2.1 Notation and basic definitions

Throughout this article algebras are considered as models of first-order languages without relations. For example, abelian $\ell $ -groups are algebras in the language $\tau _{\mathcal {G}} := \{+,-,0,\vee ,\wedge \}$ . As is customary we use bold letters ( $\mathbf {A}, \mathbf {B}, \mathbf {C}, \ldots $ ) for algebraic structures and italic letters ( $A, B, C, \ldots $ ) for the underlying sets. For algebras $\mathbf {A}$ and $\mathbf {B}$ we write $\mathbf {A} \subseteq \mathbf {B}$ whenever $\mathbf {A}$ is a subalgebra of $\mathbf {B}$ .

Given a structure $\mathbf {A}$ in a language $\tau $ and a term $t(x_1,\ldots ,x_n)$ in the same language, we write $t^{\mathbf {A}}(\bar {a})$ for the value of the term upon assigning elements $a_1,\ldots ,a_n$ from A to the variables $x_1,\ldots ,x_n$ . We may omit the superscript $\mathbf {A}$ if there is no risk of confusion.

Given a (first-order) formula $\varphi $ , we say that $\varphi $ is:

• • an identity if it has the form $\forall \bar {x} (p(\bar {x}) = q(\bar {x}))$ , where p and q are terms,

• • a quasi-identity if it has the form $\forall \bar {x} (\alpha (\bar {x}) \mathbin {\boldsymbol {\rightarrow }} \beta (\bar {x}))$ ,Footnote ¹ where $\alpha $ is a finite conjunction of term-equalities and $\beta $ is a term-equality,

• • universal if it has the form $\forall \bar {x} \psi $ , where $\psi $ is quantifier-free,

• • existential if it has the form $\exists \bar {x} \psi $ , where $\psi $ is quantifier-free.

A sentence is a formula with no free variables. If $\Sigma $ is a set of sentences, $\mathop {\mathrm {Mod}}(\Sigma )$ denotes the class of all models that satisfy the sentences in $\Sigma $ .

Whenever we consider a class $\mathcal {K}$ of algebras, we assume that all algebras in $\mathcal {K}$ have the same language. Given a class $\mathcal {K}$ of algebras, we define the usual class operators:

• • $\mathsf {I}(\mathcal {K})$ denotes the class of isomorphic images of members of $\mathcal {K}$ ,

• • $\mathsf {H}(\mathcal {K})$ denotes the class of homomorphic images of members of $\mathcal {K}$ ,

• • $\mathsf {S}(\mathcal {K})$ denotes the class of subalgebras of members of $\mathcal {K}$ ,

• • $\mathsf {P}(\mathcal {K})$ denotes the class of direct products with factors in $\mathcal {K}$ ,

• • $\mathsf {P_U}(\mathcal {K})$ denotes the class of ultraproducts with factors in $\mathcal {K}$ .

If $\mathsf {O}$ is one of the above operators and $\mathcal {K} = \{\mathbf {A}_1,\ldots , \mathbf {A}_n\}$ , we write $\mathsf {O}(\mathbf {A}_1,\ldots ,\mathbf {A}_n)$ instead of $\mathsf {O}(\mathcal {K})$ .

Let $\mathcal {K}$ be a class of algebras of the same language. We say that $\mathcal {K}$ is a variety (or equational class) if it can be axiomatized using a set of identities; equivalently, by Birkhoff’s theorem, $\mathcal {K}$ is a variety if and only if it is closed under $\mathsf {H}$ , $\mathsf {S}$ , and $\mathsf {P}$ . The smallest variety containing $\mathcal {K}$ is $\mathsf {HSP}(\mathcal {K})$ and is denoted by $\mathsf {V}(\mathcal {K})$ . A quasivariety is a class of algebras that can be axiomatized by a set of quasi-identities. By Mal’cev’s theorem, the class $\mathcal {K}$ is a quasivariety if and only if $\mathcal {K}$ is closed under $\mathsf {I}$ , $\mathsf {S}$ , $\mathsf {P}$ , and $\mathsf {P_U}$ ; the smallest quasivariety containing $\mathcal {K}$ is $\mathsf {ISPP_U}(\mathcal {K})$ , also denoted by $\mathsf {Q}(\mathcal {K})$ . Finally, recall that $\mathcal {K}$ is universal if it can be axiomatized by a set of universal sentences, which is equivalent to $\mathcal {K}$ being closed under $\mathsf {I}$ , $\mathsf {S}$ , and $\mathsf {P_U}$ . Moreover, the smallest universal class containing $\mathcal {K}$ is given by $\mathsf {ISP_U}(\mathcal {K})$ .

Given a class $\mathcal {K}$ and two sentences $\varphi $ , $\psi $ , we say that $\varphi $ and $\psi $ are equivalent in $\mathcal {K}$ , and write $\varphi \mathrel {\sim } \psi $ in $\mathcal {K}$ , if for every $\mathbf {A} \in \mathcal {K}$ we have that $\mathbf {A} \vDash \varphi $ if and only if $\mathbf {A} \vDash \psi $ .

2.2 Algebraically expandable classes

In order to define algebraically expandable classes [Reference Campercholi and Vaggione10], one of the fundamental notions in this article, we need to introduce the special type of sentences that axiomatize them. An equational function definition sentence (EFD-sentence for short) in the language $\tau $ is a sentence of the form

(1) $$ \begin{align} \forall x_1 \ldots x_n \exists! z_1 \ldots z_m \, \mathop{\boldsymbol{\bigwedge}}\limits_{i=1}^k s_i(\bar{x},\bar{z}) = t_i(\bar{x},\bar{z}), \end{align} $$

where $s_i,t_i$ are $\tau $ -terms, $n \geq 0$ , and $m \geq 1$ . Suppose $\varphi $ is the EFD-sentence in (1). Observe that $\varphi $ is valid in a structure $\mathbf {A}$ if and only if the system of equations $\displaystyle \mathop {\boldsymbol {\bigwedge }}\limits _{i=1}^k s_i(\bar {x},\bar {z}) = t_i(\bar {x},\bar {z})$ defines a (total) function $F\colon A^n \to A^m$ . If $\pi _j\colon A^m \to A$ is the j-th projection function, we write $[\varphi ]_j^{\mathbf {A}} := \pi _j \circ F$ for $j \in \{1,\ldots ,m\}$ .

Let $\varphi $ be as in (1). We define:

• • $E(\varphi ) := \displaystyle \forall \bar {x} \exists \bar {z} \, \mathop {\boldsymbol {\bigwedge }}\limits _{i=1}^k s_i(\bar {x},\bar {z}) = t_i(\bar {x},\bar {z})$ ,

• • $U(\varphi ) := \displaystyle \forall \bar {x} \bar {y} \bar {z} \mathop {\boldsymbol {\bigwedge }}\limits _{i=1}^k s_i(\bar {x},\bar {y}) = t_i(\bar {x},\bar {y}) \boldsymbol {\land } \mathop {\boldsymbol {\bigwedge }}\limits _{i=1}^k s_i(\bar {x},\bar {z}) = t_i(\bar {x},\bar {z}) \mathbin {\boldsymbol {\rightarrow }} \bar {y} = \bar {z}$ .

The following basic facts are used without explicit reference throughout the article.

• • $\varphi $ is equivalent to $E(\varphi ) \boldsymbol {\land } U(\varphi )$ .

• • $U(\varphi )$ is (equivalent to) a conjunction of quasi-identities.

• • $E(\varphi )$ is preserved by homomorphic images, that is, for any surjective homomorphism $f: \mathbf {A} \to \mathbf {B}$ , if $\mathbf {A} \vDash E(\varphi )$ , then $\mathbf {B} \vDash E(\varphi )$ .

A class of algebras $\mathcal {K}$ is an algebraically expandable class (AE-class for short) if there is a set of EFD-sentences $\Sigma $ such that $\mathcal {K} = \mathop {\mathrm {Mod}}(\Sigma )$ . Let $\mathcal {K}$ and $\mathcal {C}$ be classes of algebras, $\mathcal {K} \subseteq \mathcal {C}$ . We say that $\mathcal {K}$ is an AE-subclass of $\mathcal {C}$ if $\mathcal {K}$ is axiomatizable by EFD-sentences relative to $\mathcal {C}$ , that is, $\mathcal {K} = \mathcal {C} \cap \mathop {\mathrm {Mod}}(\Sigma )$ for some set $\Sigma $ of EFD-sentences. The reader should be aware that $\mathcal {K}$ may be an AE-subclass of $\mathcal {C}$ , but fail to be an AE-class itself.

Let $\mathcal {Q}$ be a quasivariety in the language $\tau $ and let $\Sigma $ be a set of EFD-sentences. There is an obvious expansion of the AE-subclass $\mathcal {K} := \mathcal {Q} \cap \mathop {\mathrm {Mod}}(\Sigma )$ of $\mathcal {Q}$ obtained by skolemizing the existential quantifiers in $\Sigma $ . More precisely, for each $\varphi \in \Sigma $ of the form $\forall x_1 \ldots x_n \exists ! z_1 \ldots z_m \, \mathop {\boldsymbol {\bigwedge }}\limits _{i=1}^k s_i(\bar {x},\bar {z}) = t_i(\bar {x},\bar {z})$ consider new n-ary function symbols $f_1^\varphi , \ldots , f_m^\varphi $ and the set of identities

$$ \begin{align*}E_\varphi := \{\forall \bar{x} \, s_i(\bar{x}, f_1^\varphi(\bar{x}), \ldots, f_m^\varphi(\bar{x})) = t_i(\bar{x}, f_1^\varphi(\bar{x}), \ldots, f_m^\varphi(\bar{x})): 1 \leq i \leq k\}.\end{align*} $$

Let $\tau _\Sigma $ be the expansion of $\tau $ obtained by adding the $f_j^\varphi $ ’s for each $\varphi \in \Sigma $ , and put

$$\begin{align*}E_\Sigma := \bigcup_{\varphi \in \Sigma} E_\varphi, \qquad U_\Sigma := \{U(\varphi): \varphi \in \Sigma\}. \end{align*}$$

Define

$$ \begin{align*}\mathcal{Q}^\Sigma := \mathop{\mathrm{Mod}}(\Gamma \cup E_\Sigma \cup U_\Sigma),\end{align*} $$

where $\Gamma $ is a set of quasi-identities axiomatizing $\mathcal {Q}$ . We call $\mathcal {Q}^\Sigma $ an algebraic expansion of $\mathcal {Q}$ . Note that $\mathcal {Q}^\Sigma $ is a quasivariety over the language $\tau _\Sigma $ whose members are precisely the expansions of the members of $\mathcal {K}$ . A fact worth mentioning is that if $\Sigma $ and $\Sigma '$ are two sets of EFD-sentences with the same models in $\mathcal {Q}$ , then the quasivarieties $\mathcal {Q}^\Sigma $ and $\mathcal {Q}^{\Sigma '}$ are term-equivalent (see [Reference Campercholi and Vaggione11, Theorem 5]).

We conclude this section with a preservation result for EFD-sentences needed in the sequel. Recall that a structure $\mathbf {A}$ is finitely subdirectly irreducible if its diagonal congruence is meet-irreducible in the congruence lattice of $\mathbf {A}$ . We write $\mathcal {K}_{\mathrm {fsi}}$ for the class of finitely subdirectly irreducible members of $\mathcal {K}$ . A variety is arithmetical provided that it is both congruence distributive and congruence permutable.

3.1 The lattice of algebraic expansions

Let L be a finitely algebraizable logic with equivalent algebraic semantics $\mathcal {Q}$ . The AE-subclasses of the quasivariety $\mathcal {Q}$ are naturally (lattice-)ordered by inclusion. In the current section we show how this ordering translates to the algebraic expansions of $\mathcal {Q}$ , and thus to the algebraic expansions of L. For this we need to look into interpretations between logics and between classes of algebras.

Fix a countably infinite set of variables $X := \{x_1,x_2,\ldots \}$ ; given a language $\tau $ we write $Tm(\tau )$ for the set of $\tau $ -terms over the variables in X. Let $\tau _1$ and $\tau _2$ be two expansions of a language $\tau $ . A $\tau $ -translation from $\tau _1$ into $\tau _2$ is a function $T\colon \tau _1 \to Tm(\tau _2)$ such that T maps each symbol of arity n to a term in the variables $x_1,\ldots ,x_n$ , and $T(f) = f(x_1,\ldots ,x_n)$ for every n-ary symbol $f \in \tau $ .

Let $\mathcal {K}_1$ and $\mathcal {K}_2$ be two classes of algebras over $\tau _1$ and $\tau _2$ , respectively. A $\tau $ -interpretation of $\mathcal {K}_1$ in $\mathcal {K}_2$ is a $\tau $ -translation $T\colon \tau _1 \to Tm(\tau _2)$ such that for every member $\mathbf {A} := (A,\{g^{\mathbf {A}}: g \in \tau _2\})$ in $\mathcal {K}_2$ , the algebra $\mathbf {A}^T := (A,\{T(f)^{\mathbf {A}}: f \in \tau _1\})$ belongs to $\mathcal {K}_1$ . If T and S are $\tau $ -interpretations of $\mathcal {K}_1$ in $\mathcal {K}_2$ and $\mathcal {K}_2$ in $\mathcal {K}_1$ , respectively, such that the maps $\mathbf {A} \mapsto \mathbf {A}^T$ and $\mathbf {A} \mapsto \mathbf {A}^S$ are mutually inverse, we say that $\mathcal {K}_1$ and $\mathcal {K}_2$ are $\tau $ -term-equivalent.

We turn now to maps between logics. A $\tau $ -translation T from $\tau _1$ into $\tau _2$ extends in a natural way to a mapping from $Tm(\tau _1)$ to $Tm(\tau _2)$ :

• • $T(x) = x$ for every variable $x \in X$ ;

• • $T(f(\varphi _1,\ldots ,\varphi _n)) = T(f)(T(\varphi _1),\ldots ,T(\varphi _n))$ for f in $\tau _1$ of arity n and $\varphi _1, \ldots \varphi _n$ in $Tm(\tau _1)$ .

Given a set $\Gamma $ of $\tau _1$ -terms we write $T(\Gamma )$ for $\{T(\varphi ): \varphi \in \Gamma \}$ .

Let $\tau _1$ and $\tau _2$ be expansions of a language $\tau $ , and suppose $L_1$ and $L_2$ are logics in $\tau _1$ and $\tau _2$ , respectively. A $\tau $ -morphism from $L_1$ to $L_2$ is a $\tau $ -translation from $\tau _1$ into $\tau _2$ such that

$$ \begin{align*}\Gamma \vdash_{L_1} \varphi \text{ implies } T(\Gamma) \vdash_{L_2} T(\varphi)\end{align*} $$

for $\Gamma \cup \{\varphi \} \subseteq Tm(\tau _1)$ . We say that $L_1$ and $L_2$ are $\tau $ -bimorphic, and write $L_1 \leftrightarrows _\tau L_2$ , provided there exist a $\tau $ -morphism from $L_1$ to $L_2$ and a $\tau $ -morphism from $L_2$ to $L_1$ .

The following result shows the connection between the above defined relations.

A word of caution: it is not true in general that two bimorphic algebraizable logics have term-equivalent algebraic semantics. For example, let $C := \langle \tau , \vdash \rangle $ be the Classical Propositional Logic and let $C_e = \langle \tau \cup \{e\}, \vdash _e \rangle $ where e is a new constant symbol and $\vdash _e$ is the least substitution invariant consequence operator containing $\vdash $ . Clearly $C_e$ is algebraizable with equivalent algebraic semantics $\mathcal {B}_e$ , the class of pointed Boolean algebras. It is straightforward to check that $C_e \leftrightarrows _{\tau } C$ ; however, $\mathcal {B}_e$ is obviously not $\tau $ -term-equivalent to the class of Boolean algebras.

Let L be a logic algebraized by a quasivariety $\mathcal {Q}$ . As is the case for any quasivariety, the AE subclasses of $\mathcal {Q}$ form a lattice $\Lambda $ under inclusion. In the light of Theorem 3.2, the algebraic expansions of L modulo $\leftrightarrows $ , ordered by morphisms, form a lattice as well, dually isomorphic to $\Lambda $ . Thus, classifying the algebraically expandable classes of $\mathcal {Q}$ yields a classification of all algebraic expansions of L up to $\leftrightarrows $ .

3.2 Some examples

When the AE-subclasses of a quasivariety are known, Theorem 3.2 immediately gives a description of the algebraic expansions of the corresponding logic. We present here three examples.

3.2.3 The implicative fragment of classical logic

Let $L_\to $ be the implicative fragment of classical propositional logic. The equivalent algebraic semantics of $L_\to $ is the variety $\mathcal {I}$ of implication algebras. Recall that disjunction is expressible in terms of $\mathbin {\rightarrow }$ , and thus for $n \geq 2$ and $1 \leq i \leq n$

$$ \begin{align*}s_i^n(x_1,\ldots,x_n) := \bigvee_{j=1,j\ne i}^n x_j\end{align*} $$

is an $\{\mathbin {\rightarrow }\}$ -term. For each $n \geq 2$ let

$$ \begin{align*}\Phi_n := \{z \mathbin{\rightarrow} s_i^n(\bar{x}): i \in \{1,\ldots,n\}\} \cup \bigg\{\bigvee_{i=1}^n (s_i^n(\bar{x}) \mathbin{\rightarrow} z)\bigg\}.\end{align*} $$

By definition, $L_\to ^{\Phi _n}$ is the least expansion of $L_\to $ that satisfies $(\mathrm {E}_{\Phi _n})$ and $(\mathrm {U}_{\Phi _n})$ . However, condition $(\mathrm {U}_{\Phi _n})$ is already true for $L_\to $ . Thus $L_\to ^{\Phi _n}$ is the expansion of $L_\to $ by the following axioms:

• $\mu _n(\bar {x}) \mathbin {\rightarrow } s_i^n(\bar {x})$ for $i \in \{1,\ldots ,n\}$ ,

• $\bigvee _{i=1}^n (s_i^n(\bar {x}) \mathbin {\rightarrow } \mu _n(\bar {x}))$ ,

where $\mu _n$ is a new n-ary symbol.

By the characterization of the AE-subclasses of $\mathcal {I}$ given in [Reference Campercholi9, Theorem 13] it follows from Theorem 3.2 that, up to $\leftrightarrows $ , the consistent algebraic expansions of $L_\to $ are

$$ \begin{align*}L_\to < \cdots < L_\to^{\Phi_3} < L_\to^{\Phi_2},\end{align*} $$

where $L < L'$ means that there is an $\{\to \}$ -morphism from L to $L'$ but there is no $\{\to \}$ -morphism from $L'$ to $L'$ . Observe that $\mu _2$ is the classical conjunction and, more generally, we have that $\mu _n(\bar {x}) = \bigwedge _{i=1}^n s_i^n(\bar {x})$ .

Example 3 of [Reference Caicedo5] shows classical negation is implicitly definable in $L_\to $ . Since none of the algebraic expansions of $L_\to $ has classical negation as a term, we have another example of an expansion by implicit connectives that is not algebraic.

4.1 AE-classes of abelian $\ell $ -groups

We proceed to characterize EFD-sentences modulo equivalence in $\mathcal {G}$ . We first reduce the problem to totally ordered abelian $\ell $ -groups.

For each positive integer k define

$$ \begin{align*}\delta_k := \forall x \exists! z \, kz = x.\end{align*} $$

Our next step is to show that every EFD-sentence is equivalent to a $\delta _k$ in $\mathcal {G}$ , which is accomplished in Theorem 4.12.

Recall that an $\ell $ -group $\mathbf {G}$ is divisible if for every $g \in G$ and every positive integer n, there exists $h \in G$ such that $g = nh$ . Given a divisible $\ell $ -group $\mathbf {D}$ , since $\ell $ -groups are torsion-free, we have that $\delta _k$ holds in $\mathbf {D}$ for all k; thus, we can define the expansion

(2) $$ \begin{align} \overline{\mathbf{D}} := (\mathbf{D}, ([\delta_k]^{\mathbf{D}})_{k\geq 1}). \end{align} $$

The next result shows that the only functions defined by EFD-sentences in these expansions are term-operations.

The above theorem can be derived from [Reference Caicedo6, Theorem 20]. We provide a different proof that relies on the characterization of existentially closed algebras in $\mathcal {G}_{\mathrm {to}}$ .

Given a class $\mathcal {K}$ of algebras closed under isomorphisms and $\mathbf {A} \in \mathcal {K}$ , we say that $\mathbf {A}$ is existentially closed in $\mathcal {K}$ if for every $\mathbf {B} \in \mathcal {K}$ such that $\mathbf {A} \subseteq \mathbf {B}$ , every existential formula $\varphi (\bar {x})$ , and every $\bar {a} \in A^n$

$$ \begin{align*}\mathbf{B} \vDash \varphi(\bar{a}) \text{ implies } \mathbf{A} \vDash \varphi(\bar{a}).\end{align*} $$

The next proposition characterizes the existentially closed members of the class of totally ordered $\ell $ -groups.

After this sequence of results we are ready to present:

Given a positive integer k and a term $t(\bar {x})$ in $\tau _{\mathcal {G}}$ , let

$$ \begin{align*}\delta_{k,t} := \forall \bar{x} \exists ! z \, kz = t(\bar{x}).\end{align*} $$

Observe that $U(\delta _{k,t})$ is valid in $\mathcal {G}$ because abelian $\ell $ -groups are torsion-free.

We denote by $\mathcal {D}$ the class of expansions $\overline {\mathbf {D}}$ of divisible $\ell $ -groups $\mathbf {D} \in \mathcal {G}$ . We write $\tau _{\mathcal {D}}$ for the language of the algebras in the class $\mathcal {D}$ .

In the following, by a system of linear inequalities we mean a finite conjunction of inequalities of the form $a_1x_1 + \dots + a_nx_n \geq 0$ where $a_1,\dots ,a_n$ are integers. (Note that such a system can be written as a conjunction of equations in $\tau _{\mathcal {G}}$ .)

We say that a system of linear inequalities $\alpha (\bar {x})$ is full-dimensional on an abelian $\ell $ -group $\mathbf {G}$ if there is no $(a_1,\dots ,a_n) \in \mathbb {Z}^n \setminus \{\bar {0}\}$ such that $\mathbf {G} \vDash \forall \bar {x} \, (\alpha (\bar {x}) \mathbin {\boldsymbol {\rightarrow }} \sum a_ix_i \,{=}\, 0)$ . That is, the system $\alpha (\bar {x})$ imposes no linear dependencies on its solutions in $\mathbf {G}$ . Observe that Lemma 4.1(4) implies that $\alpha (\bar {x})$ is full-dimensional on some nontrivial $\ell $ -group $\mathbf {G}$ if and only if it is full-dimensional on every nontrivial $\ell $ -group. Hence, we say that $\alpha (\bar {x})$ is full-dimensional provided it is full-dimensional on some nontrivial $\ell $ -group.

Lemma 4.10. Let $t(\bar {x})$ be a term in $\tau _{\mathcal {G}}$ . There are full-dimensional systems of linear inequalities $\alpha _1(\bar {x}), \dots , \alpha _m(\bar {x})$ and terms $t_1(\bar {x}), \dots , t_m(\bar {x})$ , which are integer linear combinations of the variables $x_1, \ldots , x_n$ , such that for all $\mathbf {G} \in \mathcal {G}_{\mathrm {to}}$ and all $\bar {g} \in G^n$ we have

(3) $$ \begin{align} t^{\mathbf{G}}(\bar{g}) = \begin{cases} t_1^{\mathbf{G}}(\bar{g}), & \text{if } \alpha_1(\bar{g}) \text{ holds in } \mathbf{G}, \\ \;\; \vdots \\ t_m^{\mathbf{G}}(\bar{g}), & \text{if } \alpha_m(\bar{g}) \text{ holds in } \mathbf{G}. \end{cases} \end{align} $$

Proof Fix a positive integer k and a $\tau _{\mathcal {G}}$ -term t; let $\alpha _j(\bar {x})$ and $t_j(\bar {x})$ for $j \in \{1,\ldots ,m\}$ be as in Lemma 4.10. Suppose $t_j(\bar {x}) = a_{j1}x_1 + \dots + a_{jn}x_n$ , and let d be the greatest common divisor of the set $\{k\} \cup \{a_{ji}: i \in \{1,\dots ,n\}, j \in \{1,\ldots ,m\}\}$ . Define $k'$ by $k = dk'$ ; we prove that $\delta _{k,t} \mathrel {\sim } \delta _{k'}$ in $\mathcal {G}$ . Observe that, due to Lemma 4.2, it suffices to show that $\delta _{k,t} \mathrel {\sim } \delta _{k'}$ in $\mathcal {G}_{\mathrm {to}}$ .

Take $\mathbf {G} \in \mathcal {G}_{\mathrm {to}}$ and assume $\mathbf {G} \vDash \delta _{k,t}$ . We claim that $t_j(\bar {g})$ is divisible by k for every $\bar {g} \in G^n$ and $j \in \{1,\dots ,m\}$ . Indeed, given $\bar {g} \in G^n$ and $j \in \{1,\dots ,m\}$ , by Lemma 4.9, we can write $\bar {g} = \sum b_l \bar {g}_l$ for some integers $b_l$ and some $\bar {g}_l \in G^n$ such that $\mathbf {G} \vDash \alpha _j(\bar {g}_l)$ . Note that $t(\bar {g}_l) = t_j(\bar {g}_l)$ for each l. Since $\mathbf {G} \vDash \delta _{k,t}$ , for each l there is $h_l \in G$ such that $k h_l = t(\bar {g}_l) = t_j(\bar {g}_l)$ . Thus

$$\begin{align*}t_j(\bar{g}) = t_j(\sum b_l \bar{g}_l) = \sum b_l t_j(\bar{g}_l) = \sum b_l k h_l = k \sum b_lh_l, \end{align*}$$

which proves the claim.

Now write $d = kc + \sum _{i,j} a_{ji}c_{ji}$ for suitable integers c and $c_{ji}$ . Then, for any $g \in G$ ,

$$\begin{align*}dg = kcg + \sum_j \sum_i a_{ji}c_{ji}g = kcg + \sum_j t_j(\bar{g}_j), \end{align*}$$

where $\bar {g}_j := (c_{j1}g,\dots ,c_{jn}g)$ . Since each $t_j(\bar {g}_j)$ is divisible by k, it follows that there is $g' \in G$ such that $dg = kg'$ . Thus $dg = dk'g'$ , so $d(g-k'g') = 0$ and, since $\mathbf {G}$ is torsion-free, $g = k'g'$ . This proves that $\mathbf {G} \vDash \delta _{k'}$ .

Conversely, assume any element in $\mathbf {G}$ is divisible by $k'$ , and fix $\bar {g} := (g_1,\dots ,g_n) \in G^n$ . We prove that $t(\bar {g})$ is divisible by k. Let $j \in \{1,\dots ,m\}$ be such that $t(\bar {g}) = t_j(\bar {g})$ . Since each $a_{ji}$ is divisible by d, there is $g' \in G$ such that $t_j(\bar {g}) = d g'$ . Now, since $g'$ is divisible by $k'$ , there is $g'' \in G$ such that $g' = k'g''$ . Putting all together we obtain $t(\bar {g}) = t_j(\bar {g}) = dg' = dk'g'' = kg''$ .

We are now ready to present our characterization of EFD-sentences in $\mathcal {G}$ .

Given a set S of prime numbers, let $\Sigma _S := \{\delta _p: p \in S\}$ . Since for an $\ell $ -group divisibility by k is equivalent to divisibility by the prime factors of k, we have the following:

4.2 The algebraic expansions of the Logic of Equilibrium

As shown in [Reference Galli, Lewin and Sagastume16, Theorem 17] the variety $\mathcal {G}$ of abelian $\ell $ -groups is the equivalent algebraic semantics of the Logic of Equilibrium $Bal$ in the language $\{\to ,{}^+\}$ whose consequence operator is defined in [Reference Galli, Lewin and Sagastume16, Section 2]. The derived connectives

• $0 := x \mathbin {\rightarrow } x ,\hspace{8.6pc} x \vee y := (x \mathbin {\rightarrow } y)^+ + x$ ,

• $-x := x \mathbin {\rightarrow } 0, \hspace{8pc} x \wedge y :=\,{-}\,(-x \vee -y)$ ,

• $x + y := \,{-}\,x \mathbin {\rightarrow } y$ ,

form a complete set for $Bal$ since $\varphi (x \mathbin {\rightarrow } y,\bar {z}) \dashv \vdash \varphi (-x + y,\bar {z})$ and $\varphi (x^+,\bar {z}) \dashv \vdash \varphi (x \vee 0,\bar {z})$ for any formula $\varphi $ . This allows us to say that $\mathcal {G}$ is the equivalent algebraic semantics of $Bal$ via equivalence formulas $\Delta (x,y) = \{x \mathbin {\rightarrow } y\}$ and defining equations $\varepsilon (x) = \{x = 0\}$ .

Given a prime number p, the algebraic expansion of $Bal$ corresponding to the EFD-sentence $\delta _p$ is, by definition, obtained from $Bal$ by adding a unary function symbol $\mathsf {d}_p$ to the langauge of $Bal$ , the rule $\mathrm {U}_{\{x \mathbin {\rightarrow } pz\}}$ , and the axiom:

(A_p) $$\begin{align} x \mathbin{\rightarrow} p\mathsf{d}_p(x). \end{align}$$

Since the rule $\mathrm {U}_{\{x \mathbin {\rightarrow } pz\}}$ is derivable in $Bal$ , the expansion is obtained simply by adding $A_p$ . For a set S of prime numbers define $Bal^S$ as the expansion of $Bal$ by the axioms $\{A_p: p \in S\}$ . Note that, since $Bal^S$ is an axiomatic expansion of $Bal$ , its equivalent algebraic semantics is a variety. These expansions were also considered in [Reference Caicedo6, Section 7] where it is proved that every implicit connective in the logic $Bal^{Primes}$ is explicit.

Recall that an expansion $L' := \langle \tau ', \vdash _{L'}\rangle $ of a logic $L := \langle \tau ,\vdash _L\rangle $ is called conservative provided that for each set of $\tau $ -formulas $\Gamma \cup \{\varphi \}$ we have that $\Gamma \vdash _{L'} \varphi $ implies $\Gamma \vdash _L \varphi $ .

5.1 EFD-sentences in perfect MV-algebras

As shown by Theorem 4.13, every EFD-sentence of $\ell $ -groups is equivalent to some $\delta _k$ . Of course, each of these sentences induces the inverse function of multiplication by some positive integer. Next, for each k we introduce a term whose interpretation plays the role of multiplication by k in $\mathsf {V}(\mathcal {P})$ , namely

$$ \begin{align*}\mathsf{t}_k(z) := (kz \wedge \neg 2z^2) \vee z^k.\end{align*} $$

Thus, division by k in $\mathsf {V}(\mathcal {P})$ is embodied by the function induced by the EFD-sentence

$$ \begin{align*}\varepsilon_k := \forall x \exists! z \, \mathsf{t}_k(z) = x.\end{align*} $$

Both of these operations enjoy some natural algebraic properties whose proofs we omit as they are easy exercises.

Given the close connection between $\ell $ -groups and perfect MV-algebras, it is hardly surprising that a version of Theorem 4.13 holds for $\mathsf {V}(\mathcal {P})$ when we take $\varepsilon _k$ in place of $\delta _k$ . Even though the proof we discovered is nontrivial, for reasons of brevity we only outline its key steps.

5.2 The algebraic expansions of $L_{\mathcal {P}}$

The Logic $L_{\mathcal {P}}$ of Perfect MV-Algebras [Reference Di Nola and Lettieri13] is the extension of Łukasiewicz Logic by the axiom $2x^2 \mathbin {\leftrightarrow } (2x)^2$ (recall that $x \mathbin {\leftrightarrow } y := (\neg x + y) \wedge (\neg y + x)$ ). As the name suggests, the equivalent algebraic semantics of $L_{\mathcal {P}}$ is the variety $\mathsf {V}(\mathcal {P})$ .

Given a prime number p, the algebraic expansion of $L_{\mathcal {P}}$ corresponding to the EFD-sentence $\varepsilon _p$ is, by definition, obtained from $L_{\mathcal {P}}$ by adding a unary function symbol $\mathsf {d}_p$ to the language of $L_{\mathcal {P}}$ , the axiom:

(D_p) $$\begin{align} ((p\mathsf{d}_p(x) \wedge \neg 2\mathsf{d}_p(x)^2) \vee \mathsf{d}_p(x)^p) \mathbin{\leftrightarrow} x , \end{align}$$

and the rule $\mathrm {U}_{\{((kz \wedge \neg 2z^2) \vee z^k) \leftrightarrow x\}}$ . However, since this rule is derivable in $L_{\mathcal {P}}$ , the expansion is obtained simply by adding $D_p$ .

For a set S of prime numbers define $L_{\mathcal {P}}^S$ as the expansion of $L_{\mathcal {P}}$ by the axioms $\{D_p: p \in S\}$ . Note that, by the comment above, $L_{\mathcal {P}}^S$ is the algebraic expansion of $L_{\mathcal {P}}$ corresponding to the AE-class axiomatized by $\Sigma _S := \{\varepsilon _p: p \in S\}$ . Thus, the equivalent algebraic semantics $\mathsf {V}(\mathcal {P})^{\Sigma _S}$ of $L_{\mathcal {P}}^S$ is a variety.