in\mathcal{K}$, if $x\in U\cap V$, then there exist
$W,D\in\mathcal{K}$ such that $x\notin W$, $x\in D$ and $D\subseteq(U\cap
V)\cup W$.

  4. (S4)

If $Y\in C_{\mathcal{K}}(X)$ and $\mathcal{J}\subseteq S(X)$ is a $Y$-family
such that $Y\cap A^{c}\neq\emptyset$, for all $A\in\mathcal{J}$, then
$Y\cap\bigcap\\{A^{c}\colon A\in\mathcal{J}\\}\neq\emptyset$.

Let $\langle X,\mathcal{K}\rangle$ be an $S$-space. Then the structure
$\mathbf{S}(X)=\langle S(X),\cap,X\rangle$ is a semilattice, called the _dual
semilattice of $\langle X,\mathcal{K}\rangle$_. Conversely, if $\mathbf{A}$ is
a semilattice, then $\langle X(\mathbf{A}),\mathcal{K}_{\mathbf{A}}\rangle$ is
a topological space where $\mathcal{K}_{\mathbf{A}}=\\{\beta(a)^{c}\colon a\in
A\\}$ is a subbase. It follows that $\langle
X(\mathbf{A}),\mathcal{K}_{\mathbf{A}}\rangle$ is an $S$-space, called the
_dual $S$-space of $\mathbf{A}$_, and $\beta\colon A\rightarrow
S(X(\mathbf{A}))$ is an isomorphism between semilattices. If we consider the
associated $S$-space $\langle
X(\mathbf{S}(X)),\mathcal{K}_{\mathbf{S}(X)}\rangle$ of the dual semilattice
$\mathbf{S}(X)$, then the mapping $H_{X}\colon X\rightarrow X(\mathbf{S}(X))$
given by $H_{X}(x)=\\{A\in S(X)\colon x\in A\\}$ for all $x\in X$ is a
homeomorphism between $S$-spaces.

Now, we mention the relationship existing between the subbasic closed subsets
and the filters of the dual semilattice of certain topological spaces proved
in Celani2020 .

###### Proposition 1

Let $\langle X,\mathcal{K}\rangle$ be a topological space such that
$\mathcal{K}$ is a subbase of compact open subsets, and it is closed under
finite unions and $\emptyset\in\mathcal{K}$. Then:

  1. 1.

For every $Y\in C_{\mathcal{K}}(X)$, $F_{Y}=\\{A\in S(X)\colon Y\subseteq
A\\}\in\operatorname{Fi}(\mathbf{S}(X))$.

  2. 2.

For every $F\in\operatorname{Fi}(\mathbf{S}(X))$, $Y_{F}=\bigcap F\in
C_{\mathcal{K}}(X)$.

  3. 3.

$Y=Y_{F_{Y}}$ for all $Y\in C_{\mathcal{K}}(X)$ and $F=F_{Y_{F}}$ for all
$F\in\operatorname{Fi}(\mathbf{S}(X))$.

Moreover, the posets $\langle C_{\mathcal{K}}(X),\subseteq\rangle$ and
$\langle\operatorname{Fi}(\mathbf{S}(X)),\subseteq\rangle$ are dually
isomorphic.

We conclude this part by making explicit Proposition 1 in terms of
semilattices and their dual S-spaces. Such a restatement will be extensively
used along the rest of the paper.

Let $\mathbf{A}$ be a semilattice. Then the mapping
$\varphi\colon\operatorname{Fi}(\mathbf{A})\rightarrow
C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))$ defined by

$\varphi(F)=\bigcap\\{\beta(a)\colon a\in F\\}=\\{P\in X(\mathbf{A})\colon
F\subseteq P\\}$ (1)

is a dual isomorphism, whose inverse $\psi\colon
C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))\rightarrow\operatorname{Fi}(\mathbf{A})$
is given by

$\psi(Y)=\\{a\in A\colon Y\subseteq\beta(a)\\}.$ (2)

###  2.2 Canonical extension

Inspired in PalmigianoDunn , we present the basic definitions of the theory of
canonical extensions of semilattices. These definitions also coincide with the
definition of canonical extension when restricted to bounded distributive
lattices and Boolean algebras (Jonsson y Tarski ; JT2 ; Gehrke -Jonsson2000 ;
GJ2 ).

###### Definition 4

Let $\mathbf{A}$ be a semilattice. A _completion_ of $\mathbf{A}$ is a pair
$\mathbf{E}=\langle E,e\rangle$ where $E$ is a complete lattice and the map
$e\colon A\rightarrow E$ is an embedding.

Let $\mathbf{A}$ be a semilattice and let $\mathbf{E}$ be a completion of
$\mathbf{A}$. An element $x\in E$ is called _closed_ if there is
$F\in\operatorname{Fi}(\mathbf{A})$ such that $x=\bigwedge e(F)$. An element
$x\in E$ is called _open_ if there is $I\in\operatorname{Id}(\mathbf{A})$ such
that $x=\bigvee e(F)$. Let us denote the collection of all closed elements of
$E$ by $K(\mathbf{E})$ and the collection of all open elements of $E$ by
$O(\mathbf{E})$. The completion $\mathbf{E}$ is _dense_ if $x=\bigwedge\\{y\in
O(\mathbf{E})\colon x\leq y\\}$ and $x=\bigvee\\{y\in K(\mathbf{E})\colon
y\leq x\\}$ for all $x\in E$. The completion $\mathbf{E}$ is _compact_ if for
every non-empty dually directed subset $D$ of $A$ and every non-empty directed
subset $U$ of $A$ such that $\bigwedge e(D)\leq\bigvee e(U)$, then there exist
$x\in D$ and $y\in U$ such that $x\leq y$.

From now on we will suppress the embedding $e$ and we assume that $\mathbf{A}$
is a subalgebra of a completion $\mathbf{E}$ of $\mathbf{A}$.

###### Definition 5

Let $\mathbf{A}$ be a semilattice. A completion $\mathbf{E}$ of $\mathbf{A}$
is a _canonical extension_ if it is dense and compact.

###### Theorem 2.4

Let $\mathbf{A}$ be a semilattice. Then $\mathbf{A}$ has a canonical extension
and it is unique up to isomorphism.

##  3 Canonical extensions of semilattices

The aim of this section is to exhibit a topological description of the
canonical extension of a semilattice in terms of the duality obtained in
Celani2020 . As a result we obtain that the canonical extension of a
semilattice is determined by a closure system generated by the set of subasic
saturated subsets of its dual space. Our motivations differ from those of
GPsemilattices  because we are interested in producing a duality for monontone
semilattices. Further details of the relation between our construction and the
Gouveia and Priestley construction will be provided in Section 3.1.

###### Definition 6

Let $\langle X,\mathcal{K}\rangle$ be an $S$-space. A subset $Z\subseteq X$ is
a _subbasic saturated subset_ if there exists a dually directed family
$\mathcal{L}\subseteq\mathcal{K}$ such that $Z=\bigcap\\{W\colon
W\in\mathcal{L}\\}$.

We denote by $\mathcal{Z}(X)$ the set of all subbasic saturated subsets of an
$S$-space $\langle X,\mathcal{K}\rangle$. Note that every subbasic saturated
subset is a saturated set.

###### Lemma 1

Let $\langle X,\mathcal{K}\rangle$ be an $S$-space. Then:

  1. 1.

If $\mathcal{L}\subseteq\mathcal{K}$ is a dually directed family, then
$\mathcal{A}_{\mathcal{L}}=\\{U^{c}\colon U\in\mathcal{L}\\}\subseteq S(X)$ is
a $Y$-family for all $Y\in C_{\mathcal{K}}(X)$.

  2. 2.

If $\mathcal{A}\subseteq S(X)$ is a $Y$-family for all $Y\in
C_{\mathcal{K}}(X)$, then $\mathcal{L}_{\mathcal{A}}=\\{U^{c}\colon
U\in\mathcal{A}\\}\subseteq\mathcal{K}$ is a dually directed family.

Moreover, the assignments $\mathcal{L}\mapsto\mathcal{A}_{\mathcal{L}}$ and
$\mathcal{A}\mapsto\mathcal{L}_{\mathcal{A}}$ establish a bijection between
the set of dually directed families of $\mathcal{K}$ and the families of
$C_{\mathcal{K}}(X)$ which are a $Y$-family for all $Y\in C_{\mathcal{K}}(X)$.

###### Proof

1\. Let $Y\in C_{\mathcal{K}}(X)$ and $U,V\in\mathcal{L}$. Since $\mathcal{L}$
is a dually directed family, there exists $W\in\mathcal{L}$ such that
$W\subseteq U\cap V$. From $X\in S(X)$, we get $Y\subseteq X$, $U^{c}\cap
X=U^{c}\subseteq W^{c}$ and $V^{c}\cap X=V^{c}\subseteq W^{c}$. So,
$\mathcal{A}_{\mathcal{L}}=\\{U^{c}\colon U\in\mathcal{L}\\}$ is a $Y$-family.

2\. Let $\mathcal{A}\subseteq S(X)$ be a $Y$-family for all $Y\in
C_{\mathcal{K}}(X)$. Let $U,V\in\mathcal{A}$. As $X\in C_{\mathcal{K}}(X)$,
there exists $W\in\mathcal{A}$ such that $U\cap X\subseteq W$ and $V\cap
X\subseteq W$. Therefore, $\mathcal{L}_{\mathcal{A}}=\\{U^{c}\colon
U\in\mathcal{A}\\}\subseteq\mathcal{K}$ is a dually directed family.

For the moreover part, observe that
$\mathcal{L}_{\mathcal{A}_{\mathcal{L}}}=\mathcal{L}$ and
$\mathcal{A}_{\mathcal{L}_{\mathcal{A}}}=\mathcal{A}$.

###### Remark 1

If $Z=\bigcap\\{W\colon W\in\mathcal{L}\\}\in\mathcal{Z}(X)$, then the
collection $\\{W^{c}\colon W\in\mathcal{L}\\}$ is a $Y$-family for all $Y\in
C_{\mathcal{K}}(X)$.

###### Proposition 2

Let $\langle X,\mathcal{K}\rangle$ be an $S$-space. The subbasic saturated
subsets are compact saturated subsets of the topology.

###### Proof

Let $Z\in\mathcal{Z}(X)$. We prove that $Z$ is compact. By the Alexander’s
subbasis Theorem, we only need to prove that for every cover $\\{O_{i}\colon
i\in I\\}$ of $Z$ by elements of $\mathcal{K}$, there is a finite subcover. By
assumption, there exists a dually directed family
$\mathcal{L}\subseteq\mathcal{K}$ such that $Z=\bigcap\\{W\colon
W\in\mathcal{L}\\}$. Since $Z\subseteq\bigcup\\{O_{i}\colon i\in I\\}$, we
have

$\bigcap\\{W\colon W\in\mathcal{L}\\}\cap\bigcap\\{O_{i}^{c}\colon i\in
I\\}=\emptyset.$

By (S4) and Lemma 1 there exists $W\in\mathcal{L}$ such that
$W\cap\bigcap\\{O_{i}^{c}\colon i\in I\\}=\emptyset$. Then
$W\subseteq\bigcup\\{O_{i}\colon i\in I\\}$ and as $W$ is a compact subset,
there exist $O_{i_{1}},\dots,O_{i_{n}}$ such that $W\subseteq
O_{i_{1}}\cup\ldots\cup O_{i_{n}}$. Hence, $Z\subseteq W\subseteq
O_{i_{1}}\cup\ldots\cup O_{i_{n}}$ and $Z$ is compact.

Let $\mathbf{A}$ be a semilattice and let $I\in$ $\mathrm{Id}({\mathbf{A}})$.
We consider the following subset

$\alpha(I)=\bigcap\\{\beta(a)^{c}\colon a\in I\\}=\\{P\in
X({\mathbf{A}})\colon I\cap P=\emptyset\\}.$

Then $\alpha(I)$ is a subbasic saturated subset of the dual $S$-space $\langle
X({\mathbf{A}}),\mathcal{K}_{\mathbf{A}}\rangle$. Conversely, let $Z\subseteq
X({\mathbf{A}})$ be a subbasic saturated subset and consider

$I_{\mathbf{A}}(Z)=\\{a\in A\colon\beta(a)\cap Z=\emptyset\\}.$

It is easy to see that $I_{\mathbf{A}}(Z)$ is a downset of $\mathbf{A}$.

###### Proposition 3

Let $\langle X,\mathcal{K}\rangle$ be an $S$-space and let $Z\subseteq X$ be a
saturated set such that $Z=\bigcap\\{W\colon W\in\mathcal{J}\\}$ for some
family $\mathcal{J}\subseteq\mathcal{K}$. Then $Z$ is a subbasic saturated
subset if and only if the family $\\{U\in\mathcal{K}\colon Z\subseteq U\\}$ is
dually directed.

###### Proof

If $Z$ is a subbasic saturated subset, then there exists a dually directed
family $\mathcal{L}\subseteq\mathcal{K}$ such that $Z=\bigcap\\{W\colon
W\in\mathcal{L}\\}$. Let $S,T\in\\{U\in\mathcal{K}\colon Z\subseteq U\\}$.
Thus, $Z\cap S^{c}=\emptyset$ and $Z\cap T^{c}=\emptyset$. By (S4), there
exist $S^{\prime},T^{\prime}\in\mathcal{L}$ such that $S^{\prime}\cap
S^{c}=\emptyset$ and $T^{\prime}\cap T^{c}=\emptyset$. As $\mathcal{L}$ is
dually directed, there is $W\in\mathcal{L}$ such that $W\subseteq S^{\prime}$
and $W\subseteq T^{\prime}$, i.e., $W\subseteq S$ and $W\subseteq T$. So,
$\\{U\in\mathcal{K}\colon Z\subseteq U\\}$ is dually directed.

Conversely, suppose that the family $\\{U\in\mathcal{K}\colon Z\subseteq U\\}$
is dually directed. Since $Z=\bigcap\\{W\colon W\in\mathcal{J}\\}$ for some
family $\mathcal{J}\subseteq\mathcal{K}$, then
$Z\subseteq\bigcap\\{U\in\mathcal{K}\colon Z\subseteq
U\\}\subseteq\bigcap\\{W\colon W\in\mathcal{J}\\}=Z$ and $Z$ is a subbasic
saturated subset.

In the following result we see that order-ideals are in one to one
correspondence with the subbasic saturated subsets.

###### Theorem 3.1

Let $\mathbf{A}$ be a semilattice. Then the posets
$\langle\mathrm{Id}(\mathbf{A}),\subseteq\rangle$ and
$\langle\mathcal{Z}(X(\mathbf{A})),\subseteq\rangle$ are dually isomorphic.

###### Proof

Let $Z\in\mathcal{Z}(X(\mathbf{A}))$. We see that $I_{\mathbf{A}}(Z)$ is an
order-ideal of $\mathbf{A}$ and $Z=\alpha(I_{\mathbf{A}}(Z))$. Moreover, if
$I\in{\rm{Id}}(\mathbf{A})$, then we prove that $I=I_{\mathbf{A}}(\alpha(I))$.

It is clear that $I_{\mathbf{A}}(Z)$ is a downset of $\mathbf{A}$. Let $a,b\in
I_{\mathbf{A}}(Z)$. So, $\beta(a)\cap Z=\emptyset$ and $\beta(b)\cap
Z=\emptyset$. Since
$Z=\bigcap\\{\beta(x)^{c}\colon\beta(x)^{c}\in\mathcal{L}\\}$ for some dually
directed family $\mathcal{L}\subseteq\mathcal{K}_{\mathbf{A}}$ and
$\beta(a),\beta(b)\in C_{\mathcal{K}}(X(\mathbf{A}))$, then from Remark 1
$\\{\beta(x)\colon\beta(x)^{c}\in\mathcal{L}\\}$ is a $\beta(a)$-family and a
$\beta(b)$-family. Whence, by (S4) there exist
$\beta(c)^{c},\beta(d)^{c}\in\mathcal{L}$ such that
$\beta(c)^{c}\cap\beta(a)=\emptyset$ and $\beta(d)^{c}\cap\beta(b)=\emptyset$.
Thus, there is $\beta(e)^{c}\in\mathcal{L}$ such that
$\beta(e)^{c}\subseteq\beta(c)^{c},\beta(d)^{c}$. So
$\beta(e)^{c}\cap\beta(a)=\emptyset$, $\beta(e)^{c}\cap\beta(b)=\emptyset$ and
$Z\cap\beta(e)=\emptyset$, i.e., $a,b\leq e$ and $e\in I_{\mathbf{A}}(Z)$.
Therefore, $I_{\mathbf{A}}(Z)$ is an order-ideal of $\mathbf{A}$ and

$\alpha(I_{\mathbf{A}}(Z))=\bigcap\\{\beta(a)^{c}\colon
Z\subseteq\beta(a)^{c}\\}\subseteq\bigcap\\{\beta(x)^{c}\colon\beta(x)^{c}\in\mathcal{L}\\}=Z.$

The other inclusion is immediate.

Let $I\in{\rm{Id}}(\mathbf{A})$. If $b\in I_{\mathbf{A}}(\alpha(I))$, then
$\beta(b)\cap\alpha(I)=\beta(b)\cap\bigcap\\{\beta(a)^{c}\colon a\in
I\\}=\emptyset$. Since $\beta(b)\in C_{\mathcal{K}}(X(\mathbf{A}))$ and the
family $\\{\beta(a)^{c}\colon a\in I\\}$ is dually directed, by Remark 1 and
(S4) we get that there is $c\in I$ such that
$\beta(b)\cap\beta(c)^{c}=\emptyset$, i.e., $\beta(b)\subseteq\beta(c)$. So,
$b\leq c$ and as $I$ is a downset, we have $b\in I$. The other inclusion is
straightforward.

Then we have a mapping
$\alpha\colon\mathrm{Id}(\mathbf{A})\rightarrow\mathcal{Z}(X(\mathbf{A}))$
with inverse
$I_{\mathbf{A}}\colon\mathcal{Z}(X(\mathbf{A}))\rightarrow\mathrm{Id}(\mathbf{A})$.
Let $I_{1},I_{2}\in\mathrm{Id}(\mathbf{A})$. If $I_{1}\subseteq I_{2}$ and
$P\in\alpha(I_{2})$, then $P\cap I_{2}=\emptyset$. It follows that $P\cap
I_{1}=\emptyset$ and $P\in\alpha(I_{1})$. Reciprocally, assume that
$\alpha(I_{1})\subseteq\alpha(I_{2})$. Let $a\in I_{2}$ and suppose $a\notin
I_{1}$. By Theorem 2.2 there exists $P\in X(\mathbf{A})$ such that $a\in P$
and $P\cap I_{1}=\emptyset$. So, $P\in\alpha(I_{1})$ and
$P\notin\alpha(I_{2})$ which is a contradiction. Then $a\in I_{1}$ and
$I_{2}\subseteq I_{1}$. Therefore,
$\langle\mathrm{Id}(\mathbf{A}),\subseteq\rangle$ and
$\langle\mathcal{Z}(X(\mathbf{A})),\subseteq\rangle$ are dually isomorphic.

###### Lemma 2

Let $\mathbf{A}$ be a semilattice. If $Y\in
C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))$ and
$Z\in\mathcal{Z}(X(\mathbf{A}))$, then $\psi(Y)\cap
I_{\mathbf{A}}(Z)\neq\emptyset$ if and only if $Y\cap Z=\emptyset$.

###### Proof

Suppose $Y\cap Z=\emptyset$. Since
$Z=\bigcap\\{\beta(a)^{c}\colon\beta(a)^{c}\in\mathcal{L}\\}$ for some dually
directed family $\mathcal{L}\subseteq\mathcal{K}_{\mathbf{A}}$, by Remark 1
and (S4) there is $\beta(c)^{c}\in\mathcal{L}$ such that
$Y\cap\beta(c)^{c}=\emptyset$. So, there exist $c\in A$ such that
$c\in\psi(Y)$ and $c\in I_{\mathbf{A}}(Z)$. Hence, $\psi(Y)\cap
I_{\mathbf{A}}(Z)\neq\emptyset$. The other implication is straightforward.

###### Remark 2

Observe that Lemma 2 can be rephrased in terms of order-ideals and filters of
$\mathbf{A}$ as follows: Let $\mathbf{A}$ be a semilattice. If
$F\in\mathrm{Fi}(\mathbf{A})$ and $I\in\mathrm{Id}(\mathbf{A})$, then
$\varphi(F)\cap\alpha(I)=\emptyset$ if and only if $F\cap I\neq\emptyset$.

Let $\mathbf{A}$ be a semilattice and let us consider the following closure
operator
$\Lambda\colon\mathrm{Up}(X({\mathbf{A}}))\rightarrow\mathrm{Up}(X({\mathbf{A}}))$
given by

$\Lambda(Y)=\bigcap\\{U^{c}\colon U\in\mathcal{Z}(X({\mathbf{A}}))\text{ and
}Y\subseteq U^{c}\\}.$ (3)

We call $E(X(\mathbf{A}))$ to the closure system on
$\mathrm{Up}(X(\mathbf{A}))$ generated by $\mathcal{Z}(X({\mathbf{A}}))$:

$E(X(\mathbf{A}))=\left\\{\bigcap\\{U^{c}\colon
U\in\mathcal{B}\\}\colon\mathcal{B}\subseteq\mathcal{Z}(X(\mathbf{A}))\right\\}.$

Notice that the structure $\langle
E(X(\mathbf{A})),\vee,\cap,\emptyset,X(\mathbf{A})\rangle$ is a complete
lattice, where the join of an arbitrary family $\\{C_{i}\colon i\in I\\}$ of
$E(X(\mathbf{A}))$ is given by

$\bigvee\\{C_{i}\colon i\in I\\}=\bigcap\left\\{G\in
E(X(\mathbf{A}))\colon\bigcup\\{C_{i}\colon i\in I\\}\subseteq
G\right\\}=\Lambda\left(\bigcup\\{C_{i}\colon i\in I\\}\right).$

###### Lemma 3

Let $\mathbf{A}$ be a semilattice. Then:

  1. 1.

$\beta(a)\in E(X(\mathbf{A}))$ for all $a\in A$ and
$C_{\mathcal{K}}(X(\mathbf{A}))\subseteq E(X(\mathbf{A}))$.

  2. 2.

If $a,b\in A$ such that $a\vee b$ exists, then $\beta(a\vee
b)=\beta(a)\vee\beta(b)$.

  3. 3.

Let $D\subseteq A$ be a directed subset. Then

$\bigvee\\{\beta(a)\colon a\in D\\}=\bigcup\\{\beta(a)\colon a\in D\\}.$

In particular, if $U\in\mathcal{Z}(X(\mathbf{A}))$, we have

$U^{c}=\bigcup\\{\beta(a)\colon a\in
I_{\mathbf{A}}(U)\\}=\bigvee\\{\beta(a)\colon a\in I_{\mathbf{A}}(U)\\}.$

###### Proof

1\. It follows from the fact that $\beta(a)^{c}\in\mathcal{Z}(X(\mathbf{A}))$
for all $a\in A$.

2\. Let $a,b\in A$ be such that $a\vee b$ exists. It is clear that
$\beta(a)\vee\beta(b)\subseteq\beta(a\vee b)$. On the other hand, let
$P\in\beta(a\vee b)$ and $U\in\mathcal{Z}(X(\mathbf{A}))$ be such that
$\beta(a)\cup\beta(b)\subseteq U^{c}$. Then $\beta(a)\cap U=\emptyset$ and
$\beta(b)\cap U=\emptyset$. So, $a,b\in I_{\mathbf{A}}(U)$ and $a\vee b\in
I_{\mathbf{A}}(U)$. Thus, $P\cap I_{\mathbf{A}}(U)\neq\emptyset$ and we get
$P\notin U$, i.e, $P\in U^{c}$. Therefore, $P\in\beta(a)\vee\beta(b)$.

3\. Let $D\subseteq A$ be a directed subset. Then $\bigcup\\{\beta(a)\colon
a\in D\\}\subseteq\bigvee\\{\beta(a)\colon a\in D\\}$. Let
$P\in\bigvee\\{\beta(a)\colon a\in D\\}$ and consider the set $(D]$. Then we
have $(D]\in\mathrm{Id}(\mathbf{A})$. Since $\beta(a)\subseteq\alpha((D])^{c}$
for all $a\in D$, then $\bigcup\\{\beta(a)\colon a\in
D\\}\subseteq\alpha((D])^{c}$. Thus, $P\notin\alpha((D])$ and
$P\cap(D]\neq\emptyset$. It follows that there is $a\in D$ such that $a\in P$
which implies $P\in\bigcup\\{\beta(a)\colon a\in D\\}$.

Now we are ready to prove the main result of this section.

###### Theorem 3.2

Let $\mathbf{A}$ be a semilattice. Then the pair $\langle
E(X(\mathbf{A})),\beta\rangle$ is a canonical extension of $\mathbf{A}$.

###### Proof

From Lemma 3, $\beta(a)\in E(X(\mathbf{A}))$ for all $a\in A$ and $\beta$ is
an embedding.

In order to prove that $E(X(\mathbf{A}))$ is dense, let $V\in
E(X(\mathbf{A}))$. Then there exists
$\mathcal{B}\subseteq\mathcal{Z}(X(\mathbf{A}))$ such that
$V=\bigcap\\{U^{c}\colon U\in\mathcal{B}\\}$. From Theorem 3.1, it follows
that

$U^{c}=\left(\bigcap\\{\beta(a)^{c}\colon a\in
I_{\mathbf{A}}(U)\\}\right)^{c}=\bigcup\beta[I_{\mathbf{A}}(U)]=\bigvee\beta[I_{\mathbf{A}}(U)]$

and

$V=\bigcap\left\\{\bigvee\beta[I_{\mathbf{A}}(U)]\colon
U\in\mathcal{B}\right\\}.$

On the other hand, since $V\in\mathrm{Up}(X(\mathbf{A}))$, it follows that
$V=\bigcup\\{\varphi(P)\colon P\in V\\}$. It is easy to check that

$V=\bigcup\\{\varphi(P)\colon P\in V\\}=\bigvee\\{\varphi(P)\colon P\in
V\\}=\bigvee\left\\{\bigcap\\{\beta(P)\colon P\in V\\}\right\\}.$

Finally, we prove that $E(X(\mathbf{A}))$ is compact. Let
$F\in\mathrm{Fi}(\mathbf{A})$ and $I\in\mathrm{Id}(\mathbf{A})$ be such that
$\bigcap\beta[F]\subseteq\bigcup\beta[I]$. So,
$\bigcap\beta[F]\cap\left(\bigcup\beta[I]\right)^{c}=\emptyset$ and by Theorem
2 and Remark 2, we have $F\cap I\neq\emptyset$. Hence, there exists $a\in A$
such that $\beta(a)\in\beta[F]$ and $\beta(a)\in\beta[I]$. The result follows.

As usual, we write $\mathbf{A}^{\sigma}$ to denote the canonical extension of
a semilattice $\mathbf{A}$.

###### Remark 3

Let $\mathbf{A}$ be a semilattice and $\langle
X(\mathbf{A}),\mathcal{K}_{\mathbf{A}}\rangle$ its dual $S$-space. As an
immediate consequence of Theorem 3.1 and Proposition 3.2 of Celani2020 , we
have

$K(\mathbf{A}^{\sigma})=C_{\mathcal{\mathcal{K}_{\mathbf{A}}}}(X(\mathbf{A})){\hskip
8.5359pt}\text{and}{\hskip 8.5359pt}O(\mathbf{A}^{\sigma})=\\{Z^{c}\colon
Z\in\mathcal{Z}(X(\mathbf{A}))\\},$

i.e., the closed elements of the canonical extension of $\mathbf{A}$ are
exactly the subbasic closed subsets of $\langle
X(\mathbf{A}),\mathcal{K}_{\mathbf{A}}\rangle$ and the open elements of the
canonical extension of $\mathbf{A}$ are exactly the complements of the
subbasic saturated subsets of $\langle
X(\mathbf{A}),\mathcal{K}_{\mathbf{A}}\rangle$.

###  3.1 Connection with the construction of Gouveia and Priestley

Canonical extensions of semilattices have been studied in detail by Gouveia
and Priestley in GPsemilattices . This work is connected with the topological
construction of canonical extensions made by Moshier and Jipsen MoshierJipsen
. Their approach places strong emphasis on the iterated filter completion and
the canonical extension: they consider a semilattice $\mathbf{A}$ and the
lattice of filters of filters of $\mathbf{A}$, denoted by
$\operatorname{Fi}^{2}(\mathbf{A})=\operatorname{Fi}(\operatorname{Fi}(\mathbf{A}))$.
The family of sets $\operatorname{Fi}^{2}(\mathbf{A})$ is an algebraic closure
system and forms an algebraic lattice in which arbitrary meet and directed
join are given, respectively, by intersection and directed union. The join of
an arbitrary family of elements $\\{F_{i}\colon i\in I\\}$ is given by

$\displaystyle\bigcap\left\\{\mathcal{G}\in\operatorname{Fi}^{2}(\mathbf{A})\colon\bigcup\\{F_{i}\colon
i\in I\\}\subseteq\mathcal{G}\right\\}.$

In GPsemilattices , the authors define an order embedding $e\colon
A\rightarrow\operatorname{Fi}^{2}(\mathbf{A})$ given by
$e(a)=\\{F\in\operatorname{Fi}(\mathbf{A})\colon a\in F\\}$ for all $a\in A$
and the pair $\langle\operatorname{Fi}^{2}(\mathbf{A}),e\rangle$ is a
completion of $\mathbf{A}$. Since $\operatorname{Fi}^{2}(\mathbf{A})$ may not
be a canonical extension of $\mathbf{A}$, to construct a canonical extension
the authors consider the subset of $\operatorname{Fi}^{2}(\mathbf{A})$
consisting of those of its elements which are meets of directed joins from
$e(A)$, i.e.,

$C=\left\\{\bigcap\bigcup e(I_{j})\colon\\{I_{j}\colon j\in
J\\}\subseteq\operatorname{Id}(\mathbf{A})\right\\}.$

Since the canonical extension is unique up to an isomorphism that fixes
$\mathbf{A}$, there is an isomorphism between the canonical extension given in
GPsemilattices  and the one we have developed in this work from the
topological duality introduced in Celani2020 . We now proceed to the details.
We know by Theorem 3.2 that $\langle E(X(\mathbf{A})),\beta\rangle$ is a
canonical extension for $\mathbf{A}$, where

$E(X(\mathbf{A}))=\left\\{\bigcap\\{U^{c}\colon
U\in\mathcal{B}\\}\colon\mathcal{B}\subseteq\mathcal{Z}(X(\mathbf{A}))\right\\}$

and $\beta\colon A\rightarrow E(X(\mathbf{A}))$ is given by $\beta(a)=\\{P\in
X(\mathbf{A})\colon a\in P\\}$. So, if we define the mapping $i\colon
C\rightarrow E(X(\mathbf{A}))$ by

$i(X)=\\{P\in X(\mathbf{A})\colon P\in X\\}=X\cap X(\mathbf{A})$

then it is easy to prove that $i(X)=\bigcap\\{\alpha(I)^{c}\colon
I\in\operatorname{Id}(\mathbf{A})\text{ and }X\subseteq\bigcup e(I)\\}$ and
$i(e(a))=\beta(a)$. We get that $i$ is an isomorphism of lattices that fixes
$\mathbf{A}$ with inverse $j\colon E(X(\mathbf{A}))\rightarrow C$ given by

$j(Y)=\bigcap\left\\{\bigcup e(I)\colon
I\in\operatorname{Id}(\mathbf{A})\text{ and
}Y\subseteq\alpha(I)^{c}\right\\}.$

It is also worth noting that $i(\bigcup e(I))=\alpha(I)^{c}$ for all
$I\in\operatorname{Id}(\mathbf{A})$ and $i(\bigcap e(F))=\varphi(F)=Y_{F}$ for
all $F\in\operatorname{Fi}(\mathbf{A})$.

###  3.2 Extension of monotone maps

In this section we study canonical extensions of monotone maps of semilattices
by using Theorem 3.2. We start by recalling some results about canonical
extensions of posets. For further details the reader may consult
PalmigianoDunn .

Let $P$ and $Q$ be two posets and $f\colon P\rightarrow Q$ be an order
preserving map. Let $P^{\sigma}$ and $Q^{\sigma}$ be the canonical extensions
of $P$ and $Q$, respectively. We define the maps $f^{\sigma},f^{\pi}\colon
P^{\sigma}\to Q^{\sigma}$ by

$\displaystyle f^{\sigma}(u)$
$\displaystyle=\bigvee\left\\{\bigwedge\\{f(p)\colon x\leq p\in P\\}\colon
u\geq x\in K(P^{\sigma})\right\\},$ (4) $\displaystyle f^{\pi}(u)$
$\displaystyle=\bigwedge\left\\{\bigvee\\{f(p)\colon y\geq p\in P\\}\colon
u\leq y\in O(P^{\sigma})\right\\}.$ (5)

For every order preserving map $f\colon P\to Q$, both $f^{\sigma}$ and
$f^{\pi}$ are order preserving extensions of $f$. In addition, $f^{\sigma}\leq
f^{\pi}$ with equality holding on both $K(P^{\sigma})$ and $O(P^{\sigma})$.
For every $u\in P^{\sigma}$, $x\in K(P^{\sigma})$ and $y\in O(P^{\sigma})$ we
have

$\displaystyle f^{\sigma}(x)$ $\displaystyle=\bigwedge\\{f(p)\colon x\leq p\in
P\\},$ $\displaystyle f^{\sigma}(u)$
$\displaystyle=\bigvee\\{f^{\sigma}(x)\colon u\geq x\in K(P^{\sigma})\\},$
$\displaystyle f^{\pi}(y)$ $\displaystyle=\bigvee\\{f(p)\colon y\geq p\in
P\\},$ $\displaystyle f^{\pi}(u)$ $\displaystyle=\bigwedge\\{f^{\pi}(y)\colon
u\leq y\in O(P^{\sigma})\\}.$

Also, $f^{\sigma}$ and $f^{\pi}$ map closed elements to closed elements and
open elements to open elements. In general the extensions $f^{\sigma}$ and
$f^{\pi}$ do not coincide.

Since every semilattice is a poset, we can borrow the properties described
before to this context. The following result is a straightforward consequence
of Lemma 3.4 of PalmigianoDunn , Theorem 3.2 and Remark 3.

###### Lemma 4

Let $\mathbf{A}$ and $\mathbf{B}$ be two semilattices and $f\colon
A\rightarrow B$ be an order preserving map. Consider the mappings
$f^{\sigma},f^{\pi}\colon\mathbf{A}^{\sigma}\rightarrow\mathbf{B}^{\sigma}$
defined in (4) and (5). Then for every $a\in A$,
$f^{\sigma}(\beta_{\mathbf{A}}(a))=f^{\pi}(\beta_{\mathbf{A}}(a))=\beta_{\mathbf{B}}(f(a))$.
Moreover, $f^{\sigma}\leq f^{\pi}$ with the equality holding in
$K(\mathbf{A}^{\sigma})\cup O(\mathbf{A}^{\sigma})$. For every $Y\in
C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))$, $Z\in\mathcal{Z}(X(\mathbf{A}))$
and $V\in E(X(\mathbf{A}))$ the following hold:

  1. 1.

$\displaystyle f^{\sigma}(Y)=\bigcap\\{\beta_{\mathbf{B}}(f(a))\colon a\in
A\text{ and }Y\subseteq\beta_{\mathbf{A}}(a)\\}$,

  2. 2.

$\displaystyle f^{\sigma}(V)=\bigvee\\{f^{\sigma}(Y)\colon V\supseteq Y\in
C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))\\}$,

  3. 3.

$\displaystyle f^{\pi}(Z^{c})=\bigvee\\{\beta_{\mathbf{B}}(f(a))\colon a\in
A\text{ and }\beta_{\mathbf{A}}(a)\subseteq Z^{c}\\}$,

  4. 4.

$\displaystyle f^{\pi}(V)=\bigcap\\{f^{\pi}(Z^{c})\colon V^{c}\supseteq
Z\in\mathcal{Z}(X(\mathbf{A}))\\}$.

###### Lemma 5

Let $\mathbf{A}$ and $\mathbf{B}$ be two semilattices and $f\colon A\to B$ be
an order preserving map. Then for every $Z\in\mathcal{Z}(X(\mathbf{A}))$ the
subset $\\{\beta_{\mathbf{B}}(f(a))\colon a\in I_{\mathbf{A}}(Z)\\}$ is
directed.

###### Proof

It follows from the fact that $I_{\mathbf{A}}(Z)$ is directed and both $f$ and
$\beta_{\mathbf{B}}$ are monotone.

###### Lemma 6

Let $\mathbf{A}$ and $\mathbf{B}$ be two semilattices and $f\colon
A\rightarrow B$ be an order preserving map. Then for every
$Z\in\mathcal{Z}(X(\mathbf{A}))$ we have

$f^{\pi}(Z^{c})=\\{P\in X(\mathbf{B})\colon f^{-1}[P]\cap
I_{\mathbf{A}}(Z)\neq\emptyset\\}.$

###### Proof

It follows from Lemma 4.

Let $\mathbf{A}$ and $\mathbf{B}$ be two semilattices and $f\colon
A\rightarrow B$ be an order preserving map. We define $R_{f}\subseteq
X(\mathbf{B})\times\mathcal{Z}(X(\mathbf{A}))$ by

$(P,Z)\in R_{f}\Longleftrightarrow f^{-1}[P]\cap I_{\mathbf{A}}(Z)=\emptyset.$

###### Proposition 4

Let $\mathbf{A}$ and $\mathbf{B}$ be two semilattices and $f\colon
A\rightarrow B$ be an order preserving map. Then for every $V\in
E(X(\mathbf{A}))$ we have

$f^{\pi}(V)=\\{P\in X(\mathbf{B})\colon\forall Z\in R_{f}(P)[Z\cap
V\neq\emptyset]\\}.$

###### Proof

Let $V\in E(X(\mathbf{A}))$. In order to prove the first inclusion, let $P\in
f^{\pi}(V)$ and $Z\in R_{f}(P)$. If $Z\cap V=\emptyset$, then $Z\subseteq
V^{c}$. By Lemma 4, it follows that $P\in f^{\pi}(Z^{c})$. So, by Lemma 6,
$f^{-1}[P]\cap I_{\mathbf{A}}(Z)\neq\emptyset$ which contradicts our
assumption. Hence, $Z\cap V\neq\emptyset$. For the remaining inclusion, let
$P\in X(\mathbf{B})$ be such that for every $Z\in R_{f}(P)$ we have $Z\cap
V\neq\emptyset$. Let $Z\in\mathcal{Z}(X(\mathbf{A}))$ be such that $Z\subseteq
V^{c}$. We prove that $f^{-1}[P]\cap I_{\mathbf{A}}(Z)\neq\emptyset$. If we
assume the contrary, then $Z\in R_{f}(P)$ and $Z\cap V\neq\emptyset$ which
gives us a contradiction. Therefore, $P\in f^{\pi}(V)$.

Finally, we analyze $f^{\sigma}$. Note that for every $P\in X(\mathbf{B})$ and
every $Y\in C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))$ we have $P\in
f^{\sigma}(Y)$ if and only if $\psi(Y)\subseteq f^{-1}[P]$ where
$\psi(Y)=\\{a\in A\colon Y\subseteq\beta_{\mathbf{A}}(a)\\}$. So, we can
define the relation $G_{f}\subseteq X(\mathbf{B})\times
C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))$ by

$(P,Y)\in G_{f}\Longleftrightarrow\psi(Y)\subseteq f^{-1}[P].$

###### Proposition 5

Let $\mathbf{A}$ and $\mathbf{B}$ be two semilattices and $f\colon
A\rightarrow B$ be an order preserving map and let $\Lambda$ be the closure
operator defined in (3). Then for every $V\in E(X(\mathbf{A}))$ we have

$f^{\sigma}(V)=\Lambda\left(\bigcup\\{G_{f}^{-1}(Y)\colon V\supseteq Y\in
C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))\\}\right).$

###### Proof

Let $V\in E(X(\mathbf{A}))$ and $P\in f^{\sigma}(P)$. Since
$f^{\sigma}(Y)=G_{f}^{-1}(Y)$ for all $Y\in
C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))$, we have

$\bigcup\\{f^{\sigma}(Y)\colon V\supseteq Y\in
C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))\\}=\bigcup\\{G_{f}^{-1}(Y)\colon
V\supseteq Y\in C_{\mathcal{K}_{\mathbf{A}}}(X(\mathbf{A}))\\}$

so the result follows.

Unlike the case of monotone distributive semilattices, in where the
topological descriptions of both extensions of a monotone map are quite easy
to handle (see menchon2  for details), from Proposition 5 it can be noticed
that in practice, the topological description of the $\sigma$ extension of a
monotone map $f$ between semilattices results more cumbersome to work with
than its $\pi$ extension. Motivated by this fact, in the next section we will
develop our duality by means of the topological description of $f^{\pi}$.

##  4 Topological duality

As we saw along Section 3, the topological description of canonical extensions
of semilattices can be used to obtain topological descriptions of the
canonical extensions of monotone maps of semilattices, namely, the extensions
$\pi$ and $\sigma$. In this section we take advantage of these results in
order to define a suitable category of multirelational topological spaces
which will serve as the categorical dual of the category of monotone
semilattices. The construction of such multirelations strongly relies in a
modeling process raising from the $f^{\pi}$ extension of a monotone function
$f$ between semilattices.

###### Definition 7

Let $\mathbf{A}$ be a semilattice. A map $m\colon A\rightarrow A$ is said to
be _monotone_ if it is an order preserving map, i.e, it satisfies the
following condition: if $a\leq b$, then $ma\leq mb$ for all $a\in A$. Then a
_monotone semilattice_ is a pair $\langle\mathbf{A},m\rangle$ where
$\mathbf{A}$ is a semilattice and $m$ is a monotone operation on $A$.

###### Remark 4

It is immediate that for every semilattice $\mathbf{A}$ a map $m\colon
A\rightarrow A$ is monotone if and only if $m(a\wedge b)\leq ma\wedge mb$ for
all $a,b\in A$.

It is clear from Remark 4 that monotone semilattices form a variety. We write
$\mathcal{MS}$ for the variety of monotone semilattices. Let
$\langle\mathbf{A},m\rangle$ and $\langle\mathbf{B},n\rangle$ be two monotone
semilattices. We say that a homomorphism $h\colon A\rightarrow B$ is a
_monotone homomorphism_ if $h(ma)=nh(a)$ for all $a\in A$. We write
$\mathsf{mMS}$ for the algebraic category of monotone semilattices and
monotone homomorphisms.

A _multirelation_ on set $X$ is a subset of the cartesian product
$X\times\mathcal{P}(X)$, that is, a set of ordered pairs $(x,Y)$ where $x\in
X$ and $Y\subseteq X$ (Duntsch-Orlowska-RewitzkyMultirelations2010 ;
Rewitzky2003 ). Recall that in classical monotone modal logic a neighborhood
frame is a pair $\langle X,R\rangle$ where $X$ is a set and $R\subseteq
X\times\mathcal{P}(X)$, i.e., $R$ is a multirelation (Chellas ; Hansen ).

We will represent a monotone operator $m$ on a semilattice $\mathbf{A}$ as a
multirelation on the dual space of $\mathbf{A}$, where the canonical extension
offers an advantageous point of view. To do so, we will apply the results we
obtained in Section 3.2.

From Lemmas 4, 5 and Lemma 3 we obtain a topological description of the
monotone operators on a semilattice.

###### Lemma 7

Let $\langle\mathbf{A},m\rangle\in\mathcal{MS}$. Then
$\langle\mathbf{A}^{\sigma},m^{\pi}\rangle\in\mathcal{MS}$ and for every $a\in
A$, $m^{\pi}(\beta(a))=\beta(m(a))$. Moreover, for every
$Z\in\mathcal{Z}(X(\mathbf{A}))$ and $V\in E(X(\mathbf{A}))$ the following
hold:

  1. 1.

$\displaystyle m^{\pi}(Z^{c})=\bigcup\\{\beta(ma)\colon
Z\subseteq\beta(a)^{c}\\}$,

  2. 2.

$\displaystyle m^{\pi}(V)=\bigcap\\{m^{\pi}(Z^{c})\colon V^{c}\supseteq
Z\in\mathcal{Z}(X(\mathbf{A}))\\}$.

Let $\langle\mathbf{A},m\rangle\in\mathcal{MS}$. We consider the multirelation
$R_{m}\subseteq X(\mathbf{A})\times\mathcal{Z}(X(\mathbf{A}))$ given by

$(P,Z)\in R_{m}\Longleftrightarrow m^{-1}[P]\cap I_{\mathbf{A}}(Z)=\emptyset.$

As a particular case of Proposition 4 we obtain the following result.

###### Proposition 6

Let $\langle\mathbf{A},m\rangle\in\mathcal{MS}$. Then for every $V\in
E(X(\mathbf{A}))$ we have

$m^{\pi}(V)=\\{P\in X(\mathbf{A})\colon\forall Z\in R_{m}(P)[Z\cap
V\neq\emptyset]\\}.$

Let $\langle X,\mathcal{K}\rangle$ be an $S$-space. For each $U\in S(X)$ we
define the subset $L_{U}$ of $\mathcal{Z}(X)$ as follows:

$L_{U}=\\{Z\in\mathcal{Z}(X)\colon Z\cap U\neq\emptyset\\}.$

###### Definition 8

An _$mS$ -space_ is a structure $\langle X,\mathcal{K},R\rangle$ where
$\langle X,\mathcal{K}\rangle$ is an $S$-space and $R\subseteq
X\times\mathcal{Z}(X)$ a multirelation such that:

  1. 1.

$m_{R}(U)=\\{x\in X\colon\forall Z\in R(x)[Z\cap U\neq\emptyset]\\}\in S(X)$
for all $U\in S(X)$,

  2. 2.

$R(x)=\displaystyle\bigcap\\{L_{U}\colon U\in S(X)\text{ and }x\in
m_{R}(U)\\}$ for all $x\in X$.

Notice that condition 1. of Definition 8 can be restated as

$m_{R}(U)=\\{x\in X\colon R(x)\subseteq L_{U}\\}\in S(X)$

for all $U\in S(X)$. In addition, condition 2. says that for every $mS$-space
$\langle X,\mathcal{K},R\rangle$, the map $m_{R}\colon S(X)\rightarrow S(X)$
is well defined and it is monotone.

Let $\mathsf{Rel}$ be the category of sets and binary relations and
$\mathsf{Set}$ be the category of sets and functions. It is well known that
there exists a faithful functor
$\mathcal{F}\colon\mathsf{Rel}^{op}\rightarrow\mathsf{Set}$ defined for every
set $X$ by $\mathcal{F}(X)=\mathcal{P}(X)$, and for every binary relation
$T\subseteq X\times Y$ and every $U\in\mathcal{P}(Y)$, $\mathcal{F}(T)$ is
defined by

$\mathcal{F}(T)(U)=\\{x\in X\colon T(x)\subseteq U\\}.$

A _meet-relation_ between two $S$-spaces $\langle
X_{1},\mathcal{K}_{1}\rangle$ and $\langle X_{2},\mathcal{K}_{2}\rangle$ was
defined in Celani2020  as a relation $T\subseteq X_{1}\times X_{2}$ satisfying
the following conditions:

  1. 1.

$\mathcal{F}(T)(U)\in S(X_{1})$ for all $U\in S(X_{2})$,

  2. 2.

$T(x)=\displaystyle\bigcap\\{U\in S(X_{2})\colon T(x)\subseteq U\\}$ for all
$x\in X_{1}$.

Let $\langle X_{i},\mathcal{K}_{i}\rangle$, with $i=1,2,3$ be $S$-spaces and
$R\subseteq X_{1}\times X_{2}$ and $T\subseteq X_{2}\times X_{3}$ be meet-
relations. Consider now the following relation

$T\ast R=\\{(x,z)\in X_{1}\times X_{3}\colon\forall U\in S(X_{3})[(T\circ
R)(x)\subseteq U\Rightarrow z\in U]\\}.$

In CelaniCalomino ; Celani2020  it was proved that $T\ast R$ is a meet-
relation, $\ast$ is associative and if $\sqsupseteq_{2}$ is the dual of the
specialization order of $X_{2}$, then $T\ast\sqsupseteq_{2}=T$ and
$\sqsupseteq_{2}\ast R=R$. Observe that the latter can be rephrased by means
that $S$-spaces and meet-relations form a category in which the composition is
given by $\ast$ and the identity arrow is given by the dual of the
specialization order. We write $\mathsf{mRel}$ for such a category. Let
$\mathsf{MS}$ be the category of semilattices and homomorphisms. Let us assign
to each $S$-space $\langle X,\mathcal{K}\rangle$ the semilattice $S(X)$ and to
each meet-relation $T$ the map $\mathcal{F}(T)$. In Proposition 3.16 of
Celani2020  it was proved that $\mathcal{F}(T\ast R)=\mathcal{F}(T\circ R)$
and it is no hard to see that $\mathcal{F}(\sqsupseteq)$ is the identity map
of $S(X)$. Moreover, from the definition of meet-relation, $\mathcal{F}(T)$
and Proposition 3.19 of Celani2020 , it follows that these assignments define
a functor $\Box\colon\mathsf{mRel}^{op}\rightarrow\mathsf{MS}$ which is
faithful by Lemma 3.21 of Celani2020 . We stress that this is an important
operational advantage taking into account that $(T\ast
R)(x)=\mathrm{cl}_{\mathcal{K}_{3}}((T\circ S)(x))$ for all $x\in X_{1}$. In
the literature (CelaniCalomino ; Celani2020 ; menchon2 ) $\Box(T)$ is denoted
by $\Box_{T}$. So for the sake of readability we will keep such a notation
along this paper.

###### Definition 9

Let $\langle X_{1},\mathcal{K}_{1},R_{1}\rangle$ and $\langle
X_{2},\mathcal{K}_{2},R_{2}\rangle$ be two $mS$-spaces. A meet-relation
$T\subseteq X_{1}\times X_{2}$ is a _monotone meet-relation_ if the following
diagram commutes:

$\textstyle{S(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box_{T}}$$\scriptstyle{m_{R_{2}}}$$\textstyle{S(X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{R_{1}}}$$\textstyle{S(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box_{T}}$$\textstyle{S(X_{1})}$

The following result is a topological characterization of monotone meet-
relations. Its proof is essentially the same as that of Lemma 44 in menchon2
so we will omit it.

###### Proposition 7

Let $\langle X_{1},\mathcal{K}_{1},R_{1}\rangle$ and $\langle
X_{2},\mathcal{K}_{2},R_{2}\rangle$ be two $mS$-spaces. A meet-relation
$T\subseteq X_{1}\times X_{2}$ is monotone if and only if for every $x\in
X_{1}$ and every $U\in S(X_{2})$ we have

$U^{c}\in R_{2}[T(x)]\Longleftrightarrow T^{-1}[U^{c}]\in R_{1}(x)$

where $R_{2}[T(x)]=\\{Z\in\mathcal{Z}(X_{2})\colon\exists y\in T(x)[(y,Z)\in
R_{2}]\\}$.

###### Proposition 8

Let $\langle X_{1},\mathcal{K}_{1},R_{1}\rangle$, $\langle
X_{2},\mathcal{K}_{2},R_{2}\rangle$ and $\langle
X_{2},\mathcal{K}_{3},R_{3}\rangle$ be $mS$-spaces. Let $L\subseteq
X_{1}\times X_{2}$ and $T\subseteq X_{2}\times X_{3}$ be monotone meet-
relations. Then $T\ast L$ is a monotone meet-relation. Moreover, $\ast$ is
associative.

###### Proof

Consider the diagram

$\textstyle{S(X_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box_{T}}$$\scriptstyle{m_{R_{3}}}$$\textstyle{S(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{R_{2}}}$$\scriptstyle{\Box_{L}}$$\textstyle{S(X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{R_{1}}}$$\textstyle{S(X_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box_{T}}$$\textstyle{S(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box_{L}}$$\textstyle{S(X_{1})}$

Since $\Box$ is contravariant and each of the squares is commutative by
assumption, then from the following calculation

$m_{R_{1}}\Box_{T\ast
L}=m_{R_{1}}(\Box_{L}\Box_{T})=(\Box_{L}m_{R_{2}})\Box_{T}=\Box_{L}(\Box_{T}m_{R_{3}})=\Box_{T\ast
L}m_{R_{3}}$

we obtain that $T\ast L$ is a meet-relation. The proof of the associativity is
analogue.

###### Proposition 9

Let $\langle X,\mathcal{K},R\rangle$ be an $mS$-space. The dual of the
specialization order $\sqsupseteq\subseteq X\times X$ is a monotone meet-
relation.

###### Proof

Recall that from the functoriality of $\Box$ it follows that
$\Box_{\sqsupseteq}$ is in fact the identity map of $S(X)$. So, it is clear
that $\sqsupseteq$ satisfies Definition 9.

By Propositions 8 and 9 we get that $mS$-spaces and monotone meet-relations
form a category in which the identity map is given by the dual of the
specialization order. We write $\mathsf{mSp}$ for such a category.

###### Lemma 8

Let $\langle X_{1},\mathcal{K}_{1},R_{1}\rangle$ and $\langle
X_{2},\mathcal{K}_{2},R_{2}\rangle$ be two $mS$-spaces and $f\colon
X_{1}\rightarrow X_{2}$ be a bijective map such that

  1. 1.

$\mathcal{K}_{2}=\\{f[V]\colon V\in\mathcal{K}_{1}\\}$,

  2. 2.

$(x,Z)\in R_{1}$ if and only if $(f(x),f[Z])\in R_{2}$ for all $x\in X_{1}$
and all $Z\in\mathcal{Z}(X_{1})$.

Then $f$ induces an isomorphism of $mS$-spaces.

###### Proof

Since $f$ is bijective, we can consider the relations $R_{f}\subseteq
X_{1}\times X_{2}$ and $T_{f}\subseteq X_{2}\times X_{1}$ which are given by

$(x,f(y))\in R_{f}\Longleftrightarrow f(x)\sqsupseteq_{2}f(y)$

and

$(f(y),x)\in T_{f}\Longleftrightarrow f(y)\sqsupseteq_{2}f(x).$

We prove that $R_{f}$ and $T_{f}$ are monotone-meet relations. We only exhibit
the proof for the case of $R_{f}$ because the proof for $T_{f}$ is similar.
Since $f$ is bijective and from 1. we obtain $\mathcal{Z}(X_{2})=\\{f[Z]\colon
Z\in\mathcal{Z}(X_{1})\\}$, $S(X_{2})=\\{f[W]\colon W\in S(X_{1})\\}$ and
$\Box_{R_{f}}(f[U])=U$, for every $U\in S(X_{1})$. Whence, it follows that for
every $x,y\in X_{1}$, $f(x)\sqsupseteq_{2}f(y)$ if and only if
$x\sqsupseteq_{1}y$. Notice that from these facts it is straightforward to see
that $R_{f}$ is a meet-relation. In order to proof that $R_{f}$ is monotone we
need to check that the following diagram commutes:

$\textstyle{S(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box_{R_{f}}}$$\scriptstyle{m_{R_{2}}}$$\textstyle{S(X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{R_{1}}}$$\textstyle{S(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box_{R_{f}}}$$\textstyle{S(X_{1})}$

Now we prove that $m_{R_{1}}(U)=\Box_{R_{f}}(m_{R_{2}}(f[U]))$ for all $U\in
S(X_{1})$. Let $x\in\Box_{R_{f}}(m_{R_{2}}(f[U]))$, then $R_{f}(x)\subseteq
m_{R_{2}}(f[U])$. Thus, for every $y\in X_{1}$ such that
$f(x)\sqsupseteq_{2}f(y)$ we have $f(y)\in m_{R_{2}}(f[U])$. Observe that for
every $Z\in\mathcal{Z}(X_{1})$ such that $(f(y),f[Z])\in R_{2}$, we have
$f[Z]\cap f[U]\neq\emptyset$. Hence, from 2. and since $f$ is bijective by
assumption, we get $y\in m_{R_{1}}(U)$. I.e., for every $y\in X_{1}$, if
$x\sqsupseteq_{1}y$, then $y\in m_{R_{1}}(U)$. But this is equivalent to say
that $x\in m_{R_{1}}(U)$. Thus, $\Box_{R_{f}}(m_{R_{2}}(f[U]))\subseteq
m_{R_{1}}(U)$. The proof of the reverse inclusion is analogue.

Finally, we prove that $T_{f}$ is the inverse of $R_{f}$ in $\mathsf{mSp}$. It
is clear from the definitions of $R_{f}$ and $T_{f}$ that $R_{f}\circ
T_{f}=\sqsupseteq_{2}$ and $T_{f}\circ R_{f}=\sqsupseteq_{1}$. Then, since
$\sqsupseteq_{2}$ is a meet-relation, for every $y\in X_{1}$ we have

$(R_{f}\ast T_{f})(f(y))=\mathrm{cl}_{\mathcal{K}_{2}}((R_{f}\circ
T_{f})(f(y)))=\mathrm{cl}_{\mathcal{K}_{2}}(\sqsupseteq_{2}(f(y)))=\sqsupseteq_{2}(f(y)).$

Therefore, $R_{f}\ast T_{f}=\sqsupseteq_{2}$. The proof for $T_{f}\ast
R_{f}=\sqsupseteq_{1}$ is similar.

###### Corollary 1

Let $\langle X_{1},\mathcal{K}_{1},R_{1}\rangle$ be an $mS$-space and $\langle
X_{2},\mathcal{K}_{2}\rangle$ be an $S$-space. If $f\colon X_{1}\rightarrow
X_{2}$ is a bijective map such that $\mathcal{K}_{2}=\\{f[V]\colon
V\in\mathcal{K}_{1}\\}$, then there exists a multirelation $R_{2}\subseteq
X_{2}\times\mathcal{Z}(X_{2})$ such that $\langle
X_{1},\mathcal{K}_{1},R_{1}\rangle$ and $\langle
X_{2},\mathcal{K}_{2},R_{2}\rangle$ are isomorphic $mS$-spaces.

###### Proof

We start by noticing that from the hypothesis on $f$ it is immediate that
$f[U]\in S(X_{2})$ for all $U\in S(X_{1})$ and $f[Z]\in\mathcal{Z}(X_{2})$ for
all $Z\in\mathcal{Z}(X_{1})$. Therefore, it easily follows that
$L_{f[U]}=f[L_{U}]$ for all $U\in S(X_{1})$. In addition, it is also the case
that for every $V\in S(X_{2})$ there exists a unique $U\in S(X_{1})$ such that
$V=f[U]$. We define $R_{2}\subseteq X_{2}\times\mathcal{Z}(X_{2})$ as

$(f(x),W)\in R_{2}\Longleftrightarrow(x,f^{-1}[W])\in R_{1}.$

From the bijectivity of $f$ and the facts of above it follows that $R_{2}$ is
well defined. In order proof that $\langle X_{2},\mathcal{K}_{2},R_{2}\rangle$
is an $mS$-space, note that since $f^{-1}[f[Z]]=Z$ for all $Z\in R_{1}(x)$,
then we get $f[R_{1}(x)]=R_{2}(f(x))$ for all $x\in X$. It is no hard to see
that $m_{R_{1}}(U)=f^{-1}[m_{R_{2}}(f[U])]$ for all $U\in S(X_{1})$. If $V\in
S(X_{2})$, then

$m_{R_{2}}(V)=m_{R_{2}}(f[U])=f[m_{R_{1}}(U)].$ (6)

Since $\langle X_{1},\mathcal{K}_{1},R_{1}\rangle$ is an $mS$-space, then
$m_{R_{1}}(U)\in S(X_{1})$ and from (6) we conclude $m_{R_{2}}(V)\in
S(X_{2})$. This proofs 1. of Definition 8. On the other hand, as $\langle
X_{1},\mathcal{K}_{1},R_{1}\rangle$ is an $mS$-space and $f$ is bijective, we
get

$\begin{array}[]{ccl}R_{2}(f(x))&=&f[R_{1}(x)]\\\ &=&\displaystyle
f\left[\bigcap\\{L_{U}\colon U\in S(X_{1})\text{ and }x\in
m_{R_{1}}(U)\\}\right]\\\ &=&\displaystyle\bigcap\\{L_{f[U]}\colon U\in
S(X_{1})\text{ and }f(x)\in m_{R_{2}}(f[U])\\}.\\\ \end{array}$

So, 2. of Definition 8 holds. Finally, it is clear from the definition of
$R_{2}$ that condition 2. of Lemma 8 holds. Therefore, $\langle
X_{1},\mathcal{K}_{1},R_{1}\rangle$ and $\langle
X_{2},\mathcal{K}_{2},R_{2}\rangle$ are isomorphic $mS$-spaces, as granted.

We recall that if $\langle X,\mathcal{K}\rangle$ is an $S$-space, then we have
that $\langle X(\mathbf{S}(X)),\mathcal{K}_{\mathbf{S}(X)}\rangle$ is the dual
$S$-space associated to the semilattice $\mathbf{S}(X)$. In Celani2020  it was
proved that the map

$H_{X}\colon X\rightarrow X(\mathbf{S}(X))$

given by $H_{X}(x)=\\{U\in S(X)\colon x\in U\\}$ is a homeomorphism between
$S$-spaces such that $\mathcal{K}_{\mathbf{S}(X)}=\\{H_{X}[U]\colon
U\in\mathcal{K}\\}$. Therefore, as a straightforward consequence of Corollary
1 we obtain:

###### Lemma 9

Let $\langle X,\mathcal{K},R\rangle$ be a $mS$-space. Then the map
$H_{X}\colon X\rightarrow X(\mathbf{S}(X))$ defined by $H_{X}(x)=\\{U\in
S(X)\colon x\in U\\}$ induces an isomorphism of $mS$-spaces.

The proofs of the following two Lemmas are similar to Propositions 30 and 46
of menchon2  with the only difference that in our proofs, Lemma 7 and
Proposition 6 are involved. We leave the details to the reader.

###### Lemma 10

Let $\langle\mathbf{A},m\rangle\in\mathcal{MS}$. Then $\langle
X({\mathbf{A}}),\mathcal{K}_{\mathbf{A}},R_{m}\rangle$ is an $mS$-space.
Moreover, $m_{R_{m}}=m^{\pi}$.

Let $\mathbf{A}$ and $\mathbf{B}$ be two semilattices. Let $h\colon
A\rightarrow B$ be a homomorphism. In Celani2020  it was proved that the
binary relation $R_{h}\subseteq X({\mathbf{B}})\times X({\mathbf{A}})$ defined
by

$(P,Q)\in R_{h}\Longleftrightarrow h^{-1}[P]\subseteq Q$

is a meet-relation such that
$\beta_{\mathbf{B}}h=\Box_{R_{h}}\beta_{\mathbf{A}}$.

###### Lemma 11

Let $\langle\mathbf{A},m\rangle,\langle\mathbf{B},n\rangle\in\mathcal{MS}$ and
$h\colon A\rightarrow B$ be a homomorphism. Then $h$ is a monotone
homomorphism if and only if $R_{h}$ is a monotone meet-relation. In
particular, if $h=id_{\mathbf{A}}$, then $R_{h}=\sqsupseteq_{X(\mathbf{A})}$.

###### Lemma 12

Let
$\langle\mathbf{A},m\rangle,\langle\mathbf{B},n\rangle,\langle\mathbf{C},l\rangle\in\mathcal{MS}$.
If $h\colon A\rightarrow B$ and $g\colon B\rightarrow C$ be monotone
homomorphisms, then $R_{gh}=R_{h}\ast R_{g}$.

###### Proof

From Lemma 11 $R_{gh}$, $R_{h}$ and $R_{g}$ are monotone meet-relations. Then,
since $\Box$ is faithful, for proving our claim it is enough to check
$\Box_{R_{gh}}=\Box_{R_{h}\circ R_{g}}$. But this is essentially the
Proposition 3.22 of Celani2020 .

Notice that Lemmas 10 and 11 allow to define a functor
$G\colon\mathsf{mMS}^{op}\rightarrow\mathsf{mSp}$ as follows:

$\begin{array}[]{ccc}\langle\mathbf{A},m\rangle&\mapsto&\langle
X({\mathbf{A}}),\mathcal{K}_{\mathbf{A}},R_{m}\rangle\\\
h&\mapsto&R_{h}\end{array}$

On the other hand, observe that the assignments

$\begin{array}[]{ccc}\langle X,\mathcal{K},R\rangle&\mapsto&\langle
S(X),m_{R}\rangle\\\ R&\mapsto&\Box_{R}\end{array}$

also define a functor $H\colon\mathsf{mSp}\rightarrow\mathsf{mMS}^{op}$.

Now we state our main theorem.

###### Theorem 4.1

The categories $\mathsf{mSp}$ and $\mathsf{mMS}$ are dually equivalent.

###### Proof

Let $\langle\mathbf{A},m\rangle\in\mathcal{MS}$. From Theorem 3.24 of
Celani2020  and Lemma 7 it follows that the map $\beta\colon A\rightarrow
S(X({\mathbf{A}}))$ defines a natural isomorphism between $H\circ G$ and
$Id_{\mathsf{mMS}^{op}}$. On the other hand, let $\langle
X,\mathcal{K},R\rangle$ be a $mS$-space. Then, from Lemma 9 and again from
Theorem 3.24 of Celani2020 , we get that the map $H_{X}\colon X\rightarrow
X(\mathbf{S}(X))$ leads to a natural isomorphism between $G\circ H$ and
$Id_{\mathsf{mSp}}$. This concludes the proof.

##  5 Characterization of homomorphic images and congruences by means of
lower-Vietoris-type topologies

In CelaniCalomino  Celani and Calomino proved that homomorphic images of a
distributive semilattice are characterized by one-to-one meet-relations and by
lower Vietoris families on the dual space. This characterization was motivated
by the results given in Guram-Jansana II  and by the fact that the class of
distributive semilattices is not a variety. Of course, this is not the case of
semilattices and monotone semilattices. In this section we characterize
congruences of semilattices and monotone semilattices by means of lower-
Vietoris-type topologies and monotone lower-Vietoris-type topologies of their
respective associated dual spaces. To achieve that goal, we start by studying
the existing relation between homomorphic images and lower-Vietoris-type
topologies in the variety of semilattices and later we extend this study to
the variety of monotone semilattices.

###### Definition 10

Let $\langle X_{1},\mathcal{K}_{1}\rangle$ and $\langle
X_{2},\mathcal{K}_{2}\rangle$ be two $S$-spaces. A meet-relation $R\subseteq
X_{1}\times X_{2}$ is _one-to-one_ if for each $x\in X_{1}$ and $U\in
S(X_{1})$ with $x\notin U$, there exists $V\in S(X_{2})$ such that
$U\subseteq\Box_{R}(V)$ and $x\notin\Box_{R}(V)$.

The following result is the $S$-space version of Theorem 36 (2) of
CelaniCalomino . Since its proof is similar, we omit it.

###### Theorem 5.1

Let $\langle X_{1},\mathcal{K}_{1}\rangle$ and $\langle
X_{2},\mathcal{K}_{2}\rangle$ be two $S$-spaces and $R\subseteq X_{1}\times
X_{2}$ be a meet-relation. Then the homomorphism $\Box_{R}\colon
S(X_{2})\rightarrow S(X_{1})$ is onto if and only if $R$ is one-to-one.

###### Corollary 2

Let $\mathbf{A}$ and $\mathbf{B}$ be two semilattices and $h\colon
A\rightarrow B$ be a homomorphism. Then $h$ is onto if and only if the meet-
relation $R_{h}\subseteq X(\mathbf{B})\times X(\mathbf{A})$ is one-to-one.

Let $\mathbf{A}$ be a semilattice and $\langle X,\mathcal{K}\rangle$ be an
$S$-space. Consider a one-to-one meet-relation $R\subseteq X\times
X(\mathbf{A})$ and let $\mathcal{F}_{R}=\\{R(x)\colon x\in X\\}$. It is clear
that $\mathcal{F}_{R}\subseteq C_{\mathcal{K}}(X(\mathbf{A}))$. For every
$a\in A$ we consider the set

$H_{a}=\\{R(x)\colon R(x)\cap\beta(a)^{c}\neq\emptyset\\}.$

###### Remark 5

Notice that for every $a\in A$ and $x\in X$ the following holds

$R(x)\in H_{a}\Longleftrightarrow x\in\Box_{R}(\beta(a))^{c}.$

In Ivanova-dimova , Ivanova-Dimova introduces a new lower-Vietoris-type
hypertopology that is a generalized version of the lower Vietoris topology.

###### Definition 11

(Ivanova-dimova ) Let $X$ be a set. Let $\mathcal{F}$ be a family of non-empty
subsets of $X$ and $\mathcal{O}$ be a topology on $\mathcal{F}$. For each
$U\subseteq X$ let us consider

$U^{-}_{\mathcal{F}}=\\{Y\in\mathcal{F}\colon Y\cap U\neq\emptyset\\}.$

Let $\mathcal{M}_{\mathcal{F}}=\\{U^{-}_{\mathcal{F}}\colon U\subseteq X\\}$.
We say that $\mathcal{O}$ is a _lower-Vietoris-type topology_ on $\mathcal{F}$
if $\mathcal{O}\cap\mathcal{M}_{\mathcal{F}}$ is a subbase for $\mathcal{O}$.

###### Proposition 10

Let $\mathbf{A}$ be a semilattice and $\langle X,\mathcal{K}\rangle$ be an
$S$-space. Let $R\subseteq X\times X(\mathbf{A})$ be a one-to-one meet-
relation. Then:

  1. 1.

$R(x)\neq\emptyset$ for all $x\in X$.

  2. 2.

The family $\mathcal{M}=\\{H_{a}\colon a\in A\\}$ is a subbasis for a topology
on $\mathcal{F}_{R}$.

  3. 3.

The topology $\tau_{\mathcal{M}}$ generated by $\mathcal{M}$ is a lower-
Vietoris-type topology on $\mathcal{F}_{R}$.

###### Proof

We only need to prove 1. since the proof of 2. is analogue to the proof of
Lemma 38 of CelaniCalomino  and 3. is a consequence of 2. and Corollary 2.11
of Ivanova-dimova . Let $x\in X$. Since $\mathcal{K}$ is a subbasis, there
exists $U\in S(X)$ such that $x\notin U$. As $R$ is a one-to-one meet-
relation, there exists $a\in A$ such that $U\subseteq\Box_{R}(\beta(a))$ and
$x\notin\Box_{R}(\beta(a))$. Therefore, from Remark 5, $R(x)\in H_{a}$ and
this implies that $R(x)\neq\emptyset$.

###### Remark 6

We would like to stress that although the proof of Proposition 10 (2) is
similar to the proof of Lemma 38 in CelaniCalomino  when considering S-spaces,
the family $\mathcal{M}$ in general is not a basis. To check this, let us
consider the following non-distributive lattice $\mathbf{L}$ together with the
identity map $id_{\bf{L}}\colon L\rightarrow L$:

$a$$0$$b$$c$$e$$d$$1$$\mathbf{L}$$P_{1}$$P_{2}$$P_{3}$$P_{4}$$X(\mathbf{L})$

It is no hard to see that $X(\mathbf{L})=\\{P_{1},P_{2},P_{3},P_{4}\\}$, where
$P_{1}=\\{a,e,1\\}$, $P_{2}=\\{b,e,1\\}$, $P_{3}=\\{c,e,d,1\\}$ and
$P_{4}=\\{d,1\\}$. It follows that $(P,Q)\in R_{id_{\mathbf{L}}}$ if and only
if $P\subseteq Q$. Therefore,
$\mathcal{F}_{R_{id_{\mathbf{L}}}}=\\{\\{P_{1}\\},\\{P_{2}\\},\\{P_{3}\\},\\{P_{3},P_{4}\\}\\}$.
Notice that $H_{0}=\mathcal{F}_{R_{id_{\mathbf{L}}}}$,
$H_{a}=\\{\\{P_{2}\\},\\{P_{3}\\},\\{P_{3},P_{4}\\}\\}$,
$H_{b}=\\{\\{P_{1}\\},\\{P_{3}\\},\\{P_{3},P_{4}\\}\\}$,
$H_{c}=\\{\\{P_{1}\\},\\{P_{2}\\},\\{P_{3},P_{4}\\}\\}$,
$H_{d}=\\{\\{P_{1}\\},\\{P_{2}\\}\\}$, $H_{e}=\\{\\{P_{3},P_{4}\\}\\}$ and
$H_{1}=\emptyset$. Nevertheless, $H_{a}\cap
H_{d}=\\{\\{P_{2}\\}\\}\notin\mathcal{M}$. Whence $\mathcal{M}$ is not a
basis.

We omit the proof of the following Lemma because it is similar to the proof of
Lemma 39 (3) in CelaniCalomino .

###### Lemma 13

Let $\mathbf{A}$ be a semilattice and $\langle X,\mathcal{K}\rangle$ be an
$S$-space. Let $R\subseteq X\times X(\mathbf{A})$ be a one-to-one meet-
relation and let $\varphi$ be the map defined in (1). Then
$Y\subseteq\mathcal{F}_{R}$ is a subbasic closed subset of
$\langle\mathcal{F}_{R},\mathcal{M}\rangle$ if and only if there exists
$F\in\operatorname{Fi}(\mathbf{A})$ such that $Y=\\{R(x)\colon
R(x)\subseteq\varphi(F)\\}$.

###### Theorem 5.2

Let $\mathbf{A}$ be a semilattice and $\langle X,\mathcal{K}\rangle$ be an
$S$-space. Let $R\subseteq X\times X(\mathbf{A})$ be a one-to-one meet-
relation. Then $\langle\mathcal{F}_{R},\mathcal{M}\rangle$ is an $S$-space.

###### Proof

We prove the conditions of Definition 3. By Proposition 10 (2) the family
$\mathcal{M}$ is a subbasis for a topology on $\mathcal{F}_{R}$. We see that
$\langle\mathcal{F}_{R},\mathcal{M}\rangle$ is $T_{0}$. Let $x,y\in X$ be such
that $R(x)\neq R(y)$. Since $\langle X,\mathcal{K}\rangle$ is an $S$-space,
there exists $U\in S(X)$ such that $x\in U$ and $y\notin U$. As $R$ is a one-
to-one meet-relation, there exists $a\in A$ such that
$U\subseteq\Box_{R}(\beta(a))$ and $y\notin\Box_{R}(\beta(a))$. From Remark 5
it follows that $R(y)\in H_{a}$ and $R(x)\notin H_{a}$. So, ($S_{1}$) is
satisfied.

We see ($S_{2}$). It is clear that $H_{1}=\emptyset$ and $\mathcal{M}$ is
closed under finite unions. Let $a\in A$ and consider $H_{a}$. Observe that as
a consequence of Remark 5, we have $H_{a}\subseteq\bigcup\\{H_{b}\colon b\in
B\subseteq A\\}$ if and only if
$\Box_{R}(\beta(a))^{c}\subseteq\bigcup\\{\Box_{R}(\beta(b))^{c}\colon b\in
B\subseteq A\\}$. Since $\Box_{R}(\beta(a))^{c}$ is a compact subset, by
Alexander’s subbasis Theorem we get that $H_{a}$ is compact.

Notice that ($S_{3}$) follows from Remark 5 and from the assumption that
$\langle X,\mathcal{K}\rangle$ is an $S$-space.

Finally, we prove ($S_{4}$). Let $Y\in C_{\mathcal{M}}(\mathcal{F}_{R})$ and
$\mathcal{J}\subseteq S(\mathcal{F}_{R})$ be a $Y$-family such that $Y\cap
A^{c}\neq\emptyset$ for all $A\in\mathcal{J}$. We show that
$Y\cap\bigcap\\{A^{c}\colon A\in\mathcal{J}\\}\neq\emptyset$. By Lemma 13
there exists $F\in\operatorname{Fi}(\mathbf{A})$ such that $Y=\\{R(x)\colon
R(x)\subseteq\varphi(F)\\}$. We consider $G=\bigcap\\{\Box_{R}(\beta(f))\colon
f\in F\\}$ which is a subbasic closed subset of $S(X(\mathbf{A}))$ and
$\mathcal{I}=\\{\Box_{R}(\beta(a))\colon H_{a}^{c}\in\mathcal{J}\\}$.

Let us see that $\mathcal{I}$ is a $G$-family of $S(X)$. Let
$H_{a}^{c},H_{b}^{c}\in\mathcal{J}$ be such that
$\Box_{R}(\beta(a)),\Box_{R}(\beta(b))\in\mathcal{I}$. Since $\mathcal{J}$ is
a $Y$-family, there exist $d,e\in A$ such that $H_{e}^{c}\in\mathcal{J}$,
$Y\subseteq H_{d}^{c}$, $H_{a}^{c}\cap H_{d}^{c}\subseteq H_{e}^{c}$ and
$H_{b}^{c}\cap H_{d}^{c}\subseteq H_{e}^{c}$. From $Y\subseteq H_{d}^{c}$ we
get that $\bigcap\\{\Box_{R}(\beta(f))\colon f\in
F\\}\subseteq\Box_{R}(\beta(d))$, i.e., $G\subseteq\Box_{R}(\beta(d))$.
Moreover, $\Box_{R}(\beta(e))\in\mathcal{I}$. By Remark 5,
$\Box_{R}(\beta(a))\cap\Box_{R}(\beta(d))\subseteq\Box_{R}(\beta(e))$ and
$\Box_{R}(\beta(b))\cap\Box_{R}(\beta(d))\subseteq\Box_{R}(\beta(e))$ and we
get that $\mathcal{I}$ is a $G$-family. Now we prove
$G\cap\bigcap\\{A^{c}\colon A\in\mathcal{I}\\}\neq\emptyset$. Suppose that
$G\cap\bigcap\\{A^{c}\colon A\in\mathcal{I}\\}=\emptyset$. So, there exists
$H_{a}^{c}\in\mathcal{J}$ such that $G\cap\Box_{R}(\beta(a))^{c}=\emptyset$.
Then $G\subseteq\Box_{R}(\beta(a))$ and again by Remark 5 we have $Y\cap
H_{a}=\emptyset$ which is a contradiction. So, $G\cap\bigcap\\{A^{c}\colon
A\in\mathcal{I}\\}\neq\emptyset$.

Suppose now that $Y\cap\bigcap\\{A^{c}\colon A\in\mathcal{J}\\}=\emptyset$.
Then $Y\subseteq\bigcup\\{H_{a}^{c}\colon H_{a}^{c}\in\mathcal{J}\\}$. By
Remark 5 we get $\bigcap\\{\Box_{R}(\beta(f))\colon f\in
F\\}\subseteq\bigcup\\{\Box_{R}(\beta(a))\colon H_{a}^{c}\in\mathcal{J}\\}$,
i.e., $G\subseteq\bigcup\\{A\colon A\in\mathcal{I}\\}$ which is a
contradiction. Thus, $Y\cap\bigcap\\{A^{c}\colon
A\in\mathcal{J}\\}\neq\emptyset$. Therefore,
$\langle\mathcal{F}_{R},\mathcal{M}\rangle$ is an $S$-space as claimed.

###### Corollary 3

Let $\mathbf{A}$ and $\mathbf{B}$ be two semilattices and
$h\colon\mathbf{A}\rightarrow\mathbf{B}$ be an onto homomorphism. Then
$\langle\mathcal{F}_{R_{h}},\mathcal{M}\rangle$ is an $S$-space which is
homeomorphic to $\langle X(\mathbf{B}),\mathcal{K}_{\mathbf{B}}\rangle$.

###### Proof

It follows from Theorem 5.2 and an adaptation of the proof of Lemma 40 of
CelaniCalomino .

###### Lemma 14

Let $\langle X,\mathcal{K}\rangle$ be an $S$-space and $\mathcal{F}$ be a
family of non-empty subbasic closed subsets of $\langle X,\mathcal{K}\rangle$.
Then the family $\mathcal{M}_{\mathcal{F}}$ is a subbasis for a topology on
$\mathcal{F}$.

###### Proof

Let $Y\in\mathcal{F}$. Then there exists $x\in X$ such that $x\in Y$. Since
$\mathcal{K}$ is a subbase, there exists $U\in\mathcal{K}$ such that $x\in U$.
Then $Y\in U^{-}_{\mathcal{F}}$ and
$\mathcal{F}=\bigcup\\{U^{-}_{\mathcal{F}}\colon U\in\mathcal{K}\\}$.

###### Definition 12

Let $\langle X,\mathcal{K}\rangle$ be an $S$-space. A family $\mathcal{F}$ of
non-empty subbasic closed subsets of $\langle X,\mathcal{K}\rangle$ is a
_lower-Vietoris-type family_ if the pair
$\langle\mathcal{F},\mathcal{M}_{\mathcal{F}}\rangle$ is an $S$-space where
$\mathcal{M}_{\mathcal{F}}=\\{U^{-}_{\mathcal{F}}\colon U\in\mathcal{K}\\}$.

We write $\mathcal{V}(X)$ for the set of lower-Vietoris-type families of
$\langle X,\mathcal{K}\rangle$.

###### Remark 7

If $\mathcal{F}$ is a lower-Vietoris-type family of an $S$-space $\langle
X,\mathcal{K}\rangle$, then $\mathbf{S}(\mathcal{F})=\langle
S(\mathcal{F}),\cap,\mathcal{F}\rangle$ is a semilattice where
$S(\mathcal{F})=\\{(U^{-}_{\mathcal{F}})^{c}\colon U\in\mathcal{K}\\}$.

Let $\mathbf{A}$ be a semilattice. If $\mathcal{F}\subseteq
C_{\mathcal{K}}(X(\mathbf{A}))$ is a lower-Vietoris-type family of the dual
$S$-space $\langle X(\mathbf{A}),\mathcal{K}_{\mathbf{A}}\rangle$, then we can
define a relation $R_{\mathcal{F}}\subseteq\mathcal{F}\times X(\mathbf{A})$ by

$(Y,P)\in R_{\mathcal{F}}\Longleftrightarrow P\in Y.$

We stress that the proofs of the following three results can be performed in
the same way that the proofs of Lemmas 42, 43 and Theorem 44 of CelaniCalomino
. Therefore we omit them.

###### Lemma 15

Let $\mathbf{A}$ be a semilattice and $\mathcal{F}\subseteq
C_{\mathcal{K}}(X(\mathbf{A}))$ be a lower-Vietoris-type family of the dual
$S$-space of $\mathbf{A}$. Then $R_{\mathcal{F}}\subseteq\mathcal{F}\times
X(\mathbf{A})$ is a one-to-one meet-relation.

###### Lemma 16

Let $\mathbf{A}$ be a semilattice and $\langle X,\mathcal{K}\rangle$ be an
$S$-space. Then:

  1. 1.

If $R\subseteq X\times X(\mathbf{A})$ is a one-to-one meet-relation, then for
every $x\in X$ and $P\in X(\mathbf{A})$ we have

$(x,P)\in R\Longleftrightarrow(R(x),P)\in R_{\mathcal{F}_{R}}.$

  2. 2.

If $\mathcal{F}\subseteq C_{\mathcal{K}}(X(\mathbf{A}))$ is a lower-Vietoris-
type family of the dual $S$-space of $\mathbf{A}$, then
$\mathcal{F}=\mathcal{F}_{R_{\mathcal{F}}}$.

###### Theorem 5.3

Let $\mathbf{A}$ be a semilattice. Then the homomorphic images of $\mathbf{A}$
are characterized by lower-Vietoris-type families of the dual $S$-space of
$\mathbf{A}$.

Since the class of semilattices is a variety, then by general facts from
universal algebra (see II.10.9 in Sanka ) it is known that every semilattice
is the homomorphic image of a freely generated free semilattice. This
observation allows us to obtain the following theorem.

###### Theorem 5.4

Every semilattice can be represented as an $S$-space associated to a lower-
Vietoris-type family.

We emphasize that it is also the case that in the class of semilattices,
homomorphic images are in correspondence with congruences. In what follows, we
present an explicit characterization of congruences of semilattices as lower-
Vietoris-type families: let $\mathbf{A}$ be a semilattice and $\theta\subseteq
A\times A$ be a congruence on $\mathbf{A}$. Let us consider the natural
homomorphism $q_{\theta}\colon A\to A/\theta$ assigning to every $a\in A$ the
equivalence class $q_{\theta}(a)=a/\theta$. Since $q_{\theta}$ is onto, by
Theorem 5.1 we have that $R_{q_{\theta}}\subseteq X(\mathbf{A}/\theta)\times
X(\mathbf{A})$ is a meet-relation one-to-one. Then

$\mathcal{F}_{R_{q_{\theta}}}=\\{R_{q_{\theta}}(Q)\colon Q\in
X(\mathbf{A}/\theta)\\}\subseteq C_{\mathcal{K}}(X(\mathbf{A}))$

is a lower-Vietoris-type family of the dual $S$-space $\langle
X(\mathbf{A}),\mathcal{K}_{\mathbf{A}}\rangle$, i.e., the structure
$\langle\mathcal{F}_{R_{q_{\theta}}},\mathcal{M}\rangle$ is an $S$-space.

###### Theorem 5.5

Let $\mathbf{A}$ be a semilattice and $\mathcal{F}\subseteq
C_{\mathcal{K}}(X(\mathbf{A}))$ be a lower-Vietoris-type family of the dual
$S$-space of $\mathbf{A}$. Consider the relation

$\theta_{\mathcal{F}}=\\{(a,b)\in
A^{2}\colon[\beta(a)^{c}]^{-}_{\mathcal{F}}=[\beta(b)^{c}]^{-}_{\mathcal{F}}\\}.$
(7)

  1. 1.

Then $\theta_{\mathcal{F}}$ is a congruence on $\mathbf{A}$. In particular, we
have that $\theta_{\mathcal{F}}$ is the kernel of the homomorphism
$\Box_{R_{\mathcal{F}}}\beta\colon A\rightarrow S(\mathcal{F})$.

  2. 2.

Then $Y\in\mathcal{F}$ if and only if there exists $Q\in
X(\mathbf{A}/\theta_{\mathcal{F}})$ such that
$Y=R_{{q_{\theta_{\mathcal{F}}}}}(Q)$. Moreover,
$\mathcal{F}=\mathcal{F}_{R_{{q_{\theta_{\mathcal{F}}}}}}$.

###### Proof

1\. It is routine.

2\. Let $Y\in\mathcal{F}$ and let $\psi$ be the map defined in (2). Let us
consider the set $Q=q_{\theta_{\mathcal{F}}}[\psi(Y)]$. Then
$q_{\theta_{\mathcal{F}}}(a)\in Q$ if and only if $a\in\psi(Y)$. It follows
that $Q\in\operatorname{Fi}(\mathbf{A}/\theta_{\mathcal{F}})$ and since
$\mathcal{F}$ is a lower-Vietoris-type family of $\langle
X(\mathbf{A}),\mathcal{K}_{\mathbf{A}}\rangle$, we have that
$\langle\mathcal{F},\mathcal{M}\rangle$ is an $S$-space and satisfies
condition (S3). Then it follows $Q\in X(\mathbf{A}/\theta_{\mathcal{F}})$ and
$P\in R_{q_{\theta_{\mathcal{F}}}}(Q)$ if and only if $\psi(Y)\subseteq P$.
Then $P\in Y$ and we get that $R_{q_{\theta_{\mathcal{F}}}}(Q)=Y$. Conversely,
suppose there exists $Q\in X(\mathbf{A}/\theta_{\mathcal{F}})$ such that
$Y=R_{{q_{\theta_{\mathcal{F}}}}}(Q)$. We consider the homeomorphism
$H_{\mathcal{F}}\colon\mathcal{F}\to X(\mathbf{S}(\mathcal{F}))$ between
$S$-spaces and let
$\delta\colon\mathbf{A}/\theta_{\mathcal{F}}\to\mathbf{S}(\mathcal{F})$ be the
isomorphism between semilattices given by
$\delta(a/\theta_{\mathcal{F}})=\\{Y\in\mathcal{F}\colon
Y\cap\beta(a)^{c}=\emptyset\\}$. Therefore,
$Y=H_{\mathcal{F}}^{-1}[\delta(Q)]\in\mathcal{F}$.

###### Theorem 5.6

Let $\mathbf{A}$ be a semilattice and $\theta$ be a congruence on
$\mathbf{A}$. Then $\theta=\theta_{\mathcal{F}_{R_{q_{\theta}}}}$.

###### Proof

Let $(a,b)\in\theta$. If
$Y\in[\beta(a)^{c}]^{-}_{\mathcal{F}_{R_{q_{\theta}}}}$, then there exists
$Q\in X(\mathbf{A}/\theta)$ such that $Y=R_{q_{\theta}}(Q)$ and
$R_{q_{\theta}}(Q)\cap\beta(a)^{c}\neq\emptyset$. So, there is $P\in
X(\mathbf{A})$ such that $(Q,P)\in R_{q_{\theta}}$ and $P\notin\beta(a)$,
i.e., $q_{\theta}^{-1}[Q]\subseteq P$ and $a\notin P$. It follows that
$q_{\theta}(a)\notin Q$ and since $(a,b)\in\theta$, we have
$q_{\theta}(b)\notin Q$. Then $b\notin q_{\theta}^{-1}[Q]$ and by Theorem 2.2
there exists $P^{\prime}\in X(\mathbf{A})$ such that
$q_{\theta}^{-1}[Q]\subseteq P^{\prime}$ and $b\notin P^{\prime}$. Thus,
$P^{\prime}\in R_{q_{\theta}}(Q)\cap\beta(b)^{c}$ and
$R_{q_{\theta}}(Q)=Y\in[\beta(b)^{c}]^{-}_{\mathcal{F}_{R_{q_{\theta}}}}$. The
other inclusion is similar. Hence,
$[\beta(a)^{c}]^{-}_{\mathcal{F}_{R_{q_{\theta}}}}=[\beta(b)^{c}]^{-}_{\mathcal{F}_{R_{q_{\theta}}}}$
and $(a,b)\in\theta_{\mathcal{F}_{R_{q_{\theta}}}}$. Conversely, let
$(a,b)\in\theta_{\mathcal{F}_{R_{q_{\theta}}}}$. Suppose that
$(a,b)\notin\theta$. Then $q_{\theta}(a)\neq q_{\theta}(b)$ and by Theorem 2.2
there exists $Q\in X(\mathbf{A}/\theta)$ such that $a\in q_{\theta}^{-1}[Q]$
and $b\notin q_{\theta}^{-1}[Q]$. Again by Theorem 2.2, there exists $P\in
X(\mathbf{A})$ such that $q_{\theta}^{-1}[Q]\subseteq P$ and $b\notin P$,
i.e., $P\in R_{q_{\theta}}(Q)\cap\beta(b)^{c}$ and
$R_{q_{\theta}}(Q)\in[\beta(b)^{c}]^{-}_{\mathcal{F}_{R_{q_{\theta}}}}$. Then,
by assumption, we have
$R_{q_{\theta}}(Q)\in[\beta(a)^{c}]^{-}_{\mathcal{F}_{R_{q_{\theta}}}}$ and
$R_{q_{\theta}}(Q)\cap\beta(a)^{c}\neq\emptyset$. So, there is $P^{\prime}\in
X(\mathbf{A})$ such that $q_{\theta}^{-1}[Q]\subseteq P^{\prime}$ and $a\notin
P^{\prime}$. On the other hand, since $a\in q_{\theta}^{-1}[Q]$ it follows
that $a\in P^{\prime}$, which is a contradiction. Therefore, $(a,b)\in\theta$
and $\theta=\theta_{\mathcal{F}_{R_{q_{\theta}}}}$.

Observe that Theorems 5.5 and 5.6 suggest the following result.

###### Theorem 5.7

Let $\mathbf{A}$ be a semilattice. Then the congruences on $\mathbf{A}$ are
characterized by lower-Vietoris-type families of the dual $S$-space of
$\mathbf{A}$.

Something more can be said about the structure of the lower-Vietoris-type
families of an $S$-space. Let $\langle X,\mathcal{K}\rangle$ be an $S$-space
and let $\mathcal{F}_{1},\mathcal{F}_{2}\in\mathcal{V}(X)$. Now let us regard
the following relation on $\mathcal{V}(X)$:

$\begin{array}[]{ccc}\mathcal{F}_{1}\leq\mathcal{F}_{2}\Longleftrightarrow\forall
U,V\in\mathcal{K}[U^{-}_{\mathcal{F}_{2}}=V^{-}_{\mathcal{F}_{2}}\Rightarrow
U^{-}_{\mathcal{F}_{1}}=V^{-}_{\mathcal{F}_{1}}].\end{array}$

###### Theorem 5.8

Let $\langle X,\mathcal{K}\rangle$ be an $S$-space. Then
$\langle\mathcal{V}(X),\leq\rangle$ is a complete lattice which is dually
isomorphic to the set of all congruences of $\mathbf{S}(X)$.

###### Proof

Let $\mathrm{Con}(\mathbf{S}(X))$ be the set of all congruences of
$\mathbf{S}(X)$. We start by noticing that from Theorem 5.7 and Theorem 3.24
of Celani2020 , the maps
$\mathrm{Con}(\mathbf{S}(X))\rightarrow\mathcal{V}(X)$, defined by
$\theta\mapsto\mathcal{F}_{\theta}$ and
$\mathcal{V}(X)\rightarrow\mathrm{Con}(\mathbf{S}(X))$, defined by
$\mathcal{F}\mapsto\theta_{\mathcal{F}}$, are mutually inverse of each other.
Then from Theorem 5.5, it is immediate that
$\mathcal{F}_{1}\leq\mathcal{F}_{2}$ if and only if
$\theta_{\mathcal{F}_{2}}\subseteq\theta_{\mathcal{F}_{1}}$. Hence, by general
reasons $\langle\mathcal{V}(X),\leq\rangle$ is a poset which is dually
isomorphic to the poset $\langle\mathrm{Con}(\mathbf{S}(X)),\subseteq\rangle$.
Moreover, since $\mathrm{Con}(\mathbf{S}(X))$ is a complete lattice then if
$\\{\mathcal{F}_{i}\colon i\in I\\}$ is a family of lower-Vietoris-type
families of $\langle X,\mathcal{K}\rangle$, the latter allows us to define:

$\begin{array}[]{ccc}\displaystyle\bigvee\\{\mathcal{F}_{i}\colon i\in
I\\}&=&\mathcal{F}_{\bigcap\\{\theta_{\mathcal{F}_{i}}\colon i\in
I\\}}\end{array}$

and

$\begin{array}[]{ccc}\displaystyle\bigwedge\\{\mathcal{F}_{i}\colon i\in
I\\}&=&\mathcal{F}_{\bigvee\\{\theta_{\mathcal{F}_{i}}\colon i\in
I\\}}.\end{array}$

This concludes the proof.

Now we provide a characterization of the congruences of monotone semilattices
by means of lower-Vietoris-type families. Unless necessary, we will omit the
most of the proofs of the following results due to essentially they use the
same ideas employed for the case of semilattices.

###### Definition 13

Let $\langle X_{1},\mathcal{K}_{1},R_{1}\rangle$ and $\langle
X_{2},\mathcal{K}_{2},R_{2}\rangle$ be two $mS$-spaces. A monotone meet-
relation $T\subseteq X_{1}\times X_{2}$ is _one-to-one_ if it is one-to-one as
a meet-relation.

###### Theorem 5.9

Let $\langle X_{1},\mathcal{K}_{1},R_{1}\rangle$ and $\langle
X_{2},\mathcal{K}_{2},R_{2}\rangle$ be two $mS$-spaces. Let $T\subseteq
X_{1}\times X_{2}$ be a monotone meet-relation. Then the monotone homomorphism
$\Box_{T}\colon S(X_{2})\rightarrow S(X_{1})$ is onto if and only if $T$ is a
one-to-one monotone meet-relation.

###### Corollary 4

Let $\langle\mathbf{A},m\rangle,\langle\mathbf{B},n\rangle\in\mathcal{MS}$ and
$h\colon A\rightarrow B$ be a monotone homomorphism. Then $h$ is onto if and
only if $R_{h}$ is a one-to-one monotone meet-relation.

In the conditions of Corollary 4, we denote by $\lambda$ the homeomorphism of
Corollary 3 between the $S$-spaces
$\langle\mathcal{F}_{R_{h}},\mathcal{M}\rangle$ and $\langle
X(\mathbf{B}),\mathcal{K}_{\mathbf{A}}\rangle$, i.e., $\lambda\colon
X(\mathbf{B})\to\mathcal{F}_{R_{h}}$ is the map defined by

$\lambda(P)=R_{h}(P)$

for all $P\in X(\mathbf{B})$.

###### Theorem 5.10

Let $\langle\mathbf{A},m\rangle,\langle\mathbf{B},n\rangle\in\mathcal{MS}$ and
$h\colon A\rightarrow B$ be an onto monotone homomorphism. Let
$T\subseteq\mathcal{F}_{R_{h}}\times\mathcal{Z}(\mathcal{F}_{R_{h}})$ be the
relation given by

$(R_{h}(Q),Z)\in T\Longleftrightarrow(Q,\lambda^{-1}[Z])\in R_{n}$

where $Q\in X(\mathbf{B})$ and $Z\in\mathcal{Z}(\mathcal{F}_{R_{h}})$. Then
$\langle\mathcal{F}_{R_{h}},\mathcal{M},T\rangle$ is an $mS$-space which is
homeomorphic to $\langle X(\mathbf{B}),\mathcal{K}_{\mathbf{B}},R_{n}\rangle$
and it satisfies the following condition

$\beta(a)^{c}\in R_{m}[R_{h}(P)]\Longleftrightarrow H_{a}\in T(R_{h}(P))$

for all $a\in A$ and $P\in X(\mathbf{B})$.

###### Proof

We recall that in Corollary 3 it is proved that
$\mathcal{M}=\\{\lambda[V]\colon V\in\mathcal{K}_{\mathbf{B}}\\}$. Moreover,
we stress that $T$ is precisely the relation claimed in Corollary 1. Hence,
the first result follows.

Since $R_{h}$ is a monotone meet-relation, we have

$\beta(a)^{c}\in R_{m}[R_{h}(P)]\Longleftrightarrow
R_{h}^{-1}[\beta(a)^{c}]\in R_{n}(P)$

for all $a\in A$ and $P\in X(\mathbf{B})$. It is easy to see that
$\lambda[R_{h}^{-1}[\beta(a)^{c}]]=H_{a}$. So, by definition of $T$ we have
that $R_{h}^{-1}[\beta(a)^{c}]\in R_{n}(P)$ if and only if $H_{a}\in
T(R_{h}(P))$.

In this part we study the congruences of monotone semilattices. To achieve
this goal, let $\langle X,\mathcal{K},R\rangle$ be an $mS$-space and let
$\mathcal{F}\in\mathcal{V}(X)$. For a subset $H\subseteq\mathcal{Z}(X)$ we
consider the set

$[H\cap\mathcal{K})_{\mathcal{F}}=\\{V\in\mathcal{K}\colon\exists U\in
H\cap\mathcal{K}\;[U^{-}_{\mathcal{F}}\subseteq V^{-}_{\mathcal{F}}]\\}.$

We say that a subset $H\subseteq\mathcal{Z}(X)$ is
_$\mathcal{M}_{\mathcal{F}}$ -increasing_ if
$[H\cap\mathcal{K})_{\mathcal{F}}=H\cap\mathcal{K}$.

###### Definition 14

Let $\langle X,\mathcal{K},R\rangle$ be an $mS$-space. A family $\mathcal{F}$
of non-empty subbasic closed subsets of $\langle X,\mathcal{K}\rangle$ is a
_monotone lower-Vietoris-type family_ if
$\langle\mathcal{F},\mathcal{M}_{\mathcal{F}}\rangle$ is an $S$-space and for
every $Y\in\mathcal{F}$ we have that $R[Y]=\\{Z\in\mathcal{Z}(X)\colon\exists
y\in Y[(y,Z)\in R]\\}$ is a $\mathcal{M}_{\mathcal{F}}$-increasing subset.

We write $\mathcal{V}_{m}(X)$ for the set of monotone lower-Vietoris-type
families of an $mS$-space $\langle X,\mathcal{K},R\rangle$. Note that as a
consequence of Theorem 5.10, if
$\langle\mathbf{A},m\rangle,\langle\mathbf{B},n\rangle\in\mathcal{MS}$ and
$h\colon A\rightarrow B$ is an onto monotone homomorphism, we get that
$R_{m}[R_{h}(P)]$ is $\mathcal{M}$-increasing for all $P\in X(\mathbf{B})$.
So, $\mathcal{F}_{R_{h}}$ is a monotone lower-Vietoris-type family.

###### Theorem 5.11

Let $\langle\mathbf{A},m\rangle\in\mathcal{MS}$ and $\langle
X,\mathcal{K},R\rangle$ be an $mS$-space. Let $T\subseteq X\times
X(\mathbf{A})$ be a one-to-one monotone meet-relation. Then $\mathcal{F}_{T}$
is a monotone lower-Vietoris-type family of $\langle
X(\mathbf{A}),\mathcal{K}_{\mathbf{A}},R_{m}\rangle$.

###### Proof

From Theorem 9 we have that $\langle\mathcal{F}_{T},\mathcal{M}\rangle$ is an
$S$-space. Let $x\in X$, we will prove that $R_{m}[T(x)]$ is
$\mathcal{M}$-increasing. Let $a,b\in A$ be such that $H_{a}\subseteq H_{b}$
and suppose that $\beta(a)^{c}\in R_{m}[T(x)]$. We will see that
$\beta(b)^{c}\in R_{m}[T(x)]$. Since $T$ is a monotone meet-relation we have
$T^{-1}[\beta(a)^{c}]\in R(x)$. It is easy to see that $y\in
T^{-1}[\beta(a)^{c}]$ if and only if $R(y)\in H_{a}$. From $H_{a}\subseteq
H_{b}$ we get that $T^{-1}[\beta(a)^{c}]\subseteq T^{-1}[\beta(b)^{c}]$. So,
$T^{-1}[\beta(b)^{c}]\in R(x)$ and this implies that $\beta(b)^{c}\in
R_{m}[T(x)]$.

###### Lemma 17

Let $\langle\mathbf{A},m\rangle\in\mathcal{MS}$ and $\mathcal{F}\subseteq
C_{\mathcal{K}}(X(\mathbf{A}))$ be a monotone lower-Vietoris-type family of
the dual $S$-space of $\mathbf{A}$. Let us consider the relation
$R\subseteq\mathcal{F}\times\mathcal{Z}(\mathcal{F})$ given by

$(Y,Z)\in R\Longleftrightarrow Z\in\bigcap\\{L_{H_{a}^{c}}\colon a\in A\text{
and }\beta(a)^{c}\notin R_{m}[Y]\\}.$

Then $\langle\mathcal{F},\mathcal{M},R\rangle$ is an $mS$-space and
$R_{\mathcal{F}}\subseteq\mathcal{F}\times X(\mathbf{A})$ is a one-to-one
monotone meet-relation.

###### Proof

We know that $\langle\mathcal{F},\mathcal{M}\rangle$ is an $S$-space. Let
$a\in A$. We will see that $m_{R}(H_{a}^{c})\in S(\mathcal{F})$. Since

$m_{R}(H_{a}^{c})=\\{Y\in\mathcal{F}\colon\forall Z\in R(Y)[Z\cap
H_{a}^{c}\neq\emptyset]\\},$

we have $Y\notin m_{R}(H_{a}^{c})$ if and only if there exists $Z\in R(Y)$
such that $Z\notin L_{H_{a}^{c}}$. So, by definition of $R$ we get
$\beta(a)^{c}\in R_{m}[Y]$.

On the other hand, let $\beta(a)^{c}\in R_{m}[Y]$ and suppose that $Y\in
m_{R}(H_{a}^{c})$. We have $H_{a}\notin R(Y)$. So, there exists $b\in A$ such
that $\beta(b)^{c}\notin R_{m}[Y]$ and $H_{a}\cap H_{b}^{c}=\emptyset$. Thus,
$H_{a}\subseteq H_{b}$ and since $R_{m}[Y]$ is $\mathcal{M}$-increasing we get
a contradiction. Therefore, $Y\notin m_{R}(H_{a}^{c})$ if and only if
$\beta(a)^{c}\in R_{m}[Y]$ and it is easy to see that $\beta(a)^{c}\in
R_{m}[Y]$ if and only if $Y\cap m_{R_{m}}(\beta(a))^{c}\neq\emptyset$. As
$m_{R_{m}}(\beta(a))^{c}=\beta(ma)^{c}$, we finally get that
$m_{R}(H_{a}^{c})^{c}=H_{ma}$ and then $m_{R}(H_{a}^{c})\in S(\mathcal{F})$.
It is immediate to see that $R(Y)=\bigcap\\{L_{H_{a}^{c}}\colon a\in A\text{
and }Y\in m_{R}(H_{a}^{c})\\}$. So, $\langle\mathcal{F},\mathcal{M},R\rangle$
is an $mS$-space.

We will see now that $R_{\mathcal{F}}$ is a monotone one-to-one relation. Let
$a\in A$. We only need to prove that

$\beta(a)^{c}\in R_{m}[Y]\Longleftrightarrow
R_{\mathcal{F}}^{-1}[\beta(a)^{c}]\in R(Y).$

Note that $R_{\mathcal{F}}^{-1}[\beta(a)^{c}]=H_{a}$. Then

$\beta(a)^{c}\in R_{m}[Y]\Longleftrightarrow Y\notin
m_{R}(H_{a}^{c})\Longleftrightarrow H_{a}\in R(Y)$

and the assertion follows.

###### Theorem 5.12

Let $\langle\mathbf{A},m\rangle\in\mathcal{MS}$. Then the homomorphic images
of $\langle\mathbf{A},m\rangle$ are characterized by monotone lower-Vietoris-
type families of the dual $S$-space of $\mathbf{A}$.

As we saw on Remark 4, the class of monotone semilattices is a variety so we
obtain a representation theorem analogue to Theorem 5.4.

###### Theorem 5.13

Every monotone semilattice can be represented as an $mS$-space associated to a
monotone lower-Vietoris-type family.

We conclude this section by providing a characterization of the congruences on
monotone semilattices. The proofs of the following two results are similar to
the proofs of Theorem 5.5 and 5.6, respectively, so we omit them.

###### Theorem 5.14

Let $\langle\mathbf{A},m\rangle\in\mathcal{MS}$.

  1. 1.

If $\mathcal{F}\subseteq C_{\mathcal{K}}(X(\mathbf{A}))$ is a monotone lower-
Vietoris-type family of the dual $mS$-space of $\langle\mathbf{A},m\rangle$,
then $\theta_{\mathcal{F}}$ given by (7) is a congruence on
$\langle\mathbf{A},m\rangle$ such that
$\mathcal{F}=\mathcal{F}_{R_{{q_{\theta_{\mathcal{F}}}}}}$.

  2. 2.

If $\theta$ is a congruence on $\langle\mathbf{A},m\rangle$, then
$\theta=\theta_{\mathcal{F}_{R_{q_{\theta}}}}$.

###### Theorem 5.15

Let $\langle\mathbf{A},m\rangle\in\mathcal{MS}$. Then the congruences on
$\langle\mathbf{A},m\rangle$ are characterized by monotone lower-Vietoris-type
families of the dual $mS$-space of $\langle\mathbf{A},m\rangle$.

Let $\langle X,\mathcal{K},R\rangle$ be an $mS$-space. If we endow
$\mathcal{V}_{m}(X)$ with the induced order on $\mathcal{V}(X)$ then we obtain
an analogue of Theorem 5.8 for $mS$-spaces.

###### Theorem 5.16

For every $mS$-space $\langle X,\mathcal{K},R\rangle$, the poset
$\langle\mathcal{V}_{m}(X),\leq\rangle$ is a complete lattice which is dually
isomorphic to the set of all congruences of
$\langle\mathbf{S}(X),m_{R}\rangle$.

###  5.1 Congruence lattices of semilattices and lattices of algebraic sets

In Fajtlowicz , Fajtlowicz and Schmidt developed a duality between congruences
of semilattices and the lattices of algebraic sets. Unlike us, they worked
with ideals instead of filters. By following the same line and applying our
characterization of congruences of semilattices, we conclude this paper by
showing that there exists a duality between congruences of semilattices and
families of filters.

A subset $X$ of a complete lattice $\mathbf{L}$ is an _algebraic subset_ if it
contains $1$, it is closed under arbitrary meets, and it is closed under non-
empty directed joins. Following the notation of Hyndman , we denote
${\rm{S}}_{p}({\mathbf{L}})$ the lattice of all algebraic subsets of
$\mathbf{L}$.

###### Definition 15

Let $\mathbf{A}$ be a semilattice and $\theta$ be a congruence on $A$. A
filter $F$ of $\mathbf{A}$ is _$\theta$ -closed_ if $(x,y)\in\theta$ and $y\in
F$ implies $x\in F$.

Let $\mathbf{A}$ be a semilattice and let $\\{F_{i}\colon i\in I\\}$ be a
family of filters of $\mathbf{A}$. We recall that
$\operatorname{Fi}(\mathbf{A})$ is a complete lattice whose operations are
given by

$\begin{array}[]{ccc}\displaystyle\bigvee\\{F_{i}\colon i\in
I\\}=F\left(\bigcup\\{F_{i}\colon i\in I\\}\right)&\text{ and
}&\displaystyle\bigwedge\\{F_{i}\colon i\in I\\}=\bigcap\\{F_{i}\colon i\in
I\\}.\end{array}$

The following Theorem can be found in KIRA ADARICHEVA and J. B. NATION , using
ideals instead of filters. We include the proof for completeness.

###### Theorem 5.17

Let $\mathbf{A}$ be a semilattice. Then the lattice of congruences of
$\mathbf{A}$ is dually isomorphic to
${\rm{S}}_{p}(\operatorname{Fi}(\mathbf{A}))$.

###### Proof

We write ${\rm{Con}}(\mathbf{A})$ the lattice of all congruences of
$\mathbf{A}$. We define the map
$\sigma\colon{\rm{Con}}(\mathbf{A})\rightarrow{\rm{S}}_{p}(\operatorname{Fi}(\mathbf{A}))$
by setting $\sigma(\theta)$ as the set of all $\theta$-closed filters of
$\mathbf{A}$. Now, let us consider the map
$\rho\colon{\rm{S}}_{p}(\operatorname{Fi}(\mathbf{A}))\rightarrow{\rm{Con}}(\mathbf{A})$
defined by

$\rho(U)=\\{(x,y)\in A^{2}\colon\forall F\in U(x\in F\Longleftrightarrow y\in
F)\\}.$

It is straightforward to check that $\sigma$ and $\rho$ are order-reversing,
$\sigma(\theta)\in{\rm{S}}_{p}(\operatorname{Fi}(\mathbf{A}))$ and
$\rho(U)\in{\rm{Con}}(\mathbf{A})$. In order to show that
$\theta=\rho\sigma(\theta)$, note that if $y<x$ and $(x,y)\notin\theta$, then
$\\{z\in A\colon(x\wedge z,x)\in\theta\\}$ is a $\theta$-closed filter
containing $x$ and but not $y$. Hence $(x,y)\notin\rho\sigma(\theta)$. To see
that $U=\sigma\rho(U)$, consider a filter $F\notin U$. For each $x\in A$, let
$F_{x}=\bigcap\\{F\in U\colon x\in F\\}$. So, $F_{x}\in U$. Then
$\\{F_{x}\colon x\in F\\}$ is directed, whence $\bigcup\\{F_{x}\colon x\in
F\\}\in U$. Thus, the latter union properly contains $F$. Then there exist
$y<x$ with $x\in F$, $y\in F_{x}$ and $y\notin F$, i.e., $F$ is not
$\rho(U)$-closed. Therefore, $F\notin U$ implies $F\notin\sigma\rho(U)$. This
concludes the proof.

Let $\mathbf{A}$ be a semilattice and $\theta$ be a congruence on
$\mathbf{A}$. Then for every $Q\in X(\mathbf{A}/\theta)$ we have
$R_{q_{\theta}}(Q)\in C_{\mathcal{K}}(X(\mathbf{A}))$. So, there is an
associated filter $\psi(R_{q_{\theta}}(Q))=q_{\theta}^{-1}[Q]$ and we get a
family of filters

$\Psi(\mathcal{F}_{R_{q_{\theta}}})=\\{q_{\theta}^{-1}[Q]\colon Q\in
X(\mathbf{A}/\theta)\\}.$

###### Lemma 18

Let $\mathbf{A}$ be a semilattice and $\theta$ be a congruence on
$\mathbf{A}$. Then:

  1. 1.

$\Psi(\mathcal{F}_{R_{q_{\theta}}})\subseteq\sigma(\theta)$.

  2. 2.

If $F\in\sigma(\theta)-\\{A\\}$, then

$F=\displaystyle\bigcap\\{q_{\theta}^{-1}[Q]\in\Psi(\mathcal{F}_{R_{q_{\theta}}})\colon
F\subseteq q_{\theta}^{-1}[Q]\\}.$

###### Proof

1\. It is easy to see that $q_{\theta}^{-1}[Q]$ is a $\theta$-closed filter
for all $Q\in X(\mathbf{A}/\theta)$.

2\. Let $F\in\sigma(\theta)-\\{A\\}$. Then
$q_{\theta}[F]\in\operatorname{Fi}(\mathbf{A}/\theta)$ and as a consequence of
Theorem 2.2 we get $q_{\theta}[F]=\bigcap\\{Q\in X(\mathbf{A}/\theta)\colon
q_{\theta}[F]\subseteq Q\\}$. It follows that
$F=\bigcap\\{q_{\theta}^{-1}[Q]\in\Psi(\mathcal{F}_{R_{q_{\theta}}})\colon
F\subseteq q_{\theta}^{-1}[Q]\\}$.

## References

  * (1) Adaricheva, K., Nation, J.: Lattices of quasi-equational theories as congruence lattices of semilattices with operators: part I. Int. J. Algebra Comput. 22(7), 27 p. (2012)
  * (2) Bezhanishvili, G., Jansana, R.: Priestley style duality for distributive meet-semilattices. Stud. Log. 98(1-2), 83–122 (2011)
  * (3) Bezhanishvili, G., Jansana, R.: Generalized Priestley quasi-orders. Order 28(2), 201–220 (2011)
  * (4) Bezhanishvili, G., Jansana, R.: Esakia style duality for implicative semilattices. Appl. Categ. Struct. 21(2), 181–208 (2013)
  * (5) Birkhoff, G.: Rings of sets. Duke Math. J. 3, 443–454 (1937)
  * (6) Burris, S., Sankappanavar, H.P.: A Course in Universal algebra. Springer Verlag (1981).
  * (7) Celani, S.: Topological representation of distributive semilattices. Sci. Math. Jpn. 58(1), 55–66 (2003)
  * (8) Celani, S., Calomino, I.: Some remarks on distributive semilattices. Commentat. Math. Univ. Carol. 54(3), 407–428 (2013)
  * (9) Celani, S., González, L.J.: A categorical duality for semilattices and lattices. Appl. Categ. Struct. 28(5), 853–875 (2020)
  * (10) Celani, S., Menchón, M.P.: Monotonic distributive semilattices. Order 36(3), 463–486 (2019)
  * (11) Chellas, B.: Modal Logic. An introduction. Cambridge University Press (1980).
  * (12) Dunn, J., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symb. Log. 70(3), 713–740 (2005)
  * (13) Düntsch, I., Orlowska, E., Rewitzky, I.: Structures with multirelations, their discrete dualities and applications. Fundam. Inform. 100(1-4), 77–98 (2010)
  * (14) Fajtlowicz, S., Schmidt J.: Bezout families, join-congruences, and meet-irreducible ideals. Lattice Theory, Szeged, 1974, Colloq. Math. Soc. Janos Bolyai 14, 51–76 (1976)
  * (15) Gehrke, M., Jónsson, B.: Monotone bounded distributive lattice expansions. Math. Jap. 52(2), 197–213 (2000)
  * (16) Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94(1), 13–45 (2004)
  * (17) Gouveia, M.J., Priestley, H.A.: Canonical extensions and profinite completions of semilattices and lattices. Order 31(2), 189–216 (2014)
  * (18) Grätzer, G.: General Lattice Theory. Birkhäuser Verlag (1998)
  * (19) Hansen, H.H.: Monotonic Modal Logic. Master’s thesis, University of Amsterdam (2003)
  * (20) Hyndman, J., Nation, J.B., Nishida, J.: Congruence lattices of semilattices with operators. Stud. Log. 104(2), 305–316 (2016)
  * (21) Ivanova-Dimova, E.: Lower-Vietoris-type topologies on hyperspaces. Topology Appl. 220, 100–110 (2017)
  * (22) Jónsson, B., Tarski, A.: Boolean algebras with operators. I. Am. J. Math. 73, 891–939 (1951)
  * (23) Jónsson, B., Tarski, A.: Boolean algebras with operators. II. Am. J. Math. 74, 127–162 (1952)
  * (24) Moshier, M.A., Jipsen, P.: Topological duality and lattice expansions. I: A topological construction of canonical extensions. Algebra Univers. 71(2), 109–126 (2014)
  * (25) Priestley H.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)
  * (26) Rewitzky, I.: Binary multirelations. Theory and applications of relational structures as knowledge instruments. Lect. Notes Comput. Sci. 2929, 256–271 (2003)
  * (27) Stone M.: Topological representations of distributive lattices and Brouwerian logics. Čas. Mat. Fys. 67, 1–25 (1937)