# Central elements and the Gaeta topos: An algebraic and functorial overview
on coextensive varieties

William Javier Zuluaga Botero

###### Abstract

In this paper we show that within the context of coextensive varieties, the
functor of central elements is representable. In addition, we use the theory
of central elements to establish a criterion for fp-coextensive varieties that
allows to decide whether the Gaeta Topos classifies indecomposable objects in
terms of the indecomposability of the free algebra on one generator.

##  1 Introduction

Extensive categories were introduced in [24] as categories $\mathsf{C}$ with
finite coproducts and pullbacks in which the canonical functor
$+:\mathsf{C}/X\times\mathsf{C}/Y\rightarrow\mathsf{C}/(X+Y)$ is an
equivalence, for every pair of objects $X$ and $Y$ of $\mathsf{C}$. A category
is said to be coextensive if its opposite is extensive. Coextensive varieties
(as algebraic categories) are of interest because according to [25] and more
recently [29], they bring an appropriate setting to develop algebraic
geometry. In [36] it was shown that the theory central elements ([3], [31],
[34]) can be taken as an accurate tool to study coextensive varieties. This
perspective comes from the intuition that varieties with well behaved products
can be algebraically described by analyzing the peculiarities on the theory
that describes the behavior of the elements which concentrate the information
about finite product decompositions of its algebras. Under certain
circumstances, it turns out that central elements can be treated functorially.
As far as we know, this approach has not yet been exploited. Most of all,
because the theory of central elements was originally constrained into the
realm of universal algebra.

Small extensive categories admit a particular subcanonical topology called the
Gaeta topology (seemingly named in honor of the spanish algebraic geometer
Federico Gaeta). This topology has to do with all the possible decompositions
of objects into finite coproducts. In concrete examples ([35], [29]), it has
been proved that the Gaeta topos is the classifying topos of the theory of
connected objects, which can be considered as the ones who does not admit non-
trivial binary coproduct decompositions. Naturally, when considering
coextensive categories, the Gaeta topology and the Gaeta topos are related
with decompositions into finite products and indecomposable objects.
Nevertheless, in the practice it seems not quite easy to provide an
axiomatization of the theory of indecomposable objects when regarding
varieties in a more general setting. Considering the characterization given in
[36], it seems natural to wander if from the tools provided by universal
algebra and the theory of central elements it is possible to determine, given
a coextensive variety, whether the Gaeta topos classifies indecomposable
objects.

This paper is organized as follows. Section 2 presents the most part of the
definitions and basic results required for reading this work. Section 3 is
devoted to the functorial treating of central elements in coextensive
varieties. Such an approach will allows to prove that the functor of central
elements $Z$ is representable by the algebra $\mathbf{0}\times\mathbf{0}$.
This result will leads us to characterize those varieties with $\vec{0}$ and
$\vec{1}$, boolean factor congruences and center stable which are coextensive.
This section concludes with an application that connects coBoolean varieties
with the Beth (definability) property for algebraizable logics. Section 4
deals with the characterization of those fp-coextensive varieties such that
the Gaeta topos classifies indecomposable objects. The paper comes to an end
with the application of this result to some particular classes of interest in
general algebra and non-classical logic.

The reader is assumed to be familiar with some standard topos theory as
presented in [26] and [21]. For standard notions in universal algebra the
reader may consult [28].

##  2 Preliminaries

###  2.1 Notation and basic results

Let $A$ be a set and $k$ be a natural number. We write $\vec{a}$ for an
element $(a_{1},\dots,a_{k})\in A^{k}$. If $f:A\rightarrow B$ is a function
and $\vec{a}\in A^{k}$, then we write $f(\vec{a})$ for the element
$(f(a_{1}),\dots,f(a_{k}))\in B^{k}$. If $X\subseteq A$ we write $f|_{X}$ for
the restriction of $f$ to $X$, $\mathcal{P}(X)$ for the power set of $X$ and
$f[X]$ for the image of $X$ through $f$. If $\vec{a}\in A^{k}$ and $\vec{b}\in
B^{k}$, we write $[\vec{a},\vec{b}]$ for the k-tuple
$((a_{1},b_{1}),\dots,(a_{k},b_{k}))\in(A\times B)^{k}$. If $g:A\times
B\rightarrow C$ is a function and $[\vec{a},\vec{b}]\in(A\times B)^{k}$ then
we write $g(\vec{a},\vec{b})$ for the element
$(g(a_{1},b_{1}),\dots,g(a_{k},b_{k}))\in C^{k}$. If $\mathbf{A}$ is an
algebra of a given type we denote its universe by $A$ and its congruence
lattice by $\mathsf{Con}(\mathbf{A})$. If $\theta\in\mathsf{Con}(\mathbf{A})$,
and $\vec{a}\in A^{k}$ we write $\vec{a}/\theta$ for the k-tuple
$(a_{1}/\theta,\dots,a_{k}/\theta)\in(A/\theta)^{k}$. The universal congruence
on $\mathbf{A}$ and identity congruence on $\mathbf{A}$ are denoted by
$\nabla^{\mathbf{A}}$ and $\Delta^{\mathbf{A}}$, respectively. If $S\subseteq
A\times A$, we write $\mathsf{Cg}^{\mathbf{A}}(S)$ for the congruence
generated by $S$. We also write $\mathsf{Cg}^{\mathbf{A}}(\vec{a},\vec{b})$,
for the congruence generated by all pairs $(a_{1},b_{1}),\dots,(a_{k},b_{k})$
where $\vec{a},\vec{b}\in A^{k}$. We say that a congruence $\theta$ on
$\mathbf{A}$ is _finitely generated_ if $\theta=\mathsf{Cg}^{\mathbf{A}}(F)$
for some finite set $F\subseteq A\times A$. We use $\mathsf{FC}(\mathbf{A})$
to denote the set of factor congruences of $\mathbf{A}$. We write
$\theta\diamond\delta$ in $\mathsf{Con}(\mathbf{A})$ to denote that $\theta$
and $\delta$ are complementary factor congruences of $\mathbf{A}$. A variety
$\mathcal{V}$ has the Fraser-Horn property [18] if for every
$\mathbf{A}_{1},\mathbf{A}_{2}\in\mathcal{V}$, it is the case that every
congruence $\theta$ in $\mathbf{A}_{1}\times\mathbf{A}_{2}$ is the product
congruence $\theta_{1}\times\theta_{2}$ for some congruences $\theta_{1}$ of
$\mathbf{A}_{1}$ and $\theta_{2}$ of $\mathbf{A}_{2}$. If
$\theta,\lambda\in\mathsf{Con}(\mathbf{A})$ and $\theta\subseteq\lambda$, we
write $\lambda/\theta$ for the set of pairs $(x/\theta,y/\theta)$ of
$A/\theta$ such that $(x,y)\in\lambda$. If $g:\mathbf{A}\rightarrow\mathbf{B}$
is a homomorphism, we write $\mathsf{Ker}(g)$ for the _kernel_ of $g$. I.e.
the congruence on $\mathbf{A}$ defined by the set $\\{(a,b)\in A^{2}\colon
g(a)=g(b)\\}$. If $\mathbf{A}$ is an algebra of type
$\mathcal{F}=\\{f_{1},\ldots,f_{m}\\}$, when required, we will write its type
as an $n$-tuple $(a_{1},\ldots,a_{m})$, where $a_{j}$ denotes the arity of
$f_{j}$, with $1\leq j\leq m$.

The following result is probably folklore but, since we have not found it in
the literature, we give some of the details of its proof. It provides a
description of the factor congruences of the quotients of an algebra of a
given type.

######  Lemma 1.

Let $\mathbf{A}$ be an algebra of a given type and let
$\theta\in\mathsf{Con}(\mathbf{A})$. Consider the sets

$P_{\theta}=\\{(\lambda,\mu)\mid\theta\subseteq\lambda,\mu;\;\lambda\cap\mu=\theta\;\lambda\circ\mu=\nabla^{\mathbf{A}}\\}$

and

$Z_{\theta}=\\{(\alpha,\beta)\in\mathsf{FC}(\mathbf{A}/\theta)^{2}\colon\alpha\diamond\beta\\}.$

Then, the assignment $(\lambda,\mu)\mapsto(\lambda/\theta,\mu/\theta)$ defines
a bijection between $P_{\theta}$ and $Z_{\theta}$.

###### Proof.

It is a straightforward consequence of Theorems 7.5, 6.15 and 6.20 of [10]. ∎

Given a variety $\mathcal{V}$ and a set $X$ of variables we use
$\mathbf{T}_{\mathcal{V}}(X)$ for the term algebra of $\mathcal{V}$ over $X$
and $\mathbf{F}_{\mathcal{V}}(X)$ for the free algebra of $\mathcal{V}$ freely
generated by $X$. In particular, if $X=\\{x_{1},\dots,x_{m}\\}$ with $m$ a
non-negative integer and if no clarification is needed, then we write
$\mathbf{T}_{\mathcal{V}}(m)$ and $\mathbf{F}_{\mathcal{V}}(m)$ instead of
$\mathbf{T}_{\mathcal{V}}(\\{x_{1},\dots,x_{m}\\})$ and
$\mathbf{F}_{\mathcal{V}}(\\{x_{1},\dots,x_{m}\\})$, respectively. We recall
that an algebra $\mathbf{A}$ in $\mathcal{V}$ is a _finitely generated free
algebra_ if it is isomorphic to $\mathbf{F}_{\mathcal{V}}(m)$ for some finite
$m$, and _finitely presented_ if it is isomorphic to an algebra of the form
$\mathbf{F}_{\mathcal{V}}(k)/\theta$, for some $k$ finite and $\theta$
finitely generated congruence on $\mathbf{F}_{\mathcal{V}}(k)$.

The following Lemma is a key result that we will employ repeatedly along
Section 4. For its proof, the reader may consult [33].

######  Lemma 2.

Let $\mathcal{V}$ be a variety and let $X$ be a set of variables. Let
$r,r_{1},\ldots,r_{m},$ $s,s_{1},\ldots,s_{m}\in T_{\mathcal{V}}(X)$. Then,
the following are equivalent:

  * (1)

$(r,s)\in\mathsf{Cg}^{\mathbf{F}_{\mathcal{V}}(X)}(\vec{r},\vec{s})$;

  * (2)

$\mathcal{V}\models\vec{r}=\vec{s}\Longrightarrow r=s$.

######  Lemma 3.

Let $\mathcal{V}$ be a variety and let
$p_{i}(\vec{x},\vec{y}),q_{i}(\vec{x},\vec{y})$, $1\leq i\leq n$ be terms in
the language of $\mathcal{V}$. Let $X=\\{\vec{x},\vec{y},\vec{z}\\}$,

$\theta=\bigvee_{i=1}^{n}\mathsf{Cg}^{\mathbf{F}_{\mathcal{V}}(X)}(p_{i}(\vec{x},\vec{y}),q_{i}(\vec{x},\vec{y}))$

and $\mathbf{H}=\mathbf{F}_{\mathcal{V}}(X)/\theta$. Let
$\mathbf{A}\in\mathcal{V}$ and suppose that
$p_{i}^{\mathbf{A}}(\vec{a},\vec{b})=q_{i}^{\mathbf{A}}(\vec{a},\vec{b})$, for
$1\leq i\leq n$. Then, for every $\vec{c}\in A^{N}$ there exists a unique
$\Omega:\mathbf{H}\rightarrow\mathbf{A}$ such that
$\Omega(\vec{x}/\theta)=\vec{a}$, $\Omega(\vec{y}/\theta)=\vec{b}$ and
$\Omega(\vec{z}/\theta)=\vec{c}$.

###### Proof.

It is straightforward. ∎

Let $\mathcal{L}$ be a first order language. If a $\mathcal{L}$-formula
$\varphi(\vec{x})$ has the form

$\bigwedge_{j=1}^{n}p_{j}(\vec{x})=q_{j}(\vec{x}),$

for some positive number $n$ and terms $p_{j}(\vec{x})$ and $q_{j}(\vec{x})$
in $\mathcal{L}$, then we say that $\varphi(\vec{x})$ is a ($\bigwedge
p=q$)-formula. If $\mathcal{K}$ is a class of $\mathcal{L}$-structures and
$R\in\mathcal{L}$ is a $n$-ary relation symbol, we say that a formula
$\varphi(x_{1},\dots,x_{n})$ _defines_ $R$ _in_ $\mathcal{K}$ if

$\mathcal{K}\vDash\varphi(\vec{x})\Longleftrightarrow R(\vec{x})\text{.}$

In particular, if a ($\bigwedge p=q$)-formula defines $R$, we say that $R$ is
_equationally definable_.

Finally, we stress that all varieties considered along this paper always will
be assumed as varieties with at least a constant symbol.

###  2.2 Central Elements

By _a variety with $\vec{0}$ and $\vec{1}$_ we understand a variety
$\mathcal{V}$ in which there are $0$-ary terms $0_{1}$, $\ldots$ , $0_{N}$,
$1_{1}$, $\ldots$ , $1_{N}$ such that
$\mathcal{V}\models\vec{0}\approx\vec{1}\Longrightarrow x\approx y$, where
$\vec{0}=(0_{1},...,0_{N})$ and $\vec{1}=(1_{1},...,1_{N})$. If
$\mathbf{A}\in\mathcal{V}$ then we say that $\vec{e}=(e_{1},...,e_{N})\in
A^{N}$ is a _central element_ of $\mathbf{A}$ if there exists an isomorphism
$\tau:\mathbf{A}\rightarrow\mathbf{A}_{1}\times\mathbf{A}_{2}$, such that
$\tau(\vec{e})=[\vec{0}_{\mathbf{A}_{1}},\vec{1}_{\mathbf{A}_{2}}]$. Also, we
say that $\vec{e}$ and $\vec{f}$ are a _pair of complementary central
elements_ of $\mathbf{A}$ if there exists an isomorphism
$\tau:\mathbf{A}\rightarrow\mathbf{A}_{1}\times\mathbf{A}_{2}$ such that
$\tau(\vec{e})=[\vec{0}_{\mathbf{A}_{1}},\vec{1}_{\mathbf{A}_{2}}]$ and
$\tau(\vec{f})=[\vec{1}_{\mathbf{A}_{1}},\vec{0}_{\mathbf{A}_{2}}]$. We write
$Z(\mathbf{A})$ to denote the set of central elements of $A$ and
$\vec{e}\diamond_{\mathbf{A}}\vec{f}$ to denote that $\vec{e}$ and $\vec{f}$
are complementary central elements of $\mathbf{A}$. We say that a variety
$\mathcal{V}$ with $\vec{0}$ and $\vec{1}$ has _Boolean Factor Congruences_
(BFC) if the set of factor congruences of any algebra of $\mathcal{V}$ is a
Boolean sublattice of its congruence lattice. Let $\mathcal{V}$ be a variety
with BFC and $\mathbf{A}\in\mathcal{V}$. If $\vec{e}\in Z(\mathbf{A})$, we
write $\theta_{\vec{0},\vec{e}}^{\mathbf{A}}$ and
$\theta_{\vec{1},\vec{e}}^{\mathbf{A}}$ for the unique pair of complementary
factor congruences satisfying
$[\vec{e},\vec{0}]\in\theta_{\vec{0},\vec{e}}^{\mathbf{A}}$ and
$[\vec{e},\vec{1}]\in\theta_{\vec{1},\vec{e}}^{\mathbf{A}}$. In Theorem 1 of
[31] it was proved that the assignment which sends $\vec{e}$ into
$\theta_{\vec{0},\vec{e}}^{\mathbf{A}}$ establishes a bijection between
$Z(\mathbf{A})$ and $\mathsf{FC}(\mathbf{A})$. Such a bijection, allows to
define some operations in $Z(\mathbf{A})$ as follows: given $\vec{e}\in
Z(\mathbf{A})$, the _complement $\vec{e}^{c_{\mathbf{A}}}$_ of $\vec{e}$, is
the only solution to the equations
$[\vec{z},\vec{1}]\in\theta^{\mathbf{A}}_{\vec{0},\vec{e}}$ and
$[\vec{z},\vec{0}]\in\theta^{\mathbf{A}}_{\vec{1},\vec{e}}$. Given
$\vec{e},\vec{f}\in Z(\mathbf{A})$, the _infimum_
$\vec{e}\wedge_{\mathbf{A}}\vec{f}$ is the only solution to the equations
$[\vec{z},\vec{0}]\in\theta^{\mathbf{A}}_{\vec{0},\vec{e}}\cap\theta^{\mathbf{A}}_{\vec{0},\vec{f}}$
and
$[\vec{z},\vec{1}]\in\theta^{\mathbf{A}}_{\vec{1},\vec{e}}\vee\theta^{\mathbf{A}}_{\vec{1},\vec{f}}$.
Finally, the _supremum_ $\vec{e}\vee_{\mathbf{A}}\vec{f}$ is the only solution
to the equations
$[\vec{z},\vec{0}]\in\theta^{\mathbf{A}}_{\vec{0},\vec{e}}\vee\theta^{\mathbf{A}}_{\vec{0},\vec{f}}$
and
$[\vec{z},\vec{1}]\in\theta^{\mathbf{A}}_{\vec{1},\vec{e}}\cap\theta^{\mathbf{A}}_{\vec{1},\vec{f}}$.
Observe that these operations makes
$\textbf{Z}(\mathbf{A})=(Z(\mathbf{A}),\wedge_{\mathbf{A}},\vee_{\mathbf{A}},^{c_{\mathbf{A}}},\vec{0}^{\mathbf{A}},\vec{1}^{\mathbf{A}})$
a Boolean algebra which is isomorphic to
$(\mathsf{FC}(\mathbf{A}),\vee,\cap,^{\ast},\Delta^{\mathbf{A}},\nabla^{\mathbf{A}})$.

The following result was proved in Lemma 2.1.1. of [2] for the case $N=1$.
Nevertheless, since the arguments used for the case of an arbitrary $N$ does
not change the essence of the proof, we omit the details.

######  Lemma 4.

Let $\mathcal{V}$ be a variety with BFC and $\mathbf{A}\in\mathcal{V}$. For
every $\vec{e},\vec{f}\in Z(\mathbf{A})$, the following holds:

  1. 1.

$\vec{a}=\vec{e}\wedge_{\mathbf{A}}\vec{f}$ if and only if
$[\vec{0},\vec{a}]\in\theta_{\vec{0},\vec{e}}^{\mathbf{A}}$ and
$[\vec{a},\vec{f}]\in\theta_{\vec{1},\vec{e}}^{\mathbf{A}}$.

  2. 2.

$\vec{a}=\vec{e}\vee_{\mathbf{A}}\vec{f}$ if and only if
$[\vec{1},\vec{a}]\in\theta_{\vec{1},\vec{e}}^{\mathbf{A}}$ and
$[\vec{a},\vec{f}]\in\theta_{\vec{0},\vec{e}}^{\mathbf{A}}$.

Let $\mathcal{V}$ be a variety with BFC. If
$\mathbf{A},\mathbf{B}\in\mathcal{V}$ and $f:\mathbf{A}\rightarrow\mathbf{B}$
is a homomorphism, we say that $f$ _preserves central elements_ if the map
$f:Z(\mathbf{A})\rightarrow Z(\mathbf{B})$ is well defined; that is to say,
for every $\vec{e}\in Z(\mathbf{A}),$ it follows that $f(\vec{e})\in
Z(\mathbf{B})$. We say that $f$ _preserves complementary central elements_ if
it preserves central elements and for every $\vec{e}_{1},\vec{e}_{2}\in
Z(\mathbf{A})$,

$\vec{e}_{1}\diamond_{\mathbf{A}}\vec{e}_{2}\Rightarrow
f(\vec{e}_{1})\diamond_{\mathbf{B}}f(\vec{e}_{2}).$

We say that a variety with BFC is _center stable_ if every homomorphism
preserves central elements and we say that it is _stable by complements_ if
every homomorphism preserves complementary central elements. In [36] it was
shown that these notions are not equivalent.

###  2.3 Algebraizable Logics

The terminology and definitions of this section are based on those of [7],
[6], [30] and all the references therein. Let $\mathcal{L}$ be a language of
algebras and let $X$ be a countable-infinite set. We write
$T_{\mathcal{L}}(X)$ for the set of terms in $\mathcal{L}$. A _logic_ over $X$
is a pair $\mathbb{L}=\langle\mathcal{L},\vdash_{\mathbb{L}}\rangle$ where
$\vdash_{\mathbb{L}}\subseteq
T_{\mathcal{L}}(X)\times\mathcal{P}(T_{\mathcal{L}}(X))$ is a _substitution
invariant consequence relation_. I.e. $\vdash_{\mathbb{L}}$ satisfies:

$\begin{array}[]{ccc}\varphi\in\Gamma&\Rightarrow&\Gamma\vdash_{\mathbb{L}}\varphi;\\\
\Gamma\vdash_{\mathbb{L}}\varphi\;\text{and}\;\Gamma\subseteq\Delta&\Rightarrow&\Delta\vdash_{\mathbb{L}}\varphi;\\\
\Gamma\vdash_{\mathbb{L}}\varphi\;\text{and}\;\Delta\vdash_{\mathbb{L}}\psi\;\text{for
every}\;\psi\in\Gamma&\Rightarrow&\Delta\vdash_{\mathcal{L}}\varphi.\\\
\Gamma\vdash_{\mathbb{L}}\varphi&\Rightarrow&\sigma[\Gamma]\vdash_{\mathbb{L}}\sigma(\varphi).\end{array}$

for every endomorphism $\sigma$ of $\mathbf{T}_{\mathcal{L}}(X)$. In this
context $X$ is usually referred as the set of _variables of $\mathcal{L}$_.
The set $T_{\mathcal{L}}(X)\times T_{\mathcal{L}}(X)$ is called the set of
_equations_ of $\mathbb{L}$ and we denote it by $Eq_{\mathcal{L}}$. The
elements $(\varphi,\psi)$ of $Eq_{\mathcal{L}}$ are noted as
$\varphi\approx\psi$. A _transformer from formulas to equations_ is a function
$\tau\colon T_{\mathcal{L}}(X)\rightarrow\mathcal{P}(Eq_{\mathcal{L}})$. A
_transformer from equations to formulas_ is a function $\rho\colon
Eq_{\mathcal{L}}\rightarrow\mathcal{P}(T_{\mathcal{L}}(X))$. In this case,
$\tau$ is said to be _structural_ if for any endomorphism $\sigma$ of
$\mathbf{T}_{\mathcal{L}}(X)$ and every $\varphi\in T_{\mathcal{L}}(X)$ we
have $\tau(\sigma(\varphi))=\sigma[\tau(\varphi)]$. On the other hand, $\rho$
is said to be _structural_ if there is a set of formulas $\Delta(x,y)$ in at
most two variables such that for any $\varphi,\psi\in T_{\mathcal{L}}(X)$ the
condition $\rho(\varphi\approx\psi)=\Delta(\varphi,\psi)$ holds. A logic
$\mathbb{L}$ is _algebraizable (in the sense of Blok and Pigozzi)_ with
equivalent variety semantics $\sf\mathcal{V}$ if there are structural
transformers $\tau\colon
T_{\mathcal{L}}(X)\rightarrow\mathcal{P}(Eq_{\mathcal{L}})$ and $\rho\colon
Eq_{\mathcal{L}}\rightarrow\mathcal{P}(T_{\mathcal{L}}(X))$ such that for all
$\Gamma\cup\\{\varphi\\}\subseteq T_{\mathcal{L}}(X)$ and
$\Theta\cup\\{\epsilon\approx\delta\\}\subseteq Eq_{\mathcal{L}}$, we have:

  1. 1.

$\Gamma\vdash_{\mathbb{L}}\varphi\Longleftrightarrow\tau\Gamma\models_{\mathcal{V}}\tau\varphi$,
and

  2. 2.

$\epsilon\approx\delta\Dashv\vDash_{\mathcal{V}}\tau\rho(\epsilon\approx\delta)$.

Let $\mathbb{L}$ be a logic and let $V$ and $W$ be a disjoint pair of sets of
variables such that $T_{\mathcal{L}}(V)\neq\emptyset$. Let $\Gamma\subseteq
T_{\mathcal{L}}(V\cup W)$. We say that _$\Gamma$ defines $W$ implicitly in
terms of $V$ in $\mathbb{L}$_ if for every set of variables $Y$, each $z\in W$
and homomorphism $h:\mathbf{T}_{\mathcal{L}}(V\cup
W)\rightarrow\mathbf{T}_{\mathcal{L}}(Y)$, such that $h(x)=x$ for all $x\in
V$, it follows that $\Gamma\cup
h[\Gamma]\models_{\mathsf{Mod}^{\ast}(\vdash_{\mathbb{L}})}z\approx h(z)$. We
say that _$\Gamma$ defines $W$ explicitly in terms of $V$ in $\mathbb{L}$_ if
for each $z\in W$, there exists $\varphi_{z}\in T_{\mathcal{L}}(V)$ such that
$\Gamma\models_{\mathsf{Mod}^{\ast}(\vdash_{\mathbb{L}})}z\approx\varphi_{z}$.
Here $\mathsf{Mod}^{\ast}(\vdash_{\mathbb{L}})$ denotes the class of all
reduced matrix models of $\vdash_{\mathbb{L}}$ (i.e.,
$\Gamma\vdash_{\mathbb{L}}\varphi$ iff
$\Gamma\models_{\mathsf{Mod}^{\ast}(\vdash_{\mathbb{L}})}\varphi$). We say
that a logic $\mathbb{L}$ has the _Beth (definability) property_ if for every
disjoint pair of sets of variables $V$ and $W$ and $\Gamma\subseteq
T_{\mathcal{L}}(V\cup W)$, if $\Gamma$ defines $W$ implicitly in terms of $V$
in $\mathbb{L}$ then $\Gamma$ defines $W$ explicitly in terms of $V$ in
$\mathbb{L}$.

If $\mathbb{L}$ is an algebraizable logic with equivalent variety semantics
$\mathcal{V}$, the underlying intuition of Beth’s (definability) property is
that epimorphisms in the (algebraic) category $\mathcal{V}$ correspond to
implicit definitions in $\mathbb{L}$ and surjections in $\mathcal{V}$
correspond to explicit definitions in $\mathbb{L}$. The following result,
originally proved in [20], establishes that such an intuition is in fact an
equivalence.

######  Theorem 1.

Let $\mathbb{L}$ be an algebraizable logic with equivalent variety semantics
$\mathcal{V}$. Then $\mathbb{L}$ has the Beth (definability) property if and
only if all the epimorphisms of $\mathcal{V}$ are surjective.

##  3 Representability of the functor Z

We recall that a category with finite products $\mathsf{C}$ is called
coextensive if for each pair of objects $X$ and $Y$ of $\mathsf{C}$ the
canonical functor
$\times:\mathsf{C}/X\times\mathsf{C}/Y\rightarrow\mathsf{C}/(X\times Y)$ is an
equivalence. Classical examples of coextensive categories are the categories
${\mathbf{Ring}}$ and $\mathbf{dLat}$ of commutative rings with unit and
bounded distributive lattices, respectively. If $\mathcal{V}$ is a coextensive
variety, the associated algebraic category will be also denoted by
$\mathcal{V}$. In what follows, we write $\mathbf{0}$ and $\mathbf{1}$ for the
initial and terminal algebras of $\mathcal{V}$, respectively. If
$\mathbf{A}\in\mathcal{V}$ we write
$\text{!`}_{\mathbf{A}}:\mathbf{0}\rightarrow\mathbf{A}$ for the unique
morphism from $\mathbf{0}$ to $\mathbf{A}$ in $\mathcal{V}$. If $\vec{e}\in
Z(\mathbf{A})$ we write $\text{!`}_{\vec{0},\vec{e}}$ and
$\text{!`}_{\vec{1},\vec{e}}$ for the unique morphisms from $\mathbf{0}$ to
$\mathbf{A}/\mathsf{Cg}^{\mathbf{A}}(\vec{0},\vec{e})$ and from $\mathbf{0}$
to $\mathbf{A}/\mathsf{Cg}^{\mathbf{A}}(\vec{1},\vec{e})$, respectively.
Finally, we recall that due to $\mathcal{V}$ is assumed with at least a
constant symbol then $\mathbf{0}$ is isomorphic to
$\mathbf{F}_{\mathcal{V}}(\emptyset)$.

Given a variety $\mathcal{V}$ with $\vec{0}$ and $\vec{1}$, BFC and center
stable, it is the case that for every $\mathbf{A}\in\mathcal{V}$ the
assignment $\mathbf{A}\mapsto Z(\mathbf{A})$ defines a functor
$Z:\mathcal{V}\rightarrow\mathsf{Set}$ in a obvious way. In this section we
prove that when $\mathcal{V}$ is coextensive, such a functor is in fact
representable by the algebra $\mathbf{0}\times\mathbf{0}$. This result leads
us to show that the functor $Z$ can also be extended to a functor from
$\mathcal{V}$ to the category $\mathsf{Boole}$ of Boolean algebras. Moreover,
a characterization of coextensivity by means of the functors $Z$ and $\times$
is provided. The section concludes with an application on Beth (definability)
property for logics that has coBoolean varieties as algebraic semantics.

We will begin by recalling some facts about coextensive varieties which will
be essential for proving the results of this section. It is well known that
every variety (as an algebraic category) has all limits. Therefore, as a
restricted dual of Propositions 2.2 and 4.1 of [11] we obtain the following
result.

######  Proposition 1.

A variety $\mathcal{V}$ is coextensive if and only if it has pushouts along
projections and every commutative diagram

$\textstyle{\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{!`}_{\mathbf{A}_{0}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbf{0}\times\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{f}$$\textstyle{\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{!`}_{\mathbf{A}_{1}}}$$\textstyle{\mathbf{A}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbf{A}_{0}\times\mathbf{A}_{1}}$$\scriptstyle{g_{0}}$$\scriptstyle{g_{1}}$$\textstyle{\mathbf{A}_{1}}$

comprises a pair of pushout squares in $\mathcal{V}$ just when the bottom row
is a product diagram in $\mathcal{V}$.

We recall that a variety $\mathcal{V}$ is a _Pierce variety_ [32] if there
exist a positive natural number $N$ , $0$-ary terms $0_{1}$, $\ldots$,
$0_{N}$, $1_{1}$, $\ldots$, $1_{N}$ and a term $U(x,y,\vec{z},\vec{w})$ such
that the identities

$\begin{array}[]{ccc}U(x,y,\vec{0},\vec{1})=x&\text{and}&U(x,y,\vec{1},\vec{0})=y\end{array}$

hold in $\mathcal{V}$. It is worth mentioning that in a Pierce variety
$\mathcal{V}$, it is also true that
$\theta^{\mathbf{A}}_{\vec{0},\vec{e}}=\mathsf{Cg}^{\mathbf{A}}(\vec{0},\vec{e})$,
for every $\mathbf{A}\in\mathcal{V}$ and every $\vec{e}\in\mathbf{A}$ (see [4]
for details).

In [36] the following characterization of coextensive varieties by means of
Pierce varieties, the equational definability of the relation “$\vec{e}$ and
$\vec{f}$ are complementary central elements” and the stability by
complements, was provided. It is a key result on which we will constantly rely
for carrying on the goals of this section.

######  Theorem 2.

Let $\mathcal{V}$ be a variety. Then, the following are equivalent:

  1. (1)

$\mathcal{V}$ is coextensive.

  2. (2)

$\mathcal{V}$ is a Pierce variety in which the relation
$\vec{e}\diamond_{\mathbf{A}}\vec{f}$ is equationally definable.

  3. (3)

$\mathcal{V}$ is a Pierce variety stable by complements.

Let $\mathcal{V}$ be a coextensive variety,
$\mathbf{A},\mathbf{B}\in\mathcal{V}$ and $f:\mathbf{A}\rightarrow\mathbf{B}$
be a homomorphism. Observe that from Theorem 2 (3) the assignments

$\begin{array}[]{ccc}\mathbf{A}\mapsto Z(\mathbf{A})\\\ f\mapsto
f|_{Z(\mathbf{A})}\end{array}$

determine a functor $Z:\mathcal{V}\rightarrow\mathsf{Set}$. In the following
result we prove that such a functor is in fact, representable.

######  Theorem 3.

Let $\mathcal{V}$ be a coextensive variety. Then, for every
$\mathbf{A}\in\mathcal{V}$ there is a bijection between $Z(\mathbf{A})$ and
$\mathcal{V}(\mathbf{0}\times\mathbf{0},\mathbf{A})$. Moreover, the functor
$Z:\mathcal{V}\rightarrow\mathsf{Set}$ is representable by
$\mathbf{0}\times\mathbf{0}$.

###### Proof.

Let $\mathbf{A}\in\mathcal{V}$ and consider the assignments
$\varphi_{\mathbf{A}}:\mathcal{V}(\mathbf{0}\times\mathbf{0},\mathbf{A})\rightarrow
Z(\mathbf{A})$ and
$\mu_{\mathbf{A}}:Z(\mathbf{A})\rightarrow\mathcal{V}(\mathbf{0}\times\mathbf{0},\mathbf{A})$,
defined by $\varphi_{\mathbf{A}}(g)=g[\vec{0},\vec{1}]$ and
$\mu_{\mathbf{A}}(\vec{e})=\text{!`}_{\vec{0},\vec{e}}\times\text{!`}_{\vec{1},\vec{e}}$,
respectively. We claim that $\varphi$ and $\mu$ are natural transformations
which are inverse of each other.

$\textstyle{\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{!`}_{\vec{0},\vec{e}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbf{0}\times\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\scriptstyle{\text{!`}_{\vec{0},\vec{e}}\times\text{!`}_{\vec{1},\vec{e}}}$$\scriptstyle{\pi_{1}}$$\textstyle{\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{!`}_{\vec{1},\vec{e}}}$$\textstyle{\mathbf{A}/\mathbf{Cg}^{\mathbf{A}}(\vec{0},\vec{e})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbf{A}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{0}}$$\textstyle{\mathbf{A}/\mathbf{Cg}^{\mathbf{A}}(\vec{1},\vec{e})}$

We start by showing that $\varphi_{\mathbf{A}}$ and $\mu_{\mathbf{A}}$ are
well defined. Since $[\vec{0},\vec{1}]\in Z(\mathbf{0}\times\mathbf{0})$, from
Theorem 2 (3), $g[\vec{0},\vec{1}]\in Z(\mathbf{A})$ for every
$g\in\mathcal{V}(\mathbf{0}\times\mathbf{0},\mathbf{A})$ so
$\varphi_{\mathbf{A}}$ is well defined. Similarly, due to every $\vec{e}\in
Z(\mathbf{A})$ induces a product decomposition of $\mathbf{A}$, then from the
coextensivity of $\mathcal{V}$, we get that
$\text{!`}_{\vec{0},\vec{e}}\times\text{!`}_{\vec{1},\vec{e}}$ is the unique
morphism in $\mathcal{V}(\mathbf{0}\times\mathbf{0},\mathbf{A})$ making
pushouts both squares of the diagram of above, so $\mu_{\mathbf{A}}$ is also
well defined.

Now we prove that $\varphi_{\mathbf{A}}$ and $\mu_{\mathbf{A}}$ are mutually
inverse. To do so, let $\vec{e}\in Z(\mathbf{A})$,
$h=\text{!`}_{\vec{0},\vec{e}}\times\text{!`}_{\vec{1},\vec{e}}$ and let us
consider $\varphi_{\mathbf{A}}(\mu_{\mathbf{A}}(\vec{e}))=h[\vec{0},\vec{1}]$.

$\textstyle{\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{!`}_{\vec{0},\vec{e}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbf{0}\times\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\scriptstyle{h}$$\scriptstyle{\pi_{1}}$$\textstyle{\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{!`}_{\vec{1},\vec{e}}}$$\textstyle{\mathbf{P}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbf{A}}$$\scriptstyle{j_{1}}$$\scriptstyle{j_{0}}$$\textstyle{\mathbf{P}_{1}}$

If $\mathbf{P}_{k}$ denote the pushouts of $\pi_{k}$ along $h$, with $1\leq
k\leq 2$, from Lemma 2.3 of [36], it is the case that:

$\begin{array}[]{ccc}\mathbf{P}_{0}\cong\mathbf{A}/\mathbf{Cg}^{\mathbf{A}}(\vec{0},\varphi_{\mathbf{A}}(\mu_{\mathbf{A}}(\vec{e})))\cong\mathbf{A}/\mathbf{Cg}^{\mathbf{A}}(\vec{0},\vec{e})\\\
\mathbf{P}_{1}\cong\mathbf{A}/\mathbf{Cg}^{\mathbf{A}}(\vec{1},\varphi_{\mathbf{A}}(\mu_{\mathbf{A}}(\vec{e})))\cong\mathbf{A}/\mathbf{Cg}^{\mathbf{A}}(\vec{1},\vec{e}).\end{array}$

Therefore, for general reasons we get:

$\begin{array}[]{ccc}\mathbf{Cg}^{\mathbf{A}}(\vec{0},\varphi_{\mathbf{A}}(\mu_{\mathbf{A}}(\vec{e})))=\mathbf{Cg}^{\mathbf{A}}(\vec{0},\vec{e})\\\
\mathbf{Cg}^{\mathbf{A}}(\vec{1},\varphi_{\mathbf{A}}(\mu_{\mathbf{A}}(\vec{e})))=\mathbf{Cg}^{\mathbf{A}}(\vec{1},\vec{e}),\end{array}$

Hence, from Corollary 4 of [34] it follows that
$\varphi_{\mathbf{A}}(\mu_{\mathbf{A}}(\vec{e}))=\vec{e}$. On the other hand,
let $g\in\mathcal{V}(\mathbf{0}\times\mathbf{0},\mathbf{A})$ and consider
$\mu_{\mathbf{A}}(\varphi_{\mathbf{A}}(g))=\text{!`}_{\vec{0},\varphi_{\mathbf{A}}(g)}\times\text{!`}_{\vec{1},\varphi_{\mathbf{A}}(g)}$.
Then we have
$\mathbf{A}\cong\mathbf{A}/\mathbf{Cg}^{\mathbf{A}}(\vec{0},\varphi_{\mathbf{A}}(g))\times\mathbf{A}/\mathbf{Cg}^{\mathbf{A}}(\vec{1},\varphi_{\mathbf{A}}(g))$.
Since $\mathcal{V}$ is coextensive, there exist unique
$u:\mathbf{0}\rightarrow\mathbf{A}/\mathbf{Cg}^{\mathbf{A}}(\vec{0},\varphi_{\mathbf{A}}(g))$
and
$v:\mathbf{0}\rightarrow\mathbf{A}/\mathbf{Cg}^{\mathbf{A}}(\vec{1},\varphi_{\mathbf{A}}(g))$
such that $g=u\times v$. Observe that due to $\mathbf{0}$ is initial in
$\mathcal{V}$, it must be the case that
$u=\text{!`}_{\vec{0},\varphi_{\mathbf{A}}(g)}$ and
$v=\text{!`}_{\vec{1},\varphi_{\mathbf{A}}(g)}$ so
$g=\mu_{\mathbf{A}}(\varphi_{\mathbf{A}}(g))$, as desired.

The proof of the naturality of $\varphi$ and $\mu$ is straightforward. ∎

As an immediate application of Theorem 3 and Corollary 9.33 of [1], we obtain
the following result.

######  Corollary 1.

If $\mathcal{V}$ is a coextensive variety, the functor
$Z:\mathcal{V}\rightarrow\mathsf{Set}$ preserves all limits. Therefore $Z$ has
a left adjoint.

Something more can be said about the functor $Z$.

######  Lemma 5.

Let $\mathcal{V}$ be a coextensive variety. If
$\mathbf{A},\mathbf{B}\in\mathcal{V}$ and $f:\mathbf{A}\rightarrow\mathbf{B}$
is a homomorphism, then $f|_{Z(\mathbf{A})}:\mathbf{A}\rightarrow\mathbf{B}$
is a homomorphism of Boolean algebras.

###### Proof.

We start by recalling that from Theorem 2 and Lemma 4.3 of [36], for every
$\mathbf{A}\in\mathcal{V}$ and every $\vec{e}\in Z(\mathbf{A})$ we have
$\theta^{\mathbf{A}}_{\vec{0},\vec{e}}=\mathsf{Cg}^{\mathbf{A}}(\vec{0},\vec{e})$.
Since $f$ is a homomorphism, it is clear that $f|_{Z(\mathbf{A})}$ preserves
$\vec{0}$ and $\vec{1}$. Now, if $\vec{e}_{1},\vec{e}_{2}\in Z(\mathbf{A})$
and $\vec{a}=\vec{e}_{1}\wedge_{\mathbf{A}}\vec{e}_{2}$, then from Lemma 4,
$[\vec{0},\vec{a}]\in\mathsf{Cg}^{\mathbf{A}}(\vec{0},\vec{e}_{1})$ and
$[\vec{a},\vec{e}_{2}]\in\mathsf{Cg}^{\mathbf{A}}(\vec{1},\vec{e}_{1})$. From
Theorem 2 (3), we get
$[\vec{0},f(\vec{a})]\in\mathsf{Cg}^{\mathbf{A}}(\vec{0},f(\vec{e}_{1}))$ and
$[f(\vec{a}),f(\vec{e}_{2})]\in\mathsf{Cg}^{\mathbf{A}}(\vec{1},f(\vec{e}_{1}))$.
Therefore, again by Lemma 4 we conclude that $f|_{Z(\mathbf{A})}$ preserves
the meet of $\mathbf{Z}(\mathbf{A})$. The proof that $f|_{Z(\mathbf{A})}$
preserves the join of $\mathbf{Z}(\mathbf{A})$ is analogue. This concludes the
proof. ∎

Observe that as result of Lemma 5, it is the case that the functor $Z$ can be
extended to a new functor $\mathbf{Z}:\mathcal{V}\rightarrow\mathsf{Boole}$.
We can take advantage of this fact in order to extend the representable
$\mathcal{V}(\mathbf{0}\times\mathbf{0},-)$ to a functor from $\mathcal{V}$ to
$\mathsf{Boole}$ which we will denote by $\mathcal{H}$. Indeed, if
$\mathbf{A}\in\mathcal{V}$ we can endow
$\mathcal{V}(\mathbf{0}\times\mathbf{0},\mathbf{A})$ with a Boolean algebra
structure in such a way that the algebra obtained be isomorphic to
$\mathbf{Z}(\mathbf{A})$. If
$f,g\in\mathcal{V}(\mathbf{0}\times\mathbf{0},\mathbf{A})$, by using Theorem 3
we define:

$\begin{array}[]{rcl}0&:=&\text{!`}_{\mathbf{1}}\times\text{!`}_{\mathbf{A}}\\\
1&:=&\text{!`}_{\mathbf{A}}\times\text{!`}_{\mathbf{1}}\\\
g^{c}&:=&\text{!`}_{\vec{1},\varphi_{\mathbf{A}}(g)}\times\text{!`}_{\vec{0},\varphi_{\mathbf{A}}(g)}\\\
g\wedge
h&:=&\text{!`}_{\vec{0},\varphi_{\mathbf{A}}(g)\wedge_{\mathbf{A}}\varphi_{\mathbf{A}}(h)}\times\text{!`}_{\vec{1},\varphi_{\mathbf{A}}(g)\wedge_{\mathbf{A}}\varphi_{\mathbf{A}}(h)}\\\
g\vee
h&:=&\text{!`}_{\vec{0},\varphi_{\mathbf{A}}(g)\vee_{\mathbf{A}}\varphi_{\mathbf{A}}(h)}\times\text{!`}_{\vec{1},\varphi_{\mathbf{A}}(g)\vee_{\mathbf{A}}\varphi_{\mathbf{A}}(h)}.\end{array}$

Notice that due to the natural isomorphism between $Z$ and the representable
$\mathcal{V}(\mathbf{0}\times\mathbf{0},-)$ the functoriality of $\mathcal{H}$
is granted.

######  Corollary 2.

Let $\mathcal{V}$ be a coextensive variety and consider the functors
$\mathbf{Z}$ and $\mathcal{H}$ from $\mathcal{V}$ to $\mathsf{Boole}$. Then
$\mathbf{Z}$ and $\mathcal{H}$ are naturally isomorphic.

###### Proof.

Immediate from Theorem 3 and Lemma 5. ∎

At this stage, one may be wandering if a characterization of coextensive
varieties in terms of the representability of the functor $Z$ can be
established. We baer that in [36] it was shown that not every variety with BFC
and $\vec{0}$ and $\vec{1}$ is center stable and even if it is, it may be the
case that it may not be coextensive. We claim that the next result provides an
effective answer to this question.

######  Theorem 4.

Let $\mathcal{V}$ be a variety with BFC, $\vec{0}$ and $\vec{1}$ and center
stable. Then, the following are equivalent:

  * (1)

$\mathcal{V}$ is coextensive.

  * (2)

The following conditions hold:

    * (i)

The functor $Z:\mathcal{V}\rightarrow\mathsf{Set}$ is representable by
$\mathbf{0}\times\mathbf{0}$.

    * (ii)

The functor
$\times:\mathcal{V}/\mathbf{0}\times\mathcal{V}/\mathbf{0}\rightarrow\mathcal{V}/(\mathbf{0}\times\mathbf{0})$
is full and faithful.

###### Proof.

We only proof $(2)\Rightarrow(1)$ because the converse follows from Theorem 3
and the dual of Lemma 1 of [12]. Let us assume $(2)$. We start by noticing
that from Theorem 3.4.5 of [8], $\mathcal{V}$ is cocomplete so in particular,
it has pushouts along projections. In addition by $(i)$, there exists a
natural isomorphism $\varphi$ from $\mathcal{V}(\mathbf{0}\times\mathbf{0},-)$
to $Z$. If we write $\mu$ for the inverse natural transformation of $\varphi$,
then for every $\mathbf{A},\mathbf{B}\in\mathcal{V}$,
$g\in\mathcal{V}(\mathbf{0}\times\mathbf{0},\mathbf{A})$, $\vec{e}\in
Z(\mathbf{A})$ and every homomorphism $f:\mathbf{A}\rightarrow\mathbf{B}$ the
following identities hold:

$\displaystyle f(\varphi_{\mathbf{A}}(g))=\varphi_{\mathbf{B}}(fg)$ (1)
$\displaystyle f(\mu_{\mathbf{A}}(\vec{e}))=\mu_{\mathbf{B}}(f(\vec{e}))$ (2)

Moreover, for every $\mathbf{B}_{0},\mathbf{B}_{1}\in\mathcal{V}$,
$\varphi_{\mathbf{B}_{0}\times\mathbf{B}_{1}}(\text{!`}_{\mathbf{B}_{0}}\times\text{!`}_{\mathbf{B}_{1}})=[\vec{0}_{\mathbf{B}_{0}},\vec{1}_{\mathbf{B}_{1}}]$.

Since $\mathcal{V}$ is a variety with $\vec{0}$ and $\vec{1}$, then it is a
variety admitting constant symbols. Then from Proposition 2.1 of [9], products
are codisjoint. Now, let
$\mathbf{A},\mathbf{A}_{0},\mathbf{A}_{1}\in\mathcal{V}$,
$\mathbf{A}_{0}\xleftarrow[]{p{0}}\mathbf{A}\xrightarrow[]{p{1}}\mathbf{A}_{1}$
be a span, and let $g\in\mathcal{V}(\mathbf{0}\times\mathbf{0},\mathbf{A})$.
Consider the following diagram in which the upper left and right squares are
pushouts:

$\textstyle{\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{!`}_{\mathbf{A}_{0}}}$$\scriptstyle{\text{!`}_{\mathbf{B}_{0}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbf{0}\times\mathbf{0}}$$\scriptstyle{\pi_{0}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{g}$$\textstyle{\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{!`}_{\mathbf{A}_{1}}}$$\scriptstyle{\text{!`}_{\mathbf{B}_{1}}}$$\textstyle{\mathbf{A}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbf{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{0}}$$\scriptstyle{p_{1}}$$\scriptstyle{h}$$\textstyle{\mathbf{A}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}}$$\textstyle{\mathbf{B}_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbf{B}_{0}\times\mathbf{B}_{1}}$$\scriptstyle{q_{0}}$$\scriptstyle{q_{1}}$$\textstyle{\mathbf{B}_{1}}$

We will prove that the aforementioned span is a product diagram. Since
$\varphi_{\mathbf{A}}(g)\in Z(\mathbf{A})$, then we have an isomorphism
$h:\mathbf{A}\rightarrow\mathbf{B}_{0}\times\mathbf{B}_{1}$. Now we prove that
$hg=i_{\mathbf{B}_{0}}\times i_{\mathbf{B}_{1}}$. To do so, we need to check
that $q_{0}(hg)=\text{!`}_{\mathbf{B}_{0}}\pi_{0}$ and
$q_{1}(hg)=\text{!`}_{\mathbf{B}_{1}}\pi_{1}$. We only check the first
condition because the proof of the second one is similar. Observe that:

$\begin{array}[]{ccll}q_{0}(hg)&=&q_{0}h(\mu_{\mathbf{A}}(\varphi_{\mathbf{A}}(g))),&\text{since
$\mu_{\mathbf{A}}(\varphi_{\mathbf{A}}(g))=g$.}\\\
&=&\mu_{\mathbf{B}_{0}}(q_{0}h(\varphi_{\mathbf{A}}(g))),&\text{from
(\ref{1b}) applied to $q_{0}h$}.\\\
&=&\mu_{\mathbf{B}_{0}}(q_{0}[\vec{0}_{\mathbf{B}_{0}},\vec{1}_{\mathbf{B}_{1}}]),&\text{since
$h(\varphi_{\mathbf{A}}(g))=[\vec{0}_{\mathbf{B}_{0}},\vec{1}_{\mathbf{B}_{1}}]$}.\\\
&=&\mu_{\mathbf{B}_{0}}(\vec{0}_{\mathbf{B}_{0}}).&\end{array}$

On the other hand,

$\begin{array}[]{ccll}\varphi_{\mathbf{B}_{0}}(\text{!`}_{\mathbf{B}_{0}}\pi_{0})&=&\varphi_{\mathbf{B}_{0}}(q_{0}(\text{!`}_{\mathbf{B}_{0}}\times\text{!`}_{\mathbf{B}_{1}}))&\\\
&=&q_{0}(\varphi_{\mathbf{B}_{0}\times\mathbf{B}_{1}}(\text{!`}_{\mathbf{B}_{0}}\times\text{!`}_{\mathbf{B}_{1}})),&\text{from
(\ref{1a}).}\\\
&=&q_{0}[\vec{0}_{\mathbf{B}_{0}},\vec{1}_{\mathbf{B}_{1}}]&\\\
&=&\vec{0}_{\mathbf{B}_{0}}.&\end{array}$

Therefore, from the following calculation we obtain:

$\text{!`}_{\mathbf{B}_{0}}\pi_{0}=\mu_{\mathbf{B}_{0}}(\varphi_{\mathbf{B}_{0}}(\text{!`}_{\mathbf{B}_{0}}\pi_{0}))=\mu_{\mathbf{B}_{0}}(\vec{0}_{\mathbf{B}_{0}})=q_{0}(hg).$

So $hg=\text{!`}_{\mathbf{B}_{0}}\times\text{!`}_{\mathbf{B}_{1}}$ as claimed.
Recall that the latter implies that
$q_{0}(hg)=\text{!`}_{\mathbf{B}_{0}}\pi_{0}$ and
$q_{1}(hg)=\text{!`}_{\mathbf{B}_{1}}\pi_{1}$. Thus, due to each of the upper
squares of the diagram of above are pushouts by assumption, there exist unique
$a_{i}:\mathbf{A}_{j}\rightarrow\mathbf{B}_{j}$ with $j=1,2,$ such that each
of the lower squares of the diagram commute. From $(ii)$ and the dual of Lemma
1 of [12], the outer left and right squares of the diagram are pushouts. So,
each of the lower squares of the diagram are pushouts. Since $q_{0}$ and
$q_{1}$ are epi and $h$ is an iso, then $a_{0}$ and $a_{1}$ must be iso too.
Therefore, the span
$\mathbf{A}_{0}\xleftarrow[]{p{0}}\mathbf{A}\xrightarrow[]{p{1}}\mathbf{A}_{1}$
is a product, as desired. Hence from Proposition 1 the result follows. ∎

We conclude this part by introducing a particular class of coextensive
varieties. It is motivated by the intimate relation they present with a
concrete property of the functor $Z$. Such a class will be related with some
results in Section 4.

######  Definition 1.

A coextensive variety $\mathcal{V}$ is said to be center presentable if
$\mathbf{0}\times\mathbf{0}$ is finitely presentable.

######  Lemma 6.

Let $\mathcal{V}$ be a coextensive variety. Then $\mathcal{V}$ is center
presentable if and only if the functor $Z:\mathcal{V}\rightarrow\mathsf{Set}$
preserves filtering colimits.

###### Proof.

From Theorem 3, the functor $Z$ is representable by
$\mathbf{0}\times\mathbf{0}$. The result follows from Proposition 3.8.14 of
[8]. ∎

###  3.1 coBoolean varieties and the Beth property

Let $\mathsf{C}$ be a category with finite limits. We recall that $\mathsf{C}$
has a subobject classifier if there is a mono $\top:1\rightarrow\Omega$ in
$\mathsf{C}$ such that for every object $X$ and mono $m:S\rightarrow X$ in
$\mathsf{C}$, there exists a unique $\chi_{m}:X\rightarrow\Omega$ such that
the following diagram

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{!}$$\scriptstyle{m}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\top}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{m}}$$\textstyle{\Omega}$

is a pullback.

######  Definition 2.

A category with finite colimits $\mathsf{D}$ has a quotient coclassifier if
$\mathsf{D}^{\mathrm{op}}$ has a subobject classifier.

The following definition is an adaptation of Definition 4.2 of [11].

######  Definition 3.

A coextensive variety $\mathcal{V}$ is said to be coBoolean if the first
projection $\pi:\mathbf{0}\times\mathbf{0}\rightarrow\mathbf{0}$ is a quotient
coclassifier.

Recall that in the case of a variety, quotients are completely determined by
congruences. Therefore, if $\mathcal{V}$ is coextensive and cooBoolean, by
Theorem 3 it is the case that $\mathsf{Con}(\mathbf{A})\cong Z(\mathbf{A})$,
for every $\mathbf{A}\in\mathcal{V}$. This observation motivates the following
definition.

######  Definition 4.

Let $\mathcal{V}$ be a variety with BFC. We say that $\mathcal{V}$ is
congruence-factor if $\mathsf{Con}(\mathbf{A})\cong\mathsf{FC}(\mathbf{A})$,
for every $\mathbf{A}\in\mathcal{V}$.

######  Remark 1.

We say that a variety $\mathcal{V}$ has Boolean congruences if
$\mathsf{Con}(\mathbf{A})$ is a Boolean algebra, for every
$\mathbf{A}\in\mathcal{V}$. If in particular $\mathcal{V}$ is congruence-
distributive, then from Theorem 4 of [22] it follows that $\mathcal{V}$ is
semisimple. Therefore, it is immediate from Definition 4, that congruence-
factor varieties are semisimple and arithmetical.

Let $\mathsf{C}$ be a category with binary products. We say that a morphism
$X\rightarrow Y$ of $\mathsf{C}$ is the _projection of a product_ if there
exist $X\rightarrow Z$ in $\mathsf{C}$ such that the span $Y\leftarrow
X\rightarrow Z$ is a product. Observe that when $\mathsf{C}$ is a variety, the
projections of a product are unique up to isomorphism in the following way: if
$f:\mathbf{A}\rightarrow\mathbf{B}$ is the projection of a product and
$g:\mathbf{A}\rightarrow\mathbf{C}$ and
$g^{\prime}:\mathbf{A}\rightarrow\mathbf{C}^{\prime}$ are such that
$\mathbf{A}\cong\mathbf{B}\times\mathbf{C}\cong\mathbf{B}\times\mathbf{C}^{\prime}$,
thus $\mathsf{Ker}(f)\diamond\mathsf{Ker}(g)$ and
$\mathsf{Ker}(f)\diamond\mathsf{Ker}(g^{\prime})$ so
$\mathsf{Ker}(g)=\mathsf{Ker}(g^{\prime})$. Then, by general reasons (Lemma
3.3 of [5]), there exist a unique isomorphism
$i:\mathbf{C}\rightarrow\mathbf{C}^{\prime}$ such that $ig=g^{\prime}$.

The following result provides a characterization of coBoolean varieties by
means of congruence-factor varieties.

######  Lemma 7.

Let $\mathcal{V}$ be a coextensive variety. Then, the following are
equivalent:

  * (1)

$\mathcal{V}$ is coBoolean.

  * (2)

Every epimorphism in $\mathcal{V}$ is the projection of a product.

  * (3)

$\mathcal{V}$ is congruence-factor and every epimorphism is surjective.

###### Proof.

$(1)\Leftrightarrow(2)$. This is a particular case of the dual of Proposition
4.4 of [11].

$(2)\Rightarrow(3)$. In order to check that $\mathcal{V}$ is congruence-
factor, let $\mathbf{A}\in\mathcal{V}$ and
$\theta\in\mathsf{Con}(\mathbf{A})$. By (2), the quotient map
$\mathbf{A}\rightarrow\mathbf{A}/\theta$ is the projection of a product, so
there exist $\mathbf{B}\in\mathcal{V}$ and $q:\mathbf{A}\rightarrow\mathbf{B}$
such that $\mathbf{A}/\mathsf{Ker}(q)\cong\mathbf{B}$ and
$\mathbf{A}\cong\mathbf{A}/\theta\times\mathbf{B}$. Hence, $\theta$ and
$\mathsf{Ker}(q)$ are complementary factor congruences, as desired. Finally,
if $e:\mathbf{A}\rightarrow\mathbf{B}$ is an epimorphism, by (2) $e$ is a
projection of a product. So in particular, $e$ is surjective.

$(3)\Rightarrow(2)$. If $e:\mathbf{A}\rightarrow\mathbf{B}$ is an epimorphism,
by (3) $e$ is surjective so $\mathbf{B}\cong\mathbf{A}/\mathsf{Ker}(e)$. Since
$\mathcal{V}$ is congruence-factor by assumption, $\mathsf{Ker}(e)$ has a
complementary factor congruence $\theta$. Thus we obtain that $e$ coincides
with a projection of $\mathbf{B}\times\mathbf{A}/\theta$, as claimed. ∎

We conclude this section with an application of Lemma 7 concerning
algebraizable logics in the sense of Blok and Pigozzi and the Beth
(definability) property. The following result reveals that cooBolean varieties
can be useful to decide whether an algebraizable logic has the Beth
(definability) property.

######  Theorem 5.

Let $\mathbb{L}$ be an algebraizable logic with equivalent variety semantics
$\mathcal{V}$ and assume that $\mathcal{V}$ is coextensive. Then the following
hold:

  * (1)

If $\mathcal{V}$ is coBoolean then $\mathbb{L}$ has the Beth (definability)
property.

  * (2)

If $\mathcal{V}$ is congruence-factor and $\mathbb{L}$ has the Beth
(definability) property, then $\mathcal{V}$ is coBoolean.

###### Proof.

The result follows by a straightforward application of Lemma 7 (3), and
Theorem 1. ∎

##  4 The Gaeta topos and fp-coextensive varieties

In this section we show that given a coextensive variety $\mathcal{V}$, the
characterization of coextensive varieties obtained in [36] brings a suitable
axiomatization of the theory of $\mathcal{V}$-indecomposable objects.
Thereafter, we restrict our study to fp-coextensive varieties. In this
setting, will provide a criterion which allows us to decide whether given a
fp-coextensive $\mathcal{V}$, the Gaeta topos classifies
$\mathcal{V}$-indecomposable objects. Finally, with the aim of furnishing some
examples, we apply our results to some particular coextensive varieties of
interest in general algebra and algebraic logic.

We start by proving some technical results on coextensive varieties which will
be used along this section.

######  Lemma 8.

Let $\mathcal{V}$ be a coextensive variety. Then, for every $n$-ary term
$p(\vec{z})$ and constant symbols $c_{1},...,c_{n}$ in the language of
$\mathcal{V}$, there exists a $2n$-ary term $q(\vec{x},\vec{y})$ such that

$\mathcal{V}\models p(\vec{c})=q(\vec{0},\vec{1}).$

###### Proof.

Since $\mathcal{V}$ is coextensive, by Theorem 2 (2), $\mathcal{V}$ is a
Pierce variety in which the relation $\vec{e}\diamond_{\mathbf{A}}{f}$ is
equationally definable. So in particular, $\mathcal{V}$ is a variety with
$\vec{0}$ and $\vec{1}$. Let

$\sigma(\vec{x},\vec{y})=\bigwedge_{i=1}^{n}p_{i}(\vec{x},\vec{y})=q_{i}(\vec{x},\vec{y})$

define the relation $\vec{e}\diamond_{\mathbf{A}}\vec{f}$ in $\mathcal{V}$.
Because $\vec{0}$ and $\vec{1}$ are complementary central elements in
$Z(\mathbf{0})$, then for every $\mathbf{A}\in\mathcal{V}$ and $1\leq i\leq
n$,

$p_{i}^{\mathbf{A}}(\vec{0}^{\mathbf{A}},\vec{1}^{\mathbf{A}})=q_{i}^{\mathbf{A}}(\vec{0}^{\mathbf{A}},\vec{1}^{\mathbf{A}}).$
(3)

Now we consider $X=\\{\vec{x},\vec{y},\vec{z}\\}$ and
$\theta=\bigvee_{i=1}^{n}\mathsf{Cg}^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}(p_{i}(\vec{x},\vec{y}),q_{i}(\vec{x},\vec{y}))$.
Let $\mathbf{H}=\mathbf{F}_{\mathcal{V}}(X)/\theta$. Observe that since
$p_{i}^{\mathbf{H}}(\vec{x}/\theta,\vec{y}/\theta)=q_{i}^{\mathbf{H}}(\vec{x}/\theta,\vec{y}/\theta)$
for every $1\leq i\leq n$, then
$\vec{x}/\theta\diamond_{\mathbf{H}}\vec{y}/\theta$. Therefore, due to

$(p(\vec{z})/\theta,p(\vec{z})/\theta)\in\nabla^{\mathbf{H}}=\mathsf{Cg}^{\mathbf{H}}(\vec{x}/\theta,\vec{0})\circ\mathsf{Cg}^{\mathbf{H}}(\vec{y}/\theta,\vec{0}),$

there exists a term $t(\vec{x},\vec{y},\vec{z})$ such that

$\displaystyle(p^{\mathbf{H}}(\vec{z}/\theta),t^{\mathbf{H}}(\vec{x}/\theta,\vec{y}/\theta,\vec{z}/\theta))\in\mathsf{Cg}^{\mathbf{H}}(\vec{x}/\theta,\vec{0})$
(4)

and

$(p^{\mathbf{H}}(\vec{z}/\theta),t^{\mathbf{H}}(\vec{x}/\theta,\vec{y}/\theta,\vec{z}/\theta))\in\mathsf{Cg}^{\mathbf{H}}(\vec{y}/\theta,\vec{0}).$

Let $\mathbf{A}\in\mathcal{V}$. Then, from (3) and Lemma 3, there exists a
unique $\Omega:\mathbf{H}\rightarrow\mathbf{A}$ such that
$\Omega(\vec{x}/\theta)=\vec{0}^{\mathbf{A}}$,
$\Omega(\vec{y}/\theta)=\vec{1}^{\mathbf{A}}$ and
$\Omega(\vec{z}/\theta)=\vec{c}^{\mathbf{A}}$. Thus from (4) we obtain
$p^{\mathbf{A}}(\vec{c}^{\mathbf{A}})=t^{\mathbf{A}}(\vec{0}^{\mathbf{A}},\vec{1}^{\mathbf{A}},\vec{c}^{\mathbf{A}})$.
Hence $\mathcal{V}\models p(\vec{c})=t(\vec{0},\vec{1},\vec{c})$. Finally, if
we define $q(\vec{x},\vec{y})=t(\vec{x},\vec{y},\vec{c})$, then from the
latter it is the case that $\mathcal{V}\models p(\vec{c})=q(\vec{0},\vec{1})$,
as required. ∎

Let $\mathcal{V}$ be a coextensive variety. Recall that again from Theorem 2,
the relation $\vec{e}\diamond_{\mathbf{A}}\vec{f}$ in $\mathcal{V}$ is
equationally definable. Let

$\sigma(\vec{x},\vec{y})=\bigwedge_{i=1}^{n}p_{i}(\vec{x},\vec{y})=q_{i}(\vec{x},\vec{y})$

define the relation $\vec{e}\diamond_{\mathbf{A}}\vec{f}$ in $\mathcal{V}$.
Now for the rest of this section we consider:

$\theta=\bigvee_{i=1}^{n}\mathsf{Cg}^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}(p_{i}(\vec{x},\vec{y}),q_{i}(\vec{x},\vec{y})),$
$\mu=\mathsf{Cg}^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}(\vec{x},\vec{0})\vee\mathsf{Cg}^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}(\vec{y},\vec{1}),$
$\lambda=\mathsf{Cg}^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}(\vec{x},\vec{1})\vee\mathsf{Cg}^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}(\vec{y},\vec{0}).$

######  Lemma 9.

Let $\mathcal{V}$ be a coextensive variety. Then, the following hold:

  * (1)

$\theta\subseteq\mu,\lambda$;

  * (2)

$\mu\circ\lambda=\lambda\circ\mu=\nabla^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}$;

  * (3)

$\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\mu\cong\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\lambda\cong\mathbf{0}$.

  * (4)

$\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/(\mu\cap\lambda)\cong\mathbf{0}\times\mathbf{0}$.

###### Proof.

$(1)$ Since $\mathcal{V}$ is coextensive, by Theorem 2 (2), $\mathcal{V}$ is a
Pierce variety in which the relation $\vec{e}\diamond_{\mathbf{A}}{f}$ is
equationally definable. So in particular, by Lemma 4.3 (1) of [36],
$\mathcal{V}$ is a variety with $\vec{0}$ and $\vec{1}$. Since
$\vec{0}^{\mathbf{A}},\vec{1}^{\mathbf{A}}\in Z(\mathbf{A})$ for every
$\mathbf{A}\in\mathcal{V}$, it is the case that

$\mathcal{V}\models\vec{x}=\vec{0}\wedge\vec{y}=\vec{1}\Longrightarrow\sigma(\vec{x},\vec{y})$

so in particular

$\mathcal{V}\models\vec{x}=\vec{0}\wedge\vec{y}=\vec{1}\Longrightarrow
p_{i}(\vec{x},\vec{y})=q_{i}(\vec{x},\vec{y})$

for every $1\leq i\leq n$. Then from Lemma 2

$\mathsf{Cg}^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}(p_{i}(\vec{x},\vec{y}),q_{i}(\vec{x},\vec{y}))\subseteq\mu$

for every $1\leq i\leq n$. Therefore, $\theta\subseteq\mu$ as required. The
proof of $\theta\subseteq\lambda$ is analogue.

$(2)$ Let
$(s(\vec{x},\vec{y}),t(\vec{x},\vec{y}))\in\nabla^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}$.
By Theorem 2 (2) $\mathcal{V}$ is a Pierce variety so there exists a term
$U(x,y,\vec{z},\vec{w})$ in the language of $\mathcal{V}$ such that

$\begin{array}[]{ccc}U(x,y,\vec{0},\vec{1})=x&\text{and}&U(x,y,\vec{1},\vec{0})=y.\end{array}$

Let us consider
$p(\vec{x},\vec{y})=U(s(\vec{x},\vec{y}),t(\vec{x},\vec{y}),\vec{x},\vec{y})$.
Observe that $p(\vec{0},\vec{1})=s(\vec{0},\vec{1})$ and
$p(\vec{1},\vec{0})=t(\vec{1},\vec{0})$. Thus, it follows that

$\mathcal{V}\models(\vec{x}=\vec{0}\wedge\vec{y}=\vec{1})\Longrightarrow
p(\vec{x},\vec{y})=s(\vec{x},\vec{y})$

and

$\mathcal{V}\models(\vec{x}=\vec{1}\wedge\vec{y}=\vec{0})\Longrightarrow
s(\vec{x},\vec{y})=t(\vec{x},\vec{y}).$

Thus by Lemma 2 we get

$\begin{array}[]{ccc}(s(\vec{x},\vec{y}),p(\vec{x},\vec{y}))\in\mu&\text{and}&(p(\vec{x},\vec{y}),t(\vec{x},\vec{y}))\in\lambda.\end{array}$

Hence $\mu\circ\lambda=\nabla^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}$.
Finally, we stress that if we take

$q(\vec{x},\vec{y})=U(s(\vec{x},\vec{y}),t(\vec{x},\vec{y}),\vec{y},\vec{x})$

it is no hard to see that $q(\vec{0},\vec{1})=t(\vec{0},\vec{1})$ and
$q(\vec{1},\vec{0})=s(\vec{1},\vec{0})$. Therefore, by the same argument we
employed before together with Lemma 2, we get
$\lambda\circ\mu=\nabla^{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})}$, as
claimed.

$(3)$ Let $\mathbf{H}=\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\mu$ and
$\mathbf{H}^{\prime}=\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\lambda$. We
will prove that $\mathbf{H}$ and $\mathbf{H}^{\prime}$ are both isomorphic to
$\mathbf{0}$. We will only exhibit the details about
$\mathbf{H}\cong\mathbf{0}$ because the proof of the last part is similar. We
start by considering the map $h:H\rightarrow F_{\mathcal{V}}(\emptyset)$
defined as $h(q(\vec{x},\vec{y})/\mu)=q(\vec{0},\vec{1})$. We will show that
$h$ is an isomorphism. In order to see that $h$ is well defined, let
$q(\vec{x},\vec{y}),q^{\prime}(\vec{x},\vec{y})\in
F_{\mathcal{V}}(\vec{x},\vec{y})$ and suppose that
$q(\vec{x},\vec{y})/\mu=q^{\prime}(\vec{x},\vec{y})/\mu$. Thus
$(q(\vec{x},\vec{y}),q^{\prime}(\vec{x},\vec{y}))\in\mu$. Then, from Lemma 2
we get

$\mathcal{V}\models(\vec{x}=\vec{0}\wedge\vec{y}=\vec{1})\Longrightarrow
q(\vec{x},\vec{y})=q^{\prime}(\vec{x},\vec{y}),$

so, in particular $q^{F_{\mathcal{V}}(\emptyset)}(\vec{0},\vec{1})=q^{\prime
F_{\mathcal{V}}(\emptyset)}(\vec{0},\vec{1})$ as claimed. Moreover, notice
that the same argument applied in the reverse direction allows us to prove
that $h$ is injective. The surjectivity of $h$ follows from Lemma 8. Finally,
it is clear that $h$ is a homomorphism. Hence
$\mathbf{H}\cong\mathbf{0}\cong\mathbf{H}^{\prime}$ as desired.

$(4)$ Immediate from $(2)$, $(3)$ and Lemma 1. ∎

###  4.1 $\mathcal{V}$-indecomposable objects in a topos

Let $\mathcal{V}$ be a coextensive variety. An algebra $\mathbf{A}$ of
$\mathcal{V}$ is said to be _$\mathcal{V}$ -indecomposable_ if it is
indecomposable by binary products; i.e. if
$\mathbf{A}\cong\mathbf{B}\times\mathbf{C}$, then $\mathbf{B}\cong\mathbf{1}$
or $\mathbf{C}\cong\mathbf{1}$. The following result allows will show that the
theory of central elements brings axiomatization for the theory of
$\mathcal{V}$-indecomposable objects.

######  Lemma 10.

The class of $\mathcal{V}$-indecomposable objects is axiomatizable by a first
order formula.

###### Proof.

From Theorem 2 (2) the relation $\vec{e}\diamond_{\mathbf{A}}\vec{f}$ is
equationally definable in $\mathcal{V}$. So we can take
$\sigma(\vec{x},\vec{y})$ as an equation defining such a relation. It is
immediate that $\mathbf{A}\in\mathcal{V}$ is $\mathcal{V}$-indecomposable if
and only if in $\mathbf{A}$ the following sentence holds

$\vec{0}\neq\vec{1}$ and
$(\forall_{\vec{e},\vec{f}}\sigma(\vec{e},\vec{f})\Rightarrow((\vec{e}=\vec{0}\wedge\vec{f}=\vec{1})\vee(\vec{e}=\vec{1}\wedge\vec{f}=\vec{0}))).$

∎

Observe that from Lemma 10, it follows that an algebra $\mathbf{A}$ in
$\mathcal{V}$ is $\mathcal{V}$-indecomposable if and only if the sequents

$0=1\vdash\perp$
$\sigma(\vec{x},\vec{y})\vdash_{\vec{x},\vec{y}}(\vec{x}=\vec{0}\wedge\vec{y}=\vec{1})\vee(\vec{x}=\vec{1}\wedge\vec{y}=\vec{0})$

hold in $\mathbf{A}$.

Let $\mathcal{V}$ be a coextensive variety, $f$ be a symbol in the language of
$\mathcal{V}$, $\mathsf{E}$ be a topos and $M$ be an object of $\mathsf{E}$.
If we write $a_{f}$ for the arity of $f$ (which is a natural number), recall
(see D1.2.1 of [21]) that the interpretation of $f$ in $M$ is a morphism
$f_{M}:M^{a_{f}}\rightarrow M$. Thus a _$\mathcal{V}$ -model in $\mathsf{E}$_
is an object $M$ of $\mathsf{E}$ equipped with morphisms
$f_{M}:M^{a_{f}}\rightarrow M$ for every symbol $f$ in the language of
$\mathcal{V}$ for which the defining identities of $\mathcal{V}$ (expressed by
diagrams in $\mathsf{E}$) hold. Moreover, a homomorphism between
$\mathcal{V}$-models $M$ and $R$ in $\mathsf{E}$ is an arrow $h:M\rightarrow
R$ in $\mathsf{E}$ making the diagram

$\textstyle{M^{a_{f}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{a_{f}}}$$\scriptstyle{f_{M}}$$\textstyle{R^{a_{f}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{R}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{R}$

commutes for every symbol $f$ in the language of $\mathcal{V}$ (here
$h^{a_{f}}$ denotes the product morphism of $h$ $a_{f}$-times). This
information defines the category of $\mathcal{V}$-models in $\mathsf{E}$.
Notice that in particular, when regarding the topos $\mathsf{Set}$, the
category of $\mathcal{V}$-models coincides with $\mathcal{V}$. In order to
illustrate the latter, let us consider the variety $\mathcal{DL}_{01}$ of
bounded distributive lattices. Then a $\mathcal{DL}_{01}$-model in
$\mathsf{E}$ is an object $L$ of $\mathsf{E}$ endowed with arrows

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{L}}$$\scriptstyle{0_{L}}$$\textstyle{L}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
L\times L}$$\scriptstyle{\wedge_{L}}$$\scriptstyle{\vee_{L}}$

such that the equations defining $\mathcal{DL}_{01}$ hold. For instance, the
commutativity of the meet can be expressed by the commutativity of the
following diagram in $\mathsf{E}$:

$\textstyle{L\times
L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle\pi_{1},\pi_{2}\rangle}$$\scriptstyle{\langle\pi_{2},\pi_{1}\rangle}$$\textstyle{L\times
L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge_{L}}$$\textstyle{L\times
L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge_{L}}$$\textstyle{L}$

Now, motivated by the observation made right after Lemma 10 we introduce the
following:

######  Definition 5.

Let $\mathsf{E}$ be a topos. A $\mathcal{V}$-model $M$ of $\mathsf{E}$ is
$\mathcal{V}$-indecomposable if the sequents

$0=1\vdash\perp$
$\sigma(\vec{x},\vec{y})\vdash_{\vec{x},\vec{y}}(\vec{x}=\vec{0}\wedge\vec{y}=\vec{1})\vee(\vec{x}=\vec{1}\wedge\vec{y}=\vec{0})$

hold in the internal logic of $\mathsf{E}$.

We stress that Definition 5 can be driven to categorical terms. To do so, let
$M$ be a $\mathcal{V}$-model in a topos $\mathsf{E}$ and let $\vec{0}_{M}$,
$\vec{1}_{M}$ and $[\sigma(\vec{x},\vec{y})]_{M}$ be the interpretations in
$M$ of the constants $\vec{0}$, $\vec{1}$ and a equation
$\sigma(\vec{x},\vec{y})$ defining the relation
$\vec{e}\diamond_{\mathbf{A}}\vec{f}$ in $\mathcal{V}$, respectively (for
details, see D1.2.6 of [21]). Now, let us consider the elements
$\langle\vec{0}_{M},\vec{1}_{M}\rangle:1\rightarrow M^{N}\times M^{N}$ and
$\langle\vec{1}_{M},\vec{0}_{M}\rangle:1\rightarrow M^{N}\times M^{N}$. If
$\alpha=[\langle\vec{0}_{M},\vec{1}_{M}\rangle,\langle\vec{1}_{M},\vec{0}\rangle_{M}]$
denotes the morphism from $1+1$ to $[\sigma(\vec{x},\vec{y})]_{M}$ induced by
the coproduct, then a basic exercise in the internal logic of toposes shows
the following:

######  Lemma 11.

Let $\mathsf{E}$ be a topos and let $M$ be a $\mathcal{V}$-model in
$\mathsf{E}$. The following are equivalent:

  * (1)

$M$ is $\mathcal{V}$-indecomposable,

  * (2)

The diagram below

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{!`}_{1}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vec{1}_{M}}$$\scriptstyle{\vec{0}_{M}}$$\textstyle{M^{N}}$

is an equalizer in $\mathsf{E}$, and the morphism
$\alpha:1+1\rightarrow[\sigma(\vec{x},\vec{y})]_{M}$ is an isomorphism.

  * (3)

In the internal logic of $\mathsf{E}$, the following sequents hold:

    * (C1)

$\vec{0}=\vec{1}\vdash\perp$,

    * (C2)

$\sigma(\vec{x},\vec{y})\vdash_{\vec{x},\vec{y}}(\vec{x}=\vec{0}\wedge\vec{y}=\vec{1})\vee(\vec{x}=\vec{1}\wedge\vec{y}=\vec{0}).$

###  4.2 The characterization

Let $\mathsf{C}$ be a small extensive category. For every object $X$ of
$\mathsf{C}$, we say that $\\{f_{i}:X_{i}\rightarrow X\mid i\in I\\}\in
K_{\mathcal{G}}(X)$ if and only if $I$ is finite and the induced arrow $\Sigma
X_{i}\rightarrow X$ is an isomorphism. From the extensivity condition it
follows that $K_{\mathcal{G}}$ is a basis for a Grothedieck topology over
$\mathsf{C}$ (see III.2.1 of [26]). The topology $J_{\mathcal{G}}$ generated
by such a basis is called the _Gaeta Topology_ and the _Gaeta topos_
$\mathcal{G}(\mathsf{C})$, is the topos of sheaves on the site
$(\mathsf{C},J_{\mathcal{G}})$. As observed in [12], $\mathcal{G}(\mathsf{C})$
is equivalent to the category
$\mathsf{Lex}(\mathsf{C}^{\mathrm{op}},\mathsf{Set})$ of product preserving
functors to $\mathsf{Set}$ from the category $\mathsf{C}^{\mathrm{op}}$ with
finite products. This fact implies that $J_{\mathcal{G}}$ is subcanonical. If
$\mathsf{C}$ has a terminal object $1$, from Proposition 4.1 of [11], it
follows that $\mathsf{C}$ is extensive if and only if the canonical functors
$1\rightarrow\mathsf{C}/0$ and
$\mathsf{C}/(1+1)\rightarrow\mathsf{C}\times\mathsf{C}$ are equivalences.

######  Remark 2.

Let $\mathsf{C}$ be an extensive category with a terminal object $1$ and let
$\mathsf{E}$ be a topos. Notice that a finite limit preserving functor
$G:\mathsf{C}\rightarrow\mathsf{E}$ is continuous (see VII.7 of [26]) with
respect to the Gaeta topology over $\mathsf{C}$ if and only $G(0)\cong 0$ and
$G(1+1)\cong 1+1$; i.e. it preserves binary coproducts.

Let $\mathcal{V}$ be a coextensive variety. We write
$\textsf{Mod}_{\textsf{fp}}(\mathcal{V})$ for the full subcategory of finitely
presented algebras of $\mathcal{V}$. Let $\mathsf{E}$ be a topos. Due to
Lawvere’s duality [23], it is known that the category of $\mathcal{V}$-models
in $\mathsf{E}$ is equivalent to the category of limit preserving functors
$\mathsf{Lex}(\textsf{Mod}_{\textsf{fp}}(\mathcal{V})^{\mathsf{op}},\mathsf{E})$.
So, for every $\mathcal{V}$-model $M$ in $\mathsf{E}$, there exists an
essentially unique limit preserving functor
$\phi_{M}:\textsf{Mod}_{\textsf{fp}}(\mathcal{V})^{\mathsf{op}}\rightarrow\mathsf{E}$,
such that $\phi_{M}(\textbf{F}_{\mathcal{V}}(x))\cong M$. In what follows we
will refer to $\phi_{M}$ as the _representative_ of $M$. It is worth
mentioning that in the case of $\mathsf{E}=\mathsf{Set}$, the representative
of $\textbf{F}_{\mathcal{V}}(x)$ reflects isomorphisms.

######  Theorem 6.

Let $\mathcal{V}$ be a coextensive variety. Then
$\textsf{Mod}_{\textsf{fp}}(\mathcal{V})$ is coextensive if and only if binary
products of finitely generated free algebras of $\mathcal{V}$ are finitely
presented.

###### Proof.

Let $\mathbf{F}_{\mathcal{V}}(n)/\theta$ and
$\mathbf{F}_{\mathcal{V}}(m)/\delta$ be finitely presented algebras of
$\mathcal{V}$. Then, $\theta$ and $\delta$ are finitely generated congruences
on $\mathbf{F}_{\mathcal{V}}(n)$ and $\mathbf{F}_{\mathcal{V}}(m)$,
respectively. From Lemma 4.3 (4) of [36], $\mathcal{V}$ has the Fraser Horn
property, thus

$\mathbf{F}_{\mathcal{V}}(n)/\theta\times\mathbf{F}_{\mathcal{V}}(m)/\delta\cong(\mathbf{F}_{\mathcal{V}}(n)\times\mathbf{F}_{\mathcal{V}}(m))/(\theta\times\delta).$
(5)

So, due to Theorem 3 (6) of [18], $\theta\times\delta$ is finitely generated.
Since $\mathbf{F}_{\mathcal{V}}(n)\times\mathbf{F}_{\mathcal{V}}(m)$ is
finitely presented by assumption, there exist variables $x_{1},\ldots,x_{k}$
and a finitely generated congruence $\epsilon$ on
$\mathbf{F}_{\mathcal{V}}(k)$, such that
$\mathbf{F}_{\mathcal{V}}(n)\times\mathbf{F}_{\mathcal{V}}(m)\cong\mathbf{F}_{\mathcal{V}}(k)/\epsilon$.
Thus, from Theorem 6.20 of [10], there exists a compact congruence $\gamma$ on
$\mathbf{F}_{\mathcal{V}}(k)$, with $\epsilon\subseteq\gamma$, such that
$\theta\times\delta=\gamma/\epsilon$. Therefore, from Theorem 6.15 of [10] and
(5) we conclude that
$\mathbf{F}_{\mathcal{V}}(n)/\theta\times\mathbf{F}_{\mathcal{V}}(m)/\delta$
is finitely presented, so $\textsf{Mod}_{\textsf{fp}}(\mathcal{V})$ has finite
products and consequently, it is coextensive. On the other hand, if
$\textsf{Mod}_{\textsf{fp}}(\mathcal{V})$ is coextensive, it has finite
products. Observe that since $\mathcal{V}$ is coextensive, from Lemma 4.3 of
[36], $\Delta^{\mathbf{A}}=\mathsf{Cg}^{\mathbf{A}}(\vec{0},\vec{0})$ for
every $\mathbf{A}\in\mathcal{V}$. In particular, this implies that every
finitely generated free algebra of $\mathcal{V}$ is finitely presented so,
binary products between them must be finitely presented. This concludes the
proof. ∎

A coextensive variety $\mathcal{V}$ is said to be _fp-coextensive_ if it
satisfies any of the equivalent conditions of Theorem 6. The following result,
immediately establishes a relation between fp-coextensive varieties and center
presentable varieties (see Definition 1).

######  Corollary 3.

Every fp-coextensive variety es center presentable.

At this stage one may be wandering if the finiteness of the type of
coextensive varieties plays any rôle to decide fp-coextensivity. The next
result provides an answer to this question.

######  Proposition 2.

Let $\mathcal{V}$ be a coextensive variety of finite type. If $\mathcal{V}$ is
locally finite then it is fp-coextensive. So, in particular, the functor $Z$
preserves filtering colimits.

###### Proof.

Let us assume that $\mathcal{V}$ is of finite type, coextensive and locally
finite. Let $\mathbf{F}_{\mathcal{V}}(n)$ and $\mathbf{F}_{\mathcal{V}}(m)$ be
finitely generated free algebras of $\mathcal{V}$. From Theorem 10.15 of [10],
the set $X=F_{\mathcal{V}}(n)\times F_{\mathcal{V}}(n)$ is finite. We stress
that $\mathbf{F}_{\mathcal{V}}(n)\times\mathbf{F}_{\mathcal{V}}(m)$ is
finitely presented because from Corollary II.10.11 of [10], such an algebra is
in fact isomorphic to $\mathbf{F}_{\mathcal{V}}(X)$ quotiented by the finitely
many conditions which describe the operations in
$\mathbf{F}_{\mathcal{V}}(n)\times\mathbf{F}_{\mathcal{V}}(m)$. The last part
follows from Corollary 3. ∎

Let $\mathcal{V}$ a fp-coextensive variety. Let $x_{1},\ldots,x_{k}$ be a
finite set of variables and let $p_{1},\ldots,p_{k},q_{1},\ldots,q_{k}$ be
terms in the language of $\mathcal{V}$ with variables $y_{1},\ldots,y_{l}$. If
$\delta$ denotes the congruence
$\bigvee_{i=1}^{k}\mathsf{Cg}^{\mathbf{F}_{\mathcal{V}}(\vec{y})}(p_{i}(\vec{y}),q_{i}(\vec{y}))$
and $\mathbf{A}$ denotes the algebra
$\mathbf{F}_{\mathcal{V}}(\vec{y})/\delta$, observe that the representative of
$M$ sends the finitely presentable algebra $\mathbf{A}$ to the following
equalizer in $\mathsf{E}$:

$\textstyle{\phi_{M}(\mathbf{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle
p_{M_{1}},\ldots,p_{M_{k}}\rangle}$$\scriptstyle{\langle
q_{M_{1}},\ldots,q_{M_{k}}\rangle}$$\textstyle{M^{k}}$

where $p_{M_{i}}$ and $q_{M_{i}}$ denote the interpretation in $\mathsf{E}$ of
the terms $p_{i}$ and $q_{i}$ in $M$, respectively, with $1\leq i\leq k$. I.e.
the image of $\mathbf{A}$ by $\phi_{M}$ essentially coincides in $\mathsf{E}$
with the interpretation in $M$ of the formula

$\varepsilon(\vec{y})=\bigwedge_{i=1}^{k}p_{i}(\vec{y})=q_{i}(\vec{y}).$

In what follows, we write $\mathcal{G}(\mathcal{V})$ for the Gaeta topos
determined by the extensive category
$\textsf{Mod}_{\textsf{fp}}(\mathcal{V})^{\mathsf{op}}$. We recall that from
VII.7.4 of [26] there is an equivalence between the category
$\mathsf{Geo}(\mathsf{E},\mathcal{G}(\mathcal{V}))$ of geometric morphisms
from $\mathsf{E}$ to $\mathcal{G}(\mathcal{V})$ and the category
$\mathsf{LexCon}(\textsf{Mod}_{\textsf{fp}}(\mathcal{V})^{\mathsf{op}},\mathsf{E})$
of limit preserving functors from
$\textsf{Mod}_{\textsf{fp}}(\mathcal{V})^{\mathsf{op}}$ to $\mathsf{E}$ which
are continuous with respect to the Gaeta topology over
$\textsf{Mod}_{\textsf{fp}}(\mathcal{V})^{\mathsf{op}}$.

As result of the above discussion, now we can restate Lemma 11 by means of the
representative of a $\mathcal{V}$-model in a topos $\mathsf{E}$.

######  Lemma 12.

Let $\mathcal{V}$ be a fp-coextensive variety, $\mathsf{E}$ be a topos and let
$M$ be a $\mathcal{V}$-model in $\mathsf{E}$. Let $\phi_{M}$ be the
representative of $M$. Then, the following are equivalent:

  * (1)

$M$ is $\mathcal{V}$-indecomposable in $\mathsf{E}$.

  * (2)

$\phi_{M}(\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\theta)\cong 1+1$ and
$\phi_{M}(\mathbf{1})\cong 0$.

Moreover, if $\mathsf{E}=\mathsf{Set}$ and $M=\mathbf{F}_{\mathcal{V}}(x)$,
any of the above conditions is equivalent to
$\phi_{\mathbf{F}_{\mathcal{V}}(x)}$ preserves finite coproducts.

###### Proof.

Let $M$ be a $\mathcal{V}$-model in a topos $\mathsf{E}$. Notice that it is
the case that

$[\sigma(\vec{x},\vec{y})]_{M}\cong\phi_{M}(\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\theta)$

and

$[\vec{0}=\vec{1}]_{M}\cong\phi_{M}(\mathbf{0}/\mathsf{Cg}^{\mathbf{0}}(\vec{0},\vec{1})).$

Since $\mathcal{V}$ is a variety with $\vec{0}$ and $\vec{1}$, then

$\mathbf{0}/\mathsf{Cg}^{\mathbf{0}}(\vec{0},\vec{1})\cong\mathbf{1}.$

Hence from Lemma 11 and Remark 2 it is immediate that a $\mathcal{V}$-model
$M$ in $\mathsf{E}$ is $\mathcal{V}$-indecomposable in such a topos if and
only if $(2)$ holds.

For the moreover part, we start by noticing that from Lemma 9, there exist
arrows $f$ and $g$ from $\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\theta$ to
$\mathbf{0}$. Now consider the following diagram in $\mathcal{V}$, in which
the outer vertical arrows denote the identity of $\mathbf{0}$ and the middle
vertical arrow is the arrow induced by the product.

$\textstyle{\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\theta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{0}}$$\textstyle{\mathbf{0}\times\mathbf{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{0}}$
(6)

We stress that Lemma 9 $(1)$, $(3)$ and $(4)$ allows us to say that each of
the squares of such a diagram is a pushout. Thus, if
$\mathbf{F}_{\mathcal{V}}(x)$ is $\mathcal{V}$-indecomposable in
$\mathsf{Set}$, by condition (2), we obtain that
$\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\theta)$
is isomorphic to $1+1$ and $\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{1})$ is
isomorphic to $\emptyset$. Thus, since $\phi_{\mathbf{F}_{\mathcal{V}}}(x)$
preserves finite limits, in order to proof that such a functor preserves
finite coproducts, we only need to show that
$\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{0}\times\mathbf{0})$ is isomorphic
to $1+1$. To do so, notice that from Lemma condition (1) and because
$\phi_{\mathbf{F}_{\mathcal{V}}(x)}$ preserves pullbacks, the diagram of above
turns into the following diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{0}\times\mathbf{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1+1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

in which both squares are pullbacks in $\mathsf{Set}$. Since $\mathsf{Set}$ is
extensive, then we conclude that
$\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{0}\times\mathbf{0})$ must be
isomorphic to $1+1$, as desired. On the other hand, if
$\phi_{\mathbf{F}_{\mathcal{V}}(x)}$ preserves finite coproducts, then
$\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{1})\cong\emptyset$. Thus, because
both of the squares of diagram (6) are pushouts and again by the extensivity
of $\mathsf{Set}$, we get

$\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{0}\times\mathbf{0})\cong
1+1\cong\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\theta),$

as claimed.

∎

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

######  Theorem 7.

Let $\mathcal{V}$ be a fp-coextensive variety. Then, the following are
equivalent:

  * (1)

$\mathcal{G}(\mathcal{V})$ is a classifying topos for
$\mathcal{V}$-indecomposable objects.

  * (2)

$\mathbf{F}_{\mathcal{V}}(x)$ is $\mathcal{V}$-indecomposable in
$\mathsf{Set}$.

###### Proof.

For the sake of readability of the following proofs, we start by fixing some
notation. Let $\mathsf{E}$ be a topos. We write
$\mathsf{Mod}(\mathcal{V}_{I},\mathsf{E})$ for the category of
$\mathcal{V}$-indecomposable objects in $\mathsf{E}$ and we will denote by
$\mathsf{A}$ the category $\mathsf{Mod_{fp}}(\mathcal{V})$.

$(1)\Rightarrow(2)$ Let us assume that $\mathcal{G}(\mathcal{V})$ classifies
$\mathcal{V}$-indecomposable objects. Then, for every topos $\mathsf{E}$, the
categories $\mathsf{Mod}(\mathcal{V}_{I},\mathsf{E})$ and
$\mathsf{Geo}(\mathsf{E},\mathsf{Set}^{\mathsf{A}})$ are equivalent. Thus, in
particular,
$\mathsf{Mod}(\mathcal{V}_{I},\mathsf{Set})\approx\mathsf{Geo}(\mathsf{Set},\mathsf{Set}^{\mathsf{A}})$.
On the other hand, from VII.7.2 of [26], there is an equivalence between the
category $\mathsf{Lex}(\mathsf{A}^{\mathrm{op}},\mathsf{Set})$ and the
category $\mathsf{Geo}(\mathsf{Set},\mathsf{Set}^{\mathsf{A}})$, so let
$g:\mathsf{Set}\rightarrow\mathsf{Set}^{\mathsf{A}}$ be the geometric morphism
corresponding to $\phi_{\textbf{F}_{\mathcal{V}}(x)}$ from this equivalence.
We will prove that $\phi_{\textbf{F}_{\mathcal{V}}(x)}$ is continuous with
respect $\mathcal{J}_{\mathcal{G}}$. Because $\mathcal{J}_{\mathcal{G}}$ is
subcanonical, $g$ factors through the inclusion from
$\mathcal{G}(\mathcal{V})$ to $\mathsf{Set}^{\mathsf{A}}$, therefore by
VII.7.3 of [26] we get
$g^{\ast}\circ\textbf{y}\cong\phi_{\textbf{F}_{\mathcal{V}}(x)}$ is continuous
with respect $\mathcal{J}_{\mathcal{G}}$, as claimed. Whence,

$(g^{\ast}\circ\textbf{y})(\mathbf{F}_{\mathcal{V}}(x))\cong\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{F}_{\mathcal{V}}(x))\cong\mathbf{F}_{\mathcal{V}}(x)$

is indecomposable in $\mathsf{Set}$, as desired.

$(2)\Rightarrow(1)$ If $\mathbf{F}_{\mathcal{V}}(x)$ is
$\mathcal{V}$-indecomposable in $\mathsf{Set}$, then, from Lemma 12 and Remark
2, $\phi_{\mathbf{F}_{\mathcal{V}}(x)}$ is continuous with respect to
$\mathcal{J}_{\mathcal{G}}$. So, in order to prove our claim, we need to show
that there is an equivalence between
$\mathsf{Mod}(\mathcal{V}_{I},\mathsf{E})$ and
$\mathsf{Geo}(\mathsf{E},\mathcal{G}(\mathcal{V}))$, for every topos
$\mathsf{E}$. Since $\mathsf{LexCon}(\mathsf{A}^{\mathrm{op}},\mathsf{E})$ and
$\mathsf{Geo}(\mathsf{E},G(\mathcal{V}))$ are equivalent from VII.7.4 of [26],
we only need to prove that $\mathsf{Mod}(\mathcal{V}_{I},\mathsf{E})$ and
$\mathsf{LexCon}(\mathsf{A}^{\mathrm{op}},\mathsf{E})$ are equivalent for
every topos $\mathsf{E}$. To do so, let $\mathsf{E}$ be a topos and let
$H:\mathsf{A}^{\mathrm{op}}\rightarrow\mathsf{E}$ be a finite limit preserving
functor continuous with respect to $J_{\mathcal{G}}$. From Lawvere’s duality,
it follows that $M=H(\textbf{F}_{\mathcal{V}}(x))$ is a $\mathcal{V}$-model in
$\mathsf{E}$ and also that $\phi_{M}\cong H$. Then from the following
calculation

$\displaystyle[\vec{0}=\vec{1}]_{M}\cong\phi_{M}(\mathbf{0}/\mathsf{Cg}^{\mathbf{0}}(\vec{0},\vec{1})))\cong\phi_{M}(\mathbf{1})\cong
H(\mathbf{1})\cong 0,$

we obtain that in the internal logic of $\mathsf{E}$, the sequent $(C1)$
holds. Now, observe that from $(2)$, and the moreover part of Lemma 12, it is
the case that
$\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\theta)\cong\phi_{\mathbf{F}_{\mathcal{V}}(x)}(\mathbf{0}\times\mathbf{0})$.
So since $\phi_{\mathbf{F}_{\mathcal{V}}(x)}$ reflects isomorphisms, we get
$\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\theta\cong\mathbf{0}\times\mathbf{0}$.
Therefore, from the following calculation

$\displaystyle[\sigma(\vec{x},\vec{y})]_{M}\cong\phi_{M}(\mathbf{F}_{\mathcal{V}}(\vec{x},\vec{y})/\theta)\cong
H(\mathbf{0}\times\mathbf{0})\cong 1+1,$

we get that the sequent $(C2)$ also holds in the internal logic of
$\mathsf{E}$. Hence, by Lemma 11 $(3)$, we conclude that $M$ is indecomposable
in $\mathsf{E}$. The proof that a $\mathcal{V}$-indecomposable model $M$ in
$\mathsf{E}$ determines a functor in
$\mathsf{LexCon}(\mathsf{A}^{\mathrm{op}},\mathsf{E})$ is similar. Hence, from
D3.1.9 in [21], the functor
$\mathsf{LexCon}(\mathsf{A}^{\mathrm{op}},\mathsf{E})\rightarrow\mathsf{Mod}(\mathcal{V}_{I},\mathsf{E})$
which sends $H$ to $H(\textbf{F}_{\mathcal{V}}(x))$ is an equivalence of
categories.

∎

###  4.3 Applications

####  4.3.1 Bounded distributive lattices

Let $\mathcal{DL}_{01}$ be the variety of bounded distributive lattices.
Straightforward calculations show that the term $U(x,y,z,w)=(x\vee
z)\wedge(y\vee w)$, together with the constants $0$ and $1$, makes
$\mathcal{DL}_{01}$ a Pierce variety. In addition it is also well known that
the relation $e\diamond_{\mathbf{A}}f$ in $\mathcal{DL}_{01}$ is defined by
the equations $e\wedge f=0$ and $e\vee f=1$. Hence, by Theorem 2,
$\mathcal{DL}_{01}$ is coextensive. Moreover, the only subdirectly irreducible
member of $\mathcal{DL}$ is the two element distributive lattice $\mathbf{2}$,
then from Theorem 10.16 of [10] it follows that $\mathcal{DL}_{01}$ is locally
finite. Thus, by Proposition 2, $\mathcal{DL}_{01}$ is fp-coextensive.
Finally, since $\textbf{F}_{\mathcal{DL}_{01}}(x)$ is indecomposable (it is
the three element chain) from Corollary 3 and Theorem 7 we can conclude:

######  Proposition 3.

The variety $\mathcal{DL}_{01}$ is fp-coextensive and
$\mathcal{G}(\mathcal{DL}_{01})$ classifies $\mathcal{DL}_{01}$-indecomposable
objects. In particular, the functor
$Z:\mathcal{DL}_{01}\rightarrow\mathsf{Set}$ preserves all limits and
filtering colimits.

It is worth mentioning that the first part of Proposition 3 was stated without
a proof in Section 8 of [25] and later on, in [35] a detailed proof was
provided.

####  4.3.2 Integral rigs

A _rig_ is an algebra $\mathbf{A}=(A,+,\cdot,0,1)$ of type $(2,2,0,0)$ such
that the structures $(A,\cdot,1)$ and $(A,+,0)$ are commutative monoids such
that “product distributes over addition” in the sense that $x\cdot 0=0$ and
$x\cdot(y+z)=(x\cdot y)+(x\cdot z)$ for every $x,y,z\in A$. One may think such
structures as “(commutative) rings (with unit) without negatives”. A rig is
said to be _integral_ if the equation $1+x=1$ holds, for every $x\in A$. It is
immediate from the latter that the class of integral rigs is a variety. We
denote such a variety by $\mathcal{RN}$. Observe that the term
$U(x,y,z,w)=(x+z)\cdot(y+w)$, together with the constants $0$ and $1$ makes
$\mathcal{RN}$ a Pierce variety. In [14], it was proved that the relation
$e\diamond_{\mathbf{A}}f$ in $\mathcal{RN}$ is defined by the equations
$e\cdot f=0$ and $e+f=1$. Thus by Theorem 2, $\mathcal{RN}$ is coextensive.
Moreover, in Corollary 8.1 of [29] it was proved that $\mathcal{RN}$ is fp-
coextensive. Observe that $\mathbf{F}_{\mathcal{RN}}(x)$ can be identified
with the chain ${0<...<x^{n}<...<x^{2}<x<x^{0}=1}$ with the obvious
multiplication and addition, so the free integral rig on one generator $x$ is
indecomposable. Therefore, by Corollary 3 and Theorem 7 we have proved the
following:

######  Proposition 4.

The variety $\mathcal{RN}$ is fp-coextensive and $\mathcal{G}(\mathcal{RN})$
classifies $\mathcal{RN}$-indecomposable objects. In particular, the functor
$Z:\mathcal{RN}\rightarrow\mathsf{Set}$ preserves all limits and filtering
colimits.

For a different proof of the fact that $\mathcal{G}(\mathcal{RN})$ classifies
$\mathcal{RN}$-indecomposable objects, the reader may consult [29].

####  4.3.3 Commutative rings with unit

Let $\mathcal{R}$ be the variety of commutative rings with unit. It is fairly
known that $\mathbf{A}\in\mathcal{R}$ is directly indecomposable if and only
if the only idempotents in $A$ are the trivial ones. Observe that this is
equivalent to say that the formula

$\sigma(x,y)=(xy=0)\wedge(x+y=1)$

defines the relation $e\diamond_{\mathbf{A}}f$ in $\mathcal{R}$. Moreover, we
can employ the same term used for integral rigs and the constants $0$ and $1$
in order to make $\mathcal{R}$ a Pierce variety. So from Theorem 2 we get
$\mathcal{R}$ is coextensive. It is also well known that $\mathcal{R}$ is fp-
coextensive. Recall that the free commutative rig with unit on one generator
$x$ can be identified with the ring $\mathbf{Z}[x]$ of polynomials in the
variable $x$ with coefficients in $\mathbb{Z}$ endowed with the usual sum and
product of polynomials. It is straightforward to see that $\mathbf{Z}[x]$ is
$\mathcal{R}$-indecomposable. Hence, due to Corollary 3 and Theorem 7 we
obtain the following:

######  Proposition 5.

The variety $\mathcal{R}$ is fp-coextensive and $\mathcal{G}(\mathcal{R})$
classifies $\mathcal{R}$-indecomposable objects. In particular, the functor
$Z:\mathcal{R}\rightarrow\mathsf{Set}$ preserves all limits and filtering
colimits.

####  4.3.4 Heyting algebras

A Heyting algebra is an algebra $\mathbf{A}=(A,\wedge,\vee,\rightarrow,0,1)$
of type $(2,2,2,0,0)$ such that the structure $(A,\wedge,\vee,0,1)$ is a
bounded distributive lattice satisfying $x\wedge y\leq z$ if and only if
$x\leq y\rightarrow z$. As usual, we denote $x\rightarrow 0$ by $\neg x$. Let
$\mathcal{H}$ be the class of Heyting algebras. It is fairly known that
$\mathcal{H}$ is a variety and also that this class provide an algebraic
semantics for intuitionistic logic. It is no hard to see that the constants
$0$ and $1$ and the term

$U(x,y,z,w)=(z\wedge y)\vee(\neg z\wedge x)$

makes $\mathcal{H}$ a Pierce variety and moreover, that the formula

$\sigma(x,y)=(x\wedge y=0)\wedge(x\vee y=1)$

defines the relation $e\diamond_{\mathbf{A}}f$ in $\mathcal{H}$. Then Theorem
2 yields that $\mathcal{H}$ is coextensive. Further, by the dual of
Proposition 5.5 of [19], $\mathcal{H}$ is fp-coextensive and by Theorem 3.2
[13], all free algebras of $\mathcal{H}$ are $\mathcal{H}$-indecomposable.
Therefore, we can conclude:

######  Proposition 6.

The variety $\mathcal{H}$ is fp-coextensive and $\mathcal{G}(\mathcal{H})$
classifies $\mathcal{H}$-indecomposable objects. In particular, the functor
$Z:\mathcal{H}\rightarrow\mathsf{Set}$ preserves all limits and filtering
colimits.

####  4.3.5 Gödel algebras

A Gödel algebra is an algebra $\mathbf{A}=(A,\wedge,\vee,\rightarrow,0,1)$ of
type $(2,2,2,0,0)$ such that the structure $(A,\wedge,\vee,\rightarrow,0,1)$
is a Heyting algebra satisfying the _prelinearity_ condition. I.e. the
equation $(x\rightarrow y)\vee(y\rightarrow x)=1$, holds for every $x,y\in A$.
Gödel algebras provide an algebraic semantics for Gödel logic, which can be
defined as the schematic extension of the intuitionistic propositional
calculus by the prelinearity axiom
$(\alpha\rightarrow\beta)\vee(\beta\rightarrow\alpha)$. We write
$\mathcal{P}\mathcal{H}$ for the variety of Gödel algebras. Observe that we
can apply the same arguments used for $\mathcal{H}$ in order to prove that
$\mathcal{P}\mathcal{H}$ is coextensive. Furthermore, due to
$\mathcal{P}\mathcal{H}$ is a locally finite variety and its type is finite,
from Lemma 2, $\mathcal{P}\mathcal{H}$ is fp-coextensive. In [17] it was shown
that $\mathbf{F}_{\mathcal{P}\mathcal{H}}(x)$ can be identified with the
lattice displayed in Fig. 1. Nonetheless, it is the case that $\neg x$ and
$\neg\neg x$ are non-trivial complementary central elements of
$\mathbf{F}_{\mathcal{P}\mathcal{H}}(x)$. We have proved:

$\neg x$$0$$x$$\neg x\vee x$$\neg\neg x$$1$ Figure 1:
$\mathbf{F}_{\mathcal{P}\mathcal{H}}(x)$

######  Proposition 7.

The variety $\mathcal{P}\mathcal{H}$ is fp-coextensive so in particular the
functor $Z:\mathcal{P}\mathcal{H}\rightarrow\mathsf{Set}$ preserves all limits
and filtering colimits. Nevertheless $\mathcal{G}(\mathcal{P}\mathcal{H})$
does not classifies $\mathcal{P}\mathcal{H}$-indecomposable objects.

####  4.3.6 MV-algebras

An MV-algebra is an algebra $(A,\oplus,\neg,0)$ of type $(2,1,0)$ such that
$(A,\oplus,0)$ is a commutative monoid such that the following equations hold:

  1. 1.

$\neg\neg x=x$,

  2. 2.

$x\oplus\neg 0=\neg 0$,

  3. 3.

$\neg(\neg x\oplus y)\oplus y=\neg(\neg y\oplus x)\oplus x$.

We write $\mathcal{M}\mathcal{V}$ for the variety of MV-algebras. If we define
the following operations

$\begin{array}[]{ccc}x+y=\neg(\neg x\oplus y)\oplus y;&1=\neg 0;&x\cdot
y=\neg(\neg x\oplus\neg y),\end{array}$

straightforward calculations allows to see that the constants $0$ and $1$ and
the term

$U(x,y,z,w)=(x+z)\cdot(y+w)$

makes $\mathcal{M}\mathcal{V}$ a Pierce variety. It is known that
$\mathcal{M}\mathcal{V}$ provides an algebraic semantics for Łukasiewicz logic
[15]. From Definition 1.5.2 and Theorem 1.5.3 of (op.cit.) it can be proved
that the formula

$\sigma(x,y)=(x+y=0)\wedge(x\cdot y=1)$

defines the relation $e\diamond_{\mathbf{A}}f$ in $\mathcal{M}\mathcal{V}$.
So, by Theorem 2, $\mathcal{M}\mathcal{V}$ is coextensive. As a consequence of
Theorem 3.4 of [27], it follows that $\mathcal{M}\mathcal{V}$ is fp-
coextensive. Finally, in [16] it was proved that every free MV-algebra is
semisimple and directly indecomposable. Hence we get:

######  Proposition 8.

The variety $\mathcal{M}\mathcal{V}$ is fp-coextensive and
$\mathcal{G}(\mathcal{M}\mathcal{V})$ classifies
$\mathcal{M}\mathcal{V}$-indecomposable objects. In particular, the functor
$Z:\mathcal{M}\mathcal{V}\rightarrow\mathsf{Set}$ preserves all limits and
filtering colimits.

## References

  * [1] S. Awodey. Category theory. Oxford university press. (2010)
  * [2] M. Badano. Variedades con Congruencias Factor Ecuacionalmente Definibles. (PhD Thesis). Universidad Nacional de Córdoba, 2012.
  * [3] M. Badano & D. Vaggione, Varieties with equationally definable factor congruences, Algebra Universalis. Volume 69, 139 –166, 2013.
  * [4] Badano, M., Vaggione, D.: Equational definability of (complementary) central elements. International Journal of Algebra and Computation 26, 509–532 (2016)
  * [5] G. Bezhanishvili, Locally finite varieties. Algebra universalis 46, 531 – 548, 2001.
  * [6] Blok, W.J., Hoogland, E. The Beth Property in Algebraic Logic. Stud Logica 83, 49–90 (2006). https://doi.org/10.1007/s11225-006-8298-0
  * [7] W.J. Blok and D. Pigozzi. Algebraizable Logics, Memoirs of the American Mathematical Society 396, Amer. Math. Soc., Providence, 1989.
  * [8] Borceux, F.: Handbook of Categorical Algebra: Volume 2, Categories and Structures. (Vol. 2.). Cambridge University Press, 1994.
  * [9] Broodryk, D.N.: Characterization of coextensive varieties of universal algebras II (2021, preprint) arXiv:2104.12188 [math.CT]
  * [10] S. Burris & H.P. Sankappanavar. A course in Universal Algebra, Springer-Verlag New York, 1981.
  * [11] A. Carboni, S. Lack, R.F.C. Walters. Introduction to extensive and distributive categories. Journal of Pure and Applied Algebra. Volume 84, Issue 2. 145 –158, 1993.
  * [12] A. Carboni, M. C. Pedicchio, J. Rosický. Syntactic characterizations of various classes of locally presentable categories. Journal of Pure and Applied Algebra, Volume 161, Issues 1 – 2, 65 – 90, 2001.
  * [13] D. Castaño, P. Díaz Varela, A. Torrens. Indecomposability of free algebras in some subvarieties of residuated lattices and their bounded subreducts. Soft Computing, 15, 1449 – 1455, DOI 10.1007/s00500-010-0683-y, 2011.
  * [14] J. L. Castiglioni, M. Menni, W. J. Zuluaga Botero. A representation theorem for integral rigs and its applications to residuated lattices. Journal of Pure and Applied Algebra, 220, 3533–3566, 2016.
  * [15] R.L. Cignoli, I.M. d’Ottaviano, D. Mundici: Algebraic foundations of many-valued reasoning (Vol. 7). Springer Science & Business Media. (2013).
  * [16] R. Cignoli, A. Torrens, Hájek basic fuzzy logic and Łkasiewicz infinite-valued logic. Arch. Math. Logic 42, (2003).
  * [17] Flaminio T., Godo L., Rodríguez R.O: A Representation Theorem for Finite Gödel Algebras with Operators. In: Iemhoff R., Moortgat M., de Queiroz R. (eds) Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science, vol 11541. Springer, Berlin, Heidelberg. (2019)
  * [18] G. A. Fraser & A. Horn, Congruence relations in direct products. Proc. Amer. Math. 26, 390-394, 1970.
  * [19] S. Ghilardi, M. Zawadowski, A Sheaf Representation and Duality for Finitely Presented Heyting Algebras. The Journal of Symbolic Logic 60, no. 3 911–939. (1995) https://doi.org/10.2307/2275765.
  * [20] E. Hoogland. Algebraic characterizations of various Beth definability properties. Studia Logica, 2000, vol. 65, no 1, p. 91–112.
  * [21] P. T. Johnstone. Sketches of an elephant: A topos theory compendium. Oxford University Press, 2002.
  * [22] T. Katriňák and S. El-Assar. Algebras with Boolean and Stonean congruence lattices. Acta Math Hung 48, 301–316 (1986). https://doi.org/10.1007/BF01951357.
  * [23] F. W. Lawvere, Functorial semantics of algebraic theories, Dissertation, Columbia University, 1963.
  * [24] F.W. Lawvere: Some thoughts on the future of category theory. In: Carboni A., Pedicchio M.C., Rosolini G. (eds) Category Theory. Lecture Notes in Mathematics, vol 1488. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084208
  * [25] F. W. Lawvere, Core varieties, extensivity, and rig geometry. Theory Appl. Categ. 20(14):497-503, 2008.
  * [26] S. Mac Lane & I. Moerdijk. Sheaves in geometry and logic: A first introduction to Topos Theory. Springer Science & Business Media, 2012.
  * [27] V. Marra, L. Spada: Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras. Annals of Pure and Applied Logic, Volume 164, Issue 3, 192–210. (2013) https://doi.org/10.1016/j.apal.2012.10.001.
  * [28] R. McKenzie, G. McNulty & W. Taylor, Algebras, Lattices, Varieties, Vol. 1, Wadsworth & BrooksrCole Math. Series, Monterey, CA, 1987.
  * [29] M. Menni. A Basis Theorem for 2-rigs and Rig Geometry. Cah. Topologie Géom. Différ. Catégoriques 62, No. 4 (2021), 451-490.
  * [30] T. Moraschini, J.G. Raftery and J.J. Wannenburg. Epimorphisms, Definability and Cardinalities. Stud Logica 108, 255–275 (2020). https://doi.org/10.1007/s11225-019-09846-5
  * [31] P. Sanchez Terraf & D. Vaggione, Varieties with definable factor congruences. Trans. Amer. Math. Soc. 361, 5061 – 5088, 2009.
  * [32] D. Vaggione, Varieties in which the Pierce Stalks are Directly Indecomposable. Journal of Algebra. 184, 424–434, 1996.
  * [33] Vaggione, D.: Varieties of shells. Algebra Universalis 36, 483–487 (1996)
  * [34] D. Vaggione, Central Elements in Varieties with the Fraser – Horn Property. Advances in Mathematics. Vol. 148, Issue 2, 193 – 202, 1999.
  * [35] R. Zanfa, Classifying topoi for extensions of algebraic theories with indecomposable models. MSc Thesis, Università degli studi di Milano, 2017.
  * [36] W.J. Zuluaga Botero, Coextensive varieties via central elements. 82, 50 (2021). https://doi.org/10.1007/s00012-021-00745-2