2 Abelian logic and its expansions

Let ${Fm}_{\!\mathcal L}$ be the set of all formulas built from a propositional language $\mathcal L$ and a countable set of variables, and let ${\boldsymbol {\boldsymbol {{Fm}_{\!\mathcal L}}}}$ be the absolutely free algebra of type $\mathcal L$ defined on ${Fm}_{\!\mathcal L}$ .

A consecution in a language $\mathcal L$ is a tuple written as $\Gamma \mathrel {\blacktriangleright } \varphi $ , where $\Gamma \cup \{\varphi \} \subseteq {Fm}_{\!\mathcal L}$ ; it is said to be finitary if $\Gamma $ is finite and infinitary otherwise, we also identify consecutions $\emptyset \mathrel {\blacktriangleright } \varphi $ with just the formula $\varphi $ . Note that any logic can be seen as a set of consecutions and thus whenever $\Gamma \vdash _{\mathrm{L}} \varphi $ , we say that the consecution $\Gamma \mathrel {\blacktriangleright }\varphi $ is valid in ${\mathrm{L}}$ or that ${\mathrm{L}}$ satisfies $\Gamma \mathrel {\blacktriangleright }\varphi $ .

A set of consecutions closed under arbitrary substitutions (i.e., endomorphisms on ${\boldsymbol {{Fm}_{\mathcal L}}}$ , which are extended to consecutions in an obvious way) is called an axiomatic system. An element $\Gamma \mathrel {\blacktriangleright } \varphi $ of the set is called an axiom, a finitary rule or an infinitary rule depending on the cardinality of $\Gamma $ . An axiomatic system is called infinitary if it contains at least one infinitary rule, otherwise it is called finitary.

Given an axiomatic system ${\mathcal {AS}}$ we write $\Gamma \vdash _{\mathcal {AS}} \varphi $ if there is a proof of $\varphi $ from $\Gamma $ in ${\mathcal {AS}}$ . Note that, as we allow infinitary rules, our notion of proof cannot be the usual finite sequence of formulas, but it has to be a tree with no infinite branches (for details, see [Reference Cintula and Noguera9]). We say that ${\mathcal {AS}}$ is a presentation (an axiomatic system) of logic ${\mathrm{L}}$ if ${\mathrm{L}} = {\vdash _{\mathcal {AS}}}$ . Consequently, we can trivially observe that every (finitary) logic possesses a (finitary) presentation.

Table 1 The axiomatic system of Abelian logic

In this paper we deal with several logics in different languages with rather intricate interrelations. Therefore, we need to introduce the formal notions of extension and expansion.

Now we can formally define the Abelian logic Ab as the logic in the languageFootnote ⁴ given by the axiomatic system presented in Table 1.

The following consecutions are known to be valid in Ab and thus they are valid in all its expansions:

Moreover, one can easily see that Ab is a weakly implicative logic in the sense of [Reference Cintula and Noguera9], i.e., it satisfies $\mathrm {(id)}, {\mathrm {(MP)}}, {\mathrm {(T)}}$ and, for each n-ary c in $\mathcal L_{\mathrm {Ab}}$ , also

Expansions of Ab need not in general satisfy $\mathrm {(sCNG_{\textit c})}$ for connectives added to the language $\mathcal L_{\mathrm {Ab}}$ . Since we want to focus on weakly implicative logics, we provide the following definition.

Notice the difference between the notion of superabelian logic and the well-known terminology of superintuitionistic logics, which are axiomatic extensions (instead of expansions) of intuitionistic logic. Actually, as we shall soon see, Ab has no non-trivial axiomatic (or even finitary) extension.

There are many natural examples of superabelian logics. In §3 we will focus on infinitary extensions of Ab and in §4 and §5 we focus on some extensions of pointed Abelian logic (the least expansion of Abelian logic in the language with the additional constant $\texttt {f}$ ). Other natural examples of superabelian logics are the expansions of Ab with connectives corresponding to additional arithmetical operations on real numbers, such as multiplication or division. Other examples would be various types of congruential modal Abelian logics, i.e., logics where:

$$ \begin{align*}\varphi \rightarrow \psi, \psi \rightarrow \varphi \mathrel{\blacktriangleright} \Box\varphi \rightarrow \Box \psi. \end{align*} $$

We introduce now the algebraic semantics of Abelian logic and all superabelian logics.

It is well known that the defining conditions of Abelian $\ell $ -groups can be expressed by means of equations, so they form a variety which we denote by $\mathbb {AL}$ . Moreover, the lattice reduct of any Abelian $\ell $ -group is distributive.

In order to match the language that we have chosen to introduce the logic Ab, Abelian $\ell $ -groups can be equivalently seen as structures based on the interdefinability: $-a:=a \rightarrow 0$ and $a \rightarrow b:=b-a$ . To avoid excessive parentheses, we stipulate that the unary operation $-$ has precedence over all binary operations. Natural examples of (linearly ordered) Abelian $\ell $ -groups are those of the integers ${\boldsymbol {{Z}}}$ , the rational numbers ${\boldsymbol {Q}}$ , and the real numbers ${\boldsymbol {R}}$ . Note that $a\to b \geq 0$ iff .

Let $\mathcal L$ be any propositional language containing $\mathcal L_{\mathrm {Ab}}$ and let ${\boldsymbol {A}}$ be an $\mathcal L$ -algebra with an Abelian $\ell $ -group reduct. An ${\boldsymbol {A}}$ -evaluation is a homomorphism from ${\boldsymbol {{Fm}_{\mathcal {L}}}}$ to ${\boldsymbol {A}}$ . A consecution $\Gamma \mathrel {\blacktriangleright } \varphi $ is valid in ${\boldsymbol {A}}$ , $\Gamma \vDash _{\boldsymbol {A}} \varphi $ in symbols, if for every ${\boldsymbol {A}}$ -evaluation e it holds that, whenever $e(\psi ) \geq 0$ for every $\psi \in \Gamma $ , then also $e(\varphi ) \geq 0$ . Given a class ${{\mathbb K}}$ of $\mathcal L$ -algebras with an Abelian $\ell $ -group reduct, we write $\Gamma \vDash _{{\mathbb K}} \varphi $ if $\Gamma \vDash _{\boldsymbol {{B}}} \varphi $ for each ${\boldsymbol {{B}}} \in {{\mathbb K}}$ . It is well known that $\vDash _{{\mathbb K}}$ is a superabelian logic.

Let ${\mathrm{L}}$ be a superabelian logic in the language $\mathcal L$ . By ${\mathbf {Alg}^{*}({\mathrm{L}})}$ we denote the class of ${\mathrm{L}}$ -algebras, i.e., $\mathcal L$ -algebras ${\boldsymbol {A}}$ which have an Abelian $\ell $ -group reduct and ${\mathrm{L}} \subseteq {\vDash _{\boldsymbol {A}}}$ . For the purposes of this paper, we sometimes write that ${\boldsymbol {A}}$ is a model of ${\mathrm{L}}$ instead of saying that ${\boldsymbol {A}}$ is an ${\mathrm{L}}$ -algebra.

The usual techniques of algebraic logic allow us to prove that ${\mathbf {Alg}^{*}(\mathrm {\mathrm {Ab}})}=\mathbb {AL}$ and, for any superabelian logic ${\mathrm{L}}$ , we have the following completeness theorem: for each $\Gamma \cup \{\varphi \}$ , we have

$$ \begin{align*}\Gamma \vdash_{\mathrm{L}} \varphi\qquad \text{iff}\qquad \Gamma \vDash_{\mathbf{Alg}^{*}({\mathrm{L}})} \varphi. \end{align*} $$

Furthermore, one can easily check that the following consecutions are valid in Ab:

• ,

• ,

• 

Taken together, these consecutions give the condition $\mathrm {(Alg)}$ from [Reference Cintula and Noguera9, theorem 2.9.5], which guarantees that Ab (resp. any superabelian logic ${\mathrm{L}}$ ) is algebraically implicative (i.e., weakly implicative and algebraizable in the sense of Blok and Pigozzi [Reference Blok and Pigozzi3]) and the class $\mathbb {AL}$ (resp. ${\mathbf {Alg}^{*}({\mathrm{L}})}$ ) is its equivalent algebraic semantics. Therefore, there is a dual order isomorphism between the lattices of finitary extensions of Ab and subquasivarieties of $\mathbb {AL}$ .

We are interested, for a given superabelian logic ${\mathrm{L}}$ , in completeness theorems with respect to various (classes of) ${\mathrm{L}}$ -algebras, instead of the whole equivalent algebraic semantics. For reasons apparent later, we distinguish two kinds of completeness theorems.

Definition 2.5. A superabelian logic ${\mathrm{L}}$ enjoys the (Finite) Strong ${{\mathbb K}}$ -Completeness with respect to a class ${{\mathbb K}}$ of ${\mathrm{L}}$ -algebras, $\mathrm {(F)}{\mathrm {S}{{\mathbb K}}\mathrm {C}}$ for short, if for every (finite) set $\Gamma \cup \{\varphi \}$ of formulas,

$$ \begin{align*}\Gamma \vdash_{\mathrm{L}} \varphi\qquad \text{iff}\qquad \Gamma \vDash_{{\mathbb K}} \varphi. \end{align*} $$

A particularly interesting class of ${\mathrm{L}}$ -algebras is that of those whose underlying lattice is linearly ordered; following the common notation in the literature (see [Reference Cintula and Noguera9]), we denote this class as $\mathbf {Alg}^{\ell }({\mathrm{L}})$ and introduce the following central notion.

As we will see later in Example 5.4, not all superabelian logics have to be semilinear. There is a very useful characterization of semilinearity among superabelian logics with a countable presentation (which is, of course, the case for Ab, all finitary superabelian logics and, as we will see later, even some prominent infinitary ones) using the notion of a strong disjunction (see [Reference Cintula and Noguera9, definition 5.1.2] and note that the definition also requires the validity of $\varphi \mathrel {\blacktriangleright } \varphi \vee \psi $ and $\psi \mathrel {\blacktriangleright } \varphi \vee \psi $ , which is clearly the case in any superabelian logic).

Definition 2.7. The lattice connective is a strong disjunction in a superabelian logic ${\mathrm{L}}$ if for each set $\Gamma \cup \Phi \cup \Psi \cup \{\xi \}$ of formulas,

where is short for

The next theorem follows directly from [Reference Cintula and Noguera9, Theorems 6.2.2 and 5.5.14].

The power of the theorem above is that there is an easy syntactical way (see Corollary 2.13) to check whether $\vee $ is a strong disjunction using a simple syntactical construction.

Definition 2.9. Let $\Gamma \cup \{\varphi \} \subseteq {Fm}_{\!\mathcal L}$ . The -form of a consecution $\Gamma \mathrel {\blacktriangleright } \varphi $ is the set of consecutions

The following theorem is a particular version of [Reference Cintula and Noguera9, Theorem 5.2.6] (clearly, the consecutions and mentioned in that theorem are valid in each superabelian logic).

Now we are ready to give the promised direct alternative proof of semilinearity of Ab, which originally was established using a completeness theorem (see, e.g., [Reference Cintula and Noguera9, Example 6.2.3]).

Proof. As Ab has a countable presentation, it suffices to prove that $\vee $ is a strong disjunction which, thanks to Theorem 2.11, follows if we prove that and are valid in Ab.

For the first one, let ${\boldsymbol {A}}$ be an Abelian $\ell $ -group and e an ${\boldsymbol {A}}$ -evaluation such that and . Then, by the assumptions and distributivity of the lattice operations, we have

To prove the validity of , consider an evaluation e such that and . This gives us . Using the distributivity of the lattice operations, we obtain

For simplicity, let us set , $e(\psi ) = b$ , and . As holds in $\ell $ -groups, we obtain

Note that, due to monotonicity of addition, we have

Therefore, we obtain as required.

The following corollary is a direct consequence of Theorems 2.8, 2.11, and 2.12.

In order to obtain completeness results for the logics studied in this paper, we will make use of the following two theorems and corollary which offer algebraic characterizations adapted from the general framework of the monograph [Reference Cintula and Noguera9] to the setting of superabelian logics. For clarity, we first recall the definitions of the standard class operators from universal algebra.

Since superabelian logics are algebraically implicative, we can recall the following two useful characterizations of (finite) strong completeness proved as [Reference Cintula and Noguera9, corollaries 3.8.3 and 3.8.7] (in the latter case we also use the fact that for semilinear logics $\mathbf {Alg}^*({\mathrm{L}})_{\mathrm {RFSI}}=\mathbf {Alg}^{\ell }({\mathrm{L}})$ , see [Reference Cintula and Noguera9, Theorem 6.1.7]).

Finally, we add a simple yet important corollary of the previous which follows from the fact that [Reference Bergman1, Theorem 5.6].

Thanks to [Reference Khisamiev22], we know the quasivariety of Abelian $\ell $ -groups is generated (as a quasivariety) by ${\boldsymbol {{Z}}}$ and, since ${\boldsymbol {{Z}}}$ can be embedded into any non-trivial Abelian $\ell $ -group, we obtain the following corollary.

As we will see in the next section, $\mathbb {AL}$ is not generated as a generalized quasivariety by any of ${\boldsymbol {R}}$ , ${\boldsymbol {Q}}$ , or ${\boldsymbol {{Z}}}$ , and it admits numerous non-trivial generalized subquasivarieties.

In logical terminology, we know that Ab is finitely strongly complete w.r.t. any non-trivial Abelian $\ell $ -group and has no consistent finitary extension, though it has numerous proper consistent infinitary extensions. Moreover, it is not strongly complete w.r.t. any of ${\boldsymbol {R}}$ , ${\boldsymbol {Q}}$ or ${\boldsymbol {{Z}}}$ . In the next section, we will axiomatize its proper extension strongly complete w.r.t. ${\boldsymbol {R}}$ .

3 Infinitary extensions of Abelian logic

Having established that Abelian logic (Ab) lacks non-trivial finitary extensions, a well-known result that motivates our next step, we now turn our attention to the much richer landscape of its infinitary extensions. This section explores this domain by introducing infinitary rules designed to capture specific semantic properties.

Clearly, the most prominent candidates for such infinitary logics are ${\vDash _{\boldsymbol {R}}}$ , ${\vDash _{\boldsymbol {Q}}}$ , and ${\vDash _{\boldsymbol {{Z}}}}$ . We know that:

$$ \begin{align*}{\mathrm {Ab}} \subseteq {\vDash_{\boldsymbol{R}}} \subseteq {\vDash_{\boldsymbol{Q}}} \subseteq {\vDash_{\boldsymbol {{Z}}}} \end{align*} $$

and that above ${\vDash _{\boldsymbol {{Z}}}}$ there is only the inconsistent logic (because ${\boldsymbol {{Z}}}$ obviously generates the smallest non-trivial generalized subquasivariety of Abelian $\ell $ -groups).

In this section, we obtain the following new results:

• • strictness of all three inclusions, and thus these three logics are indeed not finitary (Example 3.3, Corollary 3.6 and Theorem 3.7);

• • there are $2^{2^\omega }$ distinct infinitary logics between ${\vDash _{\boldsymbol {R}}}$ and ${\vDash _{\boldsymbol {Q}}}$ (Corollary 3.6);

• • an axiomatization of the logic $\vDash _{\boldsymbol {R}}$ (Theorem 3.4).

Moreover, we give a conjecture for an axiomatization of the logic $\vDash _{\boldsymbol {{Z}}}$ , give strong arguments to support it, and point to the impossibility of proving it by usual methods (Conjecture 3.8).

From now on, we will use the following notation: for $n \in \mathbb {N}$ and a formula $\varphi $ , we write $n \cdot \varphi $ instead of $\underbrace {\varphi\ {\&} \dots {\&}\ \varphi }_{n \text { times}}$ (we also set $0 \cdot \varphi = \texttt {t}$ ). We use the analogous notation for $+$ in the algebraic setting.

Let us first recall the infinitary rule used to axiomatize the infinitary Łukasiewicz logic (see [Reference Hay19]):

(Hay) $$ \begin{align} \{\neg \varphi \rightarrow n \cdot \varphi \mid n \in \mathbb{N} \} \mathrel{\blacktriangleright} \varphi. \end{align} $$

Since the language of Ab does not include a negation with similar properties to that of Łukasiewicz logic, we have to replace $\neg \varphi $ by $\psi $ and define

(Arch) $$ \begin{align} \{\psi \rightarrow n \cdot \varphi \mid n \in \mathbb{N} \} \mathrel{\blacktriangleright} \varphi. \end{align} $$

As its name suggests, the rule (Arch) is intended as a syntactic counterpart to the well-known algebraic property of being Archimedean. We introduce the standard definition of this property for Abelian $\ell $ -groups.

Definition 3.1. Let ${\boldsymbol {A}}$ be an Abelian $\ell $ -group. We say that ${\boldsymbol {A}}$ is Archimedean if and only if for each $a,b \in A$ it holds

The following lemma makes the connection between the rule and the property precise, showing that the (Arch) rule is valid in an Abelian $\ell $ -group if and only if that group is Archimedean.

By the next example, we know that (Arch) is not valid in Ab and so Ab is strictly contained in $\vDash _{\boldsymbol {R}}$ and thus $\vDash _{\boldsymbol {R}}$ cannot be finitary.

Let us call the logic $\vDash _{\boldsymbol {R}}$ the real Abelian logic and denote it ${\mathrm {rAb}}$ . The next theorem shows how to axiomatize it.

Note that this proof relies on the -form of the rule (Arch) which is necessary to ensure that the resulting logic is semilinear. Therefore, it remains an open question whether the rule (Arch) itself would suffice on its own to axiomatize ${\mathrm {rAb}}$ , which is equivalent to the question whether (Arch) is valid in ${\mathrm {Ab}} + \mathrm {(Arch)}$ .

Next, we show that $\vDash _{\boldsymbol {Q}}$ is strictly stronger than $\vDash _{\boldsymbol {R}}$ (in other words, ${\mathrm {rAb}}$ is not strongly complete w.r.t. ${\boldsymbol {Q}}$ ).

We use the following convention. For an $\ell $ -group ${\boldsymbol {A}}$ , elements $a,b \in A$ and ${\frac {k}{l} \in \mathbb Q \setminus \{0\}}$ we write or when . Similarly, we write $\frac {k}{l} \cdot \varphi \rightarrow \psi $ or $\varphi \rightarrow \frac {l}{k} \cdot \psi $ instead of $k \cdot \varphi \rightarrow l \cdot \psi $ .

For each $0<\gamma \in \mathbb R \setminus \mathbb Q$ , we introduce the following rule, in which is a strictly increasing sequence of rationals converging to $\gamma $ and is a strictly decreasing sequence of rationals converging to $\gamma $ :

(str_γ) $$\begin{align} & \{q_j \cdot \varphi \rightarrow \psi, \psi \rightarrow r_j \cdot \varphi \mid j \in \mathbb{N}\} \mathrel{\blacktriangleright} \varphi \rightarrow \texttt{t}. \end{align} $$

The next proposition shows that in our context $\mathrm {(str_\gamma )}$ does not depend on the choice of the sequences.

Proposition 3.5. Given $0<\gamma \in {\boldsymbol {R}} \setminus {\boldsymbol {Q}}$ , a sequence of rationals strictly increasing and converging to $\gamma $ , and a sequence of rationals strictly decreasing and converging to $\gamma $ , the logic is strongly complete w.r.t.

$$ \begin{align*}{{\mathbb K}}=\{{\boldsymbol{A}} \subseteq {\boldsymbol{R}} \mid (\forall a,b \in A) \; (b= \gamma \cdot a \implies a=0) \}.\end{align*} $$

Moreover, for any ${\boldsymbol {A}} \in \mathbf {IS}({\boldsymbol {R}})$ we have if and only if ${\boldsymbol {A}} \in {{\mathbb K}}$ .

Proof. Since the logic satisfies the conditions from Corollary 2.13, it is semilinear. Since is an extension of ${\mathrm {rAb}}$ , we obtain that is strongly complete w.r.t. some subclass of $\mathbf {S}({\boldsymbol {R}})$ .

We need the following technical observation.

Observation. Let ${\boldsymbol {A}} \subseteq {\boldsymbol {R}}$ and $a,b \in A$ . Then we have

Proof. First, since and $q_0 < r_0$ , we can conclude that . We have and . Consequently, . Therefore, we obtain $\gamma \cdot a=b$ . The other direction follows as well: if $\gamma \cdot a=b$ and $a \geq 0$ , then for each $j \in \mathbb {N}$ (since $a \geq 0$ and $q_j$ is strictly increasing) and for each $j \in \mathbb {N}$ (since $a \geq 0$ and $r_j$ is strictly decreasing).

First, let us take an arbitrary ${\boldsymbol {A}} \in \mathbb K$ . We show that ${\boldsymbol {A}}$ satisfies . By the observation and the definition of ${{\mathbb K}}$ , if, for $a,b \in A$ , we have and for each $j \in \mathbb {N}$ , then it must be that $a=0$ . Thus, any evaluation in ${\boldsymbol {A}}$ that satisfies the premise of $\mathrm {(str_\gamma )}$ must map $\varphi $ to $0$ , thereby satisfying the conclusion $\varphi \rightarrow \texttt {t}$ . Therefore, $\mathrm {(str_\gamma )}$ is valid in ${\boldsymbol {A}}$ . Since ${\boldsymbol {A}}$ is a chain, by Lemma 2.10 is also valid in ${\boldsymbol {A}}$ , and thus ${\boldsymbol {A}}$ is a model of .

To prove the other direction, assume for an algebra ${\boldsymbol {{1}}} \neq {\boldsymbol {A}} \subseteq {\boldsymbol {R}}$ that there exist $a,b \in A$ such that $b=\gamma \cdot a$ and $a \neq 0$ . Clearly, $\gamma \cdot (-a)=\ -b$ holds as well. Since ${\boldsymbol {A}}$ is totally ordered, we can assume without loss of generality that $a> 0$ . By the observation, this implies that for each $j \in \mathbb {N}$ , and . But since $a,b \in A$ and $a> 0$ , this is a counterexample to the validity of rule $\mathrm {(str_\gamma )}$ in ${\boldsymbol {A}}$ . By Lemma 2.10, fails in ${\boldsymbol {A}}$ as well. This shows that ${\boldsymbol {A}}$ is not a model of .

Clearly, for $\gamma _1,\gamma _2 \in {\boldsymbol {R}} \setminus {\boldsymbol {Q}}$ , the rules and do not necessarily define different classes of models. For example, one can easily show that is valid in an algebra ${\boldsymbol {A}}$ if and only if is valid in ${\boldsymbol {A}}$ . Nevertheless, one can show the following.

Note that Proposition 3.5 also hints at a possible axiomatization of the logic $\vDash _{\boldsymbol {Q}}$ . Indeed we can easily observe that $\vDash _{\boldsymbol {Q}}$ is axiomatized as iff this logic is semilinear (which we unfortunately do not know, as Corollary 2.13 works for countably axiomatized logics only). Therefore, the axiomatization of $\vDash _{\boldsymbol {Q}}$ remains as an open problem.

Finally, we show that $\vDash _{\boldsymbol {{Z}}}$ is strictly stronger than $\vDash _{\boldsymbol {Q}}$ . To this end, we propose the following infinitary rule IDC (standing for “infinite decreasing chain”):

(IDC) $$ \begin{align} \{ 2\cdot \varphi_{n+1} \rightarrow \varphi_{n}, \varphi_n \rightarrow 4 \cdot \varphi_{n+1} \mid n \in \mathbb{N}\} \mathrel{\blacktriangleright} \varphi_0 \rightarrow \texttt{t} \end{align} $$

and prove the following theorem.

Before we prove this theorem, let us state a conjecture based on this result.Footnote ⁵

Observe that the conjecture holds if and only if ${\mathrm {Ab}}+\mathrm {(IDC)}^\vee +\mathrm {(Arch)}^\vee $ is a semilinear logic. Unfortunately, the rule (IDC) uses infinitely many variables and thus, while we know that $\vee $ is a strong disjunction in ${\mathrm {Ab}}+\mathrm {(IDC)}^\vee +\mathrm {(Arch)}^\vee $ by Corollary 2.13, we do not know whether this logic is actually a semilinear logic.

This is, however, a systematic problem as any rule valid in $\vDash _{\boldsymbol {{Z}}}$ but not in $\vDash _{\boldsymbol {Q}}$ has to use infinitely many variables. Indeed, since any finitely generated subalgebra of ${\boldsymbol {Q}}$ is isomorphic to ${\boldsymbol {{Z}}}$ , any rule with finitely many variables valid in $\vDash _{\boldsymbol {{Z}}}$ must also be valid in $\vDash _{\boldsymbol {Q}}$ .

Let us now proceed towards the proof of Theorem 3.7. First, we need to introduce a machinery of decreasing sequences (which justifies the name of the rule).

Definition 3.9. Let ${\boldsymbol {A}}$ be an Abelian $\ell $ -group. A sequence of elements of A is strongly decreasing if, for each $i \in \mathbb {N}$ , there is an $n_{i} \geq 4$ such that:

$$ \begin{align*}n_{i} \cdot a_{i+1} \geq a_i \geq 2 \cdot a_{i+1}. \end{align*} $$

We call the sequence trivial if $a_i = 0$ for each i.

Let us note that any non-trivial strongly decreasing sequence in ${\boldsymbol {A}}$ is indeed decreasing and $a_i \geq 0$ for each $i \geq 1$ . The latter claim follows from the fact that for each i there is an $n_i\geq 4$ such that $n_i \cdot a_i \geq 2n_i \cdot a_{i+1} \geq 2 \cdot a_i$ , thus $(n_i-2) \cdot a_i \geq 0$ , from which one can derive $a_i \geq 0$ (since in ${\boldsymbol {{Z}}}$ for any $n \geq 1$ the quasiequation $n \cdot x\geq 0 \implies x\geq 0$ is valid, it holds in ${\boldsymbol {A}}$ ). The former claim then follows from the validity of $x \geq 0 \implies 2 \cdot x \geq x$ in ${\boldsymbol {{Z}}}$ .

Proof. To prove the first claim, it suffices to show the strictness. Assume that ${a_i=a_{i+1}}$ for some $i \in \mathbb {N}$ . Then the inequality $a_i \geq 2 \cdot a_{i+1}$ becomes $a_i \geq 2 \cdot a_i$ and thus $0 \geq a_i.$ Since also $a_i \geq 0$ , we obtain $a_i=0$ . For each $j> i$ we have $a_i \geq a_j \geq 0$ , thus we get $a_j=0$ as well. Similarly, for $j < i$ , there is an $n \in \mathbb {N}$ such that $n \cdot a_i \geq a_j \geq 0$ and thus again $a_j=0$ . Therefore, the sequence has to be trivial, a contradiction.

We will prove the second claim. Let us assume that ${\boldsymbol {A}}$ is an Archimedean chain and is a strictly decreasing sequence of positive elements. Without loss of generality, as ${\boldsymbol {A}}$ is Archimedean, we can assume that ${\boldsymbol {A}} \subseteq {\boldsymbol {R}}$ , by Hölder’s Theorem. We construct a non-trivial strongly decreasing sequence . Set $b_0> 0$ arbitrarily. We will define the rest of the sequence recursively. Assume that we have already constructed satisfying the conditions from the definition of a strongly decreasing sequence. Now we want a $b_{n+1}$ such that .

Since is a strictly decreasing sequence, it follows that there exists a $j \in \mathbb {N}$ such that $\frac {a_j}{b_n} \notin \mathbb Z$ (otherwise would be a strictly decreasing sequence of positive elements in ${\boldsymbol {{Z}}}$ , which is not possible).

Since ${\boldsymbol {A}}$ is Archimedean, there is a $k \in \mathbb {N}$ such that . Since $\frac {a_j}{b_n} \notin \mathbb Z$ , we have $0 \neq a_j -k \cdot b_n$ and thus $0 < a_j -k \cdot b_n < b_n$ . Hence,

Therefore, we set

$$ \begin{align*}b_{n+1}=\min\{a_j -k \cdot b_n, b_n-(a_j -k \cdot b_n)\}.\end{align*} $$

Clearly, $b_{n+1}>0$ and . The second condition for strong decreasing sequences easily follows from the fact that ${\boldsymbol {A}}$ is Archimedean. Thus, is a non-trivial strongly decreasing sequence.

The previous proposition gives us a description of Abelian $\ell $ -groups satisfying (IDC) in terms of the existence of strongly decreasing sequences. Lemma 3.10 shows that there is a connection between strictly decreasing sequences of positive elements and strongly decreasing sequences. Therefore, it is natural to ask whether the previous proposition could be stated in terms of the existence of strictly decreasing sequences of positive elements. The following example shows that this is not the case.

Example 3.13. There exist $\ell $ -groups in $\mathbf {P}({\boldsymbol {{Z}}})$ which have strictly decreasing sequences of positive elements but lack a non-trivial strongly decreasing sequence.

Consider the Abelian $\ell $ -group ${\boldsymbol {{Z}}}^\omega $ and note that the sequence of positive elements

is strictly decreasing. However, the previous proposition clearly states that only the trivial sequence is strongly decreasing in ${\boldsymbol {{Z}}}^{\omega }$ , since ${\boldsymbol {{Z}}}^{\omega } \in \mathbf {P}({\boldsymbol {{Z}}})$ , $\mathbf {P}({\boldsymbol {{Z}}}) \subseteq \mathbf {ISP}_\omega ({\boldsymbol {{Z}}})$ , and (IDC) holds in $\mathbf {ISP}_\omega ({\boldsymbol {{Z}}})$ .

Now we are finally ready to give the promised proof of Theorem 3.7.

4 Pointed Abelian logic

In the previous sections, we have explored the landscape of Abelian logic and its infinitary extensions. We now turn our attention to expansions, that is, we allow not only new rules, but also enriched logical languages. This section introduces pointed Abelian logic $({\mathrm {pAb}})$ , which extends Abelian logic with a new constant symbol $\texttt {f}$ . This seemingly simple addition of an additional “point” to the algebraic semantics dramatically alters the character of the logic. Whereas Abelian logic has no non-trivial finitary extensions, we will show that this pointed version gives rise to a plentiful landscape of logical extensions.Footnote ⁶ The central result of this section is a completeness theorem demonstrating that this new logic is finitely strongly complete with respect to just three canonical pointed structures over the real numbers: ${\boldsymbol {R}}_{-1}, {\boldsymbol {R}}_0$ , and ${\boldsymbol {R}}_1$ .

Let us consider the expansion of the language $\mathcal L_{\mathrm {Ab}}$ of Abelian logic by a new constant symbol $\texttt {f}$ . We denote this language by $\mathcal L_{\mathrm {pAb}}$ . In this section, we study the minimal expansion of Ab in this language.

Recall that Full Lambek logic with exchange, ${\mathrm {FL}}_{\mathrm {e}}$ , is a prominent substructural logic in the language of $\mathcal L_{\mathrm {pAb}}$ which is axiomatized by omitting the axiom $\mathrm {(rel)}$ from the axiomatic system given in Table 1. Thus, pointed Abelian logic is the extension of ${\mathrm {FL}}_{\mathrm {e}}$ by axiom $\mathrm {(rel)}$ .

Interestingly, we can see Ab as an extension of ${\mathrm {pAb}}$ by the axiom which collapses both constants:

$$ \begin{align*}(\texttt{f}\to\texttt{t})\wedge(\texttt{t}\to\texttt{f}). \end{align*} $$

Therefore, sometimes in the literature Ab is presented as the expansion of ${\mathrm {FL}}_{\mathrm {e}}$ by axioms $\mathrm {(rel)}$ and $(\texttt {f}\to \texttt {t})\wedge (\texttt {t}\to \texttt {f})$ . But, as we have commented in footnote 3, we prefer to see ${\mathrm {pAb}}$ as an expansion of Ab.

Clearly, ${\mathrm {pAb}}$ and all its extensions have ${\mathrm {(sCng)}}$ and hence they are superabelian logics. Therefore, it is easy to see that $\mathbf {Alg}^*({\mathrm {pAb}})$ , i.e., the equivalent algebraic semantic of ${\mathrm {pAb}}$ , is the class of pointed Abelian $\ell $ -groups, i.e., Abelian $\ell $ -groups expanded to the language $\mathcal L_{\mathrm {pAb}}$ with a new constant symbol interpreted in an arbitrary way.

To be able to easily talk about particular ${\mathrm {pAb}}$ -algebras, we present the following definition.

${\boldsymbol {R}}_{-1}$ , ${\boldsymbol {R}}_0$ , and ${\boldsymbol {R}}_1$ are natural examples of pointed Abelian $\ell $ -groups. Note that obviously, $\mathbf {Alg}^*({\mathrm {pAb}}) = \mathbf {p} \mathbb {AL}$ . It is also a straightforward, albeit rather technical, matter to show that this pointing operator $\mathbf {p}$ commutes with the standard algebraic class operators for taking isomorphisms, subalgebras, products, and ultraproducts.

Let us formulate a trivial lemma which allows us to improve the previous result for ${\boldsymbol {A}}$ being either ${\boldsymbol {R}}$ or ${\boldsymbol {Q}}$ and will also be of use later.

Observe that the same would not work, e.g., for the Abelian $\ell $ -group of the integers, since ${\boldsymbol {{Z}}}_a$ and ${\boldsymbol {{Z}}}_b$ are not isomorphic unless $a=b$ . In fact, one can show that ${\boldsymbol {{Z}}}_a$ generates a different variety for each $a \in \mathbb Z$ .

Proof. One direction is obvious. To prove the converse one, let us assume $a \neq b$ . To show that $\mathbf {HSP}({\boldsymbol {{Z}}}_a) \neq \mathbf {HSP}({\boldsymbol {{Z}}}_b)$ , it is sufficient to find an equation which is valid in one of the generating algebras, ${\boldsymbol {{Z}}}_a$ or ${\boldsymbol {{Z}}}_b$ , but not in the other. If $a<0$ and $b \geq 0$ take the equation $\texttt {f} \geq 0$ . This equation clearly holds in ${\boldsymbol {{Z}}}_b$ but it does not hold in ${\boldsymbol {{Z}}}_a$ . The case and $b>0$ is similar. Therefore, it remains to check the cases where $a,b>0$ or $a,b<0$ . We will check only the first case (the second one is analogous). Moreover, let us assume without loss of generality that $0 <a<b$ .

Consider the following equation:

(*)

This equation is not valid in ${\boldsymbol {{Z}}}_b$ , since if we take an evaluation e such that $e(x)=1$ we get . Thus the equation (*) does not hold in ${\boldsymbol {{Z}}}_b$ .

We show the same equation holds in ${\boldsymbol {{Z}}}_a$ . First let us consider an evaluation e such that . Clearly, we get $-e(x) \geq 0$ and thus It remains to consider an evaluation e such that $e(x)>0$ (and thus $e(x) \geq 1$ ). Then $a \cdot e(x) -a \geq a \cdot 1-a=0$ . Thus the equation (*) is valid in ${\boldsymbol {{Z}}}_a$ .

5 Łukasiewicz unbound logic (and other logics of reals)

In this section, we formally introduce Łukasiewicz unbound logic within our framework, motivated by the observation that its original algebraic semantics is isomorphic to the pointed Abelian $\ell $ -group ${\boldsymbol {R}}_{-1}$ . Our primary goal is to axiomatize this logic as an extension of pointed Abelian logic. We first define the finitary version, $\mathrm {Lu}$ , and establish its finite strong completeness with respect to ${\boldsymbol {R}}_{-1}$ . Then, we introduce its infinitary version, $\mathrm {Lu}_{\infty }$ , by adding the Archimedean rule to achieve strong completeness, and make the connection to standard Łukasiewicz logic precise by providing a formal translation. Generalizing the methods developed for this specific case, we broaden our scope to systematically axiomatize the logics of other prominent pointed groups, such as ${\boldsymbol {R}}_0$ and ${\boldsymbol {R}}_1$ . We establish a correspondence between algebraic properties of the point element $\texttt {f}$ and simple syntactic rules, allowing us to provide a comprehensive overview of completeness results for this family of logics.

Let us start this section by observing that the algebra ${\boldsymbol {R}}_{-1}$ is isomorphic to the algebra ${\boldsymbol {{LU}}}$ that was used in [Reference Cintula, Grimau, Noguera and Smith8] to introduce Łukasiewicz unbound logic. ${\boldsymbol {{{LU}}}}$ is a member of $\mathbf {p}\mathbb {AL}$ defined over the reals with lattice operations defined in the usual way and other operations/constants defined as

$$ \begin{align*}x \to^{\boldsymbol {{LU}}} y = 1 - x +y \qquad x\ {\&}^{\boldsymbol {{LU}}} y = x + y - 1 \qquad \texttt{t}^{\boldsymbol {{LU}}} = 1 \qquad \texttt{f}^{\boldsymbol {{LU}}} = 0. \end{align*} $$

Clearly, defined as $f(x) = x +1$ is the desired isomorphism. As already mentioned in the introduction, the operations on ${\boldsymbol {{LU}}}$ closely resemble those of the well-known standard MV-algebra which provides semantics for the Łukasiewicz logic (see more details at the end of this section). In the context of this paper we will see it as an algebra in the language of ${\mathrm {pAb}}$ (though it clearly is not a member of ${\mathbf {Alg}^{*}(\mathrm {\mathrm {pAb}})}$ )Footnote ⁷ defined on the real unit interval $[0,1]$ with lattice operations defined in the usual way and other operations/constants defined as

Recall that we know that ${\mathrm {pAb}}$ is finitely strongly complete with respect to $\{{\boldsymbol {R}}_{-1}, {\boldsymbol {R}}_{1}, {\boldsymbol {R}}_{0}\}$ , thus our goal is to find a finitary rule which allows us to extend ${\mathrm {pAb}}$ in a way that preserves semilinearity and also ensures that the rule is not valid in ${\boldsymbol {R}}_0$ and ${\boldsymbol {R}}_1$ .

Definition 5.1. Łukasiewicz unbound logic $\mathrm {Lu}$ is the extension of the logic ${\mathrm {pAb}}$ by the rule:

(Lu)

To obtain the desired completeness, let us characterize linearly ordered Abelian $\ell $ -groups satisfying the rule $\mathrm {(Lu)}$ . We give a slightly more general result, which will allow us to show that we cannot replace $\mathrm {(Lu)}$ by a simpler, weaker rule that might initially seem sufficient.

Thus, we know that in particular ${\boldsymbol {R}}_{-1} \in \mathbf {Alg}^{\ell }(\mathrm {Lu})$ but ${\boldsymbol {R}}_0, {\boldsymbol {R}}_1 \not \in \mathbf {Alg}^{\ell }(\mathrm {Lu})$ . The next example shows that the rules and $\texttt {f} \mathrel {\blacktriangleright } \varphi $ are not interderivable and thus the extension of ${\mathrm {pAb}}$ by the rule $\texttt {f} \mathrel {\blacktriangleright } \varphi $ is strictly weaker than $\mathrm {Lu}$ .

Proof. We prove the final part of the claim, the rest then follows from Lemma 4.4. Using Theorem 4.3 and Corollary 2.17, we know that

$$ \begin{align*}\mathbf{Alg}^{\ell}(\mathrm {Lu}) \subseteq \mathbf{Alg}^{\ell}(\mathrm {\mathrm {pAb}}) \subseteq \mathbf{I}({\boldsymbol {{1}}}) \cup \mathbf{ISP}_{\mathrm{U}}(\{{\boldsymbol{A}}_{a} \mid a < 0\}) \cup \mathbf{ISP}_{\mathrm{U}}(\{{\boldsymbol{A}}_{a} \mid a \geq 0\}). \end{align*} $$

Clearly, for any non-trivial element of $\mathbf {ISP}_{\mathrm {U}}(\{{\boldsymbol {A}}_{a} \mid a \geq 0\})$ , we have $\texttt {f}^{{\boldsymbol {A}}_{a}}\geq 0$ (an equation preserved by ultrapowers) and thus it is not a member of $\mathbf {Alg}^{\ell }(\mathrm {Lu})$ due to the previous lemma. Thus,

$$ \begin{align*}\mathbf{Alg}^{\ell}(\mathrm {Lu}) \subseteq \mathbf{I}({\boldsymbol {{1}}}) \cup \mathbf{ISP}_{\mathrm{U}}(\{{\boldsymbol{A}}_{a} \mid a < 0\}) \end{align*} $$

and so the proof is done by Corollary 2.17.

It is easy to see that the logic $\mathrm {Lu}$ is not strongly complete with respect to ${\boldsymbol {R}}_{-1}$ . For instance, the rule (Arch), introduced in the previous section, is valid in ${\boldsymbol {R}}_{-1}$ but is not derivable in $\mathrm {Lu}$ (by Example 3.3). This shows that the full set of validities in ${\boldsymbol {R}}_{-1}$ cannot be captured by a finitary system extending $\mathrm {Lu}$ and will therefore require an infinitary axiomatization. To this end, we define now the following extension of $\mathrm {Lu}$ .

Equivalently, $\mathrm {Lu}_{\infty }$ can be seen as the expansion of ${\mathrm {rAb}}$ to the language of ${\mathrm {pAb}}$ by the rule $\mathrm {(Lu)}$ .

To conclude our analysis of these systems, we discuss the connection between (infinitary) Łukasiewicz unbound logic and (infinitary) Łukasiewicz logic. The Łukasiewicz logic is a finitary logic axiomatized by four axioms and modus ponens and is finitely strongly complete w.r.t. the standard ${\mathrm {MV}}$ -algebra introduced at the beginning of this section. The infinitary Łukasiewicz logic is its extension by the rule (Hay) introduced in §3, this logic indeed is not finitary but is strongly complete w.r.t. the standard ${\mathrm {MV}}$ -algebra .

As we have seen in the beginning of the section, the algebra is a restriction of the algebra ${\boldsymbol {{LU}}}$ which in turn is an isomorphic copy of ${\boldsymbol {R}}_{-1}$ . We use this fact (together with the completeness results for the involved logics) to prove the following results linking (infinitary) Łukasiewicz logic and (infinitary) Łukasiewicz unbound logics via the mapping $\tau $ Footnote ⁸ defined recursively on the set of formulas:

• • for all variables p.

• • .

• • .

• • $\tau (c) = c$ for $c \in \{\texttt {f},\texttt {t}\}$ .

• • $\tau (\psi \circ \chi ) = \tau (\psi ) \circ \tau (\chi )$ for

Theorem 5.8. Let $\Gamma \cup \{\varphi \}$ be a set of formulas. Then, we have

Moreover, if $\Gamma $ is finite we have

Proof. Recall that, thanks to the known completeness properties, we can replace the syntactical consequence relations by semantical ones w.r.t. algebras and ${\boldsymbol {{\mathrm {Lu}}}}$ (because it is an isomorphic copy of ${\boldsymbol {R}}_{-1}$ ).

Let us consider any mapping e of propositional variables into reals and define the mapping $\bar e$ as its restriction to $[0,1]$ , i.e.,

We denote by and $e^{\boldsymbol {{Lu}}}$ the corresponding evaluations. By a simple proof by induction over the complexity of formula $\chi $ , we obtain

Clearly, it holds for variables and constants by definition of $\bar e$ and $\tau $ . Now assume and for formulas $\alpha ,\beta $ . We want to show that for . We show this step only for $\circ ={\&}$ . We write

Now we can prove the left-to-right direction of the first equivalence. Assume that and that e is a mapping such that for each $\gamma \in \Gamma $ we have (recall that $\texttt {t}^{\boldsymbol {{LU}}} = 1$ )

$$ \begin{align*}e^{\boldsymbol {{Lu}}}(\tau(\gamma)) \geq 1. \end{align*} $$

As we have for $\gamma \in \Gamma $ and so, by the assumption, also , which entails $e^{\boldsymbol {{\mathrm {Lu}}}}(\tau (\varphi )) = 1$ .

To prove the converse direction, it suffices to note that if the co-domain of e is a subset of $[0,1]$ , then $e = \bar e$ .

The second equivalence can either be proved in exactly the same way or actually follows from the first one and the fact that on finite sets of premises the Łukasiewicz (unbound) logic coincides with its infinitary variant.

The translation provides some information regarding computational complexity. In Łukasiewicz logic, the problem of deciding the validity of finitary consecutions is known to be coNP-complete [Reference Haniková, Cintula, Hájek and Noguera17]. As Theorem 5.8 provides a polynomial-time reduction of finitary consecutions from to $\mathrm {Lu}$ , it immediately follows that the corresponding problem for $\mathrm {Lu}$ is coNP-hard. The more involved question of whether the set of valid finitary consecutions of $\mathrm {Lu}$ is also in coNP, and thus coNP-complete, has been answered affirmatively in the paper [Reference Haniková, Jankovec, Guerrini and König18].

Having established the axiomatization for the logic of ${\boldsymbol {R}}_{-1}$ , we now turn to the complementary task of characterizing the logics of the other prominent pointed algebras, ${\boldsymbol {R}}_0$ and ${\boldsymbol {R}}_1$ . As we will show, a similar approach allows us to axiomatize these systems in a straightforward manner. Let us define a new connective $-\varphi $ as $\varphi \rightarrow \texttt {t}$ . By simple checking we prove the following lemma (note that we could prove a more complex variant akin to Lemma 5.3 to demonstrate that seemingly simpler rules would not do).

Theorem 5.10. Let ${\boldsymbol {A}}$ be any non-trivial ${\mathrm {rAb}}$ -algebra. Then the extension of ${\mathrm {pAb}}$ by the rule/axiom mentioned in a given row of Table 2 is finitely strongly complete w.r.t. the sets of algebras mentioned in the corresponding columns Set 1–3. Its extension by the rule (Arch) $^\vee $ is strongly complete w.r.t. the set of algebras mentioned in the corresponding column Set 3.

Table 2 Completeness properties of prominent extensions of ${\mathrm {pAb}}$

As mentioned before, the expansion of ${\mathrm {pAb}}$ by $\mathrel {\blacktriangleright } \texttt {f}\wedge\ {-}\texttt {f}$ can be seen as the Abelian logic itself (formally speaking, these two logics are termwise equivalent by setting $\texttt {f} = \texttt {t}$ ).

Let us conclude this section by showing a natural and simple translation between $\mathrm {Lu}$ and the expansion of ${\mathrm {pAb}}$ by , denoted here by $\mathrm {Lu}^*$ .

Theorem 5.11. Let us consider the mapping defined recursively as follows:

• • $\tau \colon \mathtt {f} \mapsto\ {-}\mathtt {f}$ .

• • $\tau \colon \mathtt {t} \mapsto \mathtt {t}$ .

• • $\tau \colon p \mapsto p $ for each variable p.

• • $\tau \colon (\psi \circ \xi ) \mapsto (\tau (\psi ) \circ \tau (\xi ))$ for all binary connectives $\circ \in \mathcal L_{\mathrm {pAb}}$ .

For any consecution $\Gamma \mathrel {\blacktriangleright } \varphi $ and for each Abelian $\ell $ -group ${\boldsymbol {A}}$ , and $a \in A$ we have

$$ \begin{align*}\Gamma\vDash_{{\boldsymbol{A}}_a} \varphi \qquad \text{iff} \qquad \tau[\Gamma]\vDash_{{\boldsymbol{A}}_{-a}} \tau(\varphi).\end{align*} $$

In particular, for any consecution $\Gamma \mathrel {\blacktriangleright } \varphi $ we have:

1. 1. $\Gamma \vdash _{\mathrm {Lu}} \varphi $ iff $\tau [\Gamma ]\vdash _{\mathrm {Lu}^*} \tau (\varphi ) $ .

2. 2. $\Gamma \vdash _{\mathrm {Lu}^*} \varphi $ iff $\tau [\Gamma ]\vdash _{\mathrm {Lu}} \tau (\varphi ) $ .

Clearly, the same translation would link the infinitary versions of these two logics (i.e., their extensions of (Arch) $^\vee $ ); the expansions of ${\mathrm {pAb}}$ by (1) by axiom $\texttt {f}$ and (2) axiom ${-}\texttt {f}$ ; and their extensions of (Arch) $^\vee $ .

6 Future work

In this section, we outline possible directions for future research on the topics covered in this paper.

• • Study of the lattice of infinitary extensions of Abelian logic: While the present paper provides an infinitary rule for the extension corresponding to the generalized quasivariety generated by ${\boldsymbol {R}}$ and establishes the existence of $2^{2^\omega }$ distinct infinitary extensions, it is clear that we have only started the topic and the structure of the full lattice of infinitary extensions of Ab remains largely unexplored. One of the goals for future research is a systematic classification of this lattice. In particular, a major open challenge is explicitly axiomatizing other fundamental logics, namely, the logics of ${\boldsymbol {{Z}}}$ and ${\boldsymbol {Q}}$ .

• • Infinitary logics of distinct total orders on ${\boldsymbol {{Z}}}^n$ : It is a well-established result [Reference Clay and Rolfsen10] that the space of total orders on ${\boldsymbol {{Z}}}^n$ is homeomorphic to a Cantor space for $n>1$ . A natural open problem is to determine whether each distinct ordering within this space generates a unique infinitary logic.

• • Study of finitary extensions of pointed Abelian logic: In the case of ${\mathrm {pAb}}$ , one should first focus on finitary extensions, as they are plentiful. In this paper we have axiomatized several prominent logics of pointed Abelian $\ell $ -groups, but a systematic study remains an interesting topic for the future. On this matter, it is relevant to mention that the relation between Łukasiewicz logic and the Łukasiewicz unbound logic proved in Theorem 5.8 also holds between the n-valued Łukasiewicz logic and the logic of ${\boldsymbol {{Z}}}_{n-1}$ (and analogously between other axiomatic extensions of Łukasiewicz logic and properly chosen extensions of $\mathrm {Lu}$ ). Algebraically speaking, the lattice of subvarieties of Łukasiewicz logic (described by Komori in [Reference Komori23]) can be embedded into the lattice of subvarieties of $\mathrm {Lu}$ (first shown in [Reference Young30]). The submitted paper [Reference Jankovec21] describes all quasivarieties generated by linearly ordered pointed Abelian $\ell $ -groups. Nevertheless, the case of non-semilinear finitary extensions of pointed Abelian logic remains unexplored.

• • Semilinearity and completeness: Another area for further investigation is to determine whether certain extensions of Abelian logic, specifically the logic ${\mathrm {Ab}}+\mathrm {(IDC)}^\vee +\mathrm {(Arch)}^\vee $ , are semilinear, which would imply strong completeness with respect to ${\boldsymbol {{Z}}}$ . We have conjectured that this logic may be strongly complete with respect to ${\boldsymbol {{Z}}}$ , but this remains to be proven.

• • Study of other superabelian logics: In this paper we have focused on Łukasiewicz unbound logic and infinitary extensions of Abelian logic, but the framework from §2 applies to any superabelian logic, e.g., certain modal logics or the expansions of Ab with connectives corresponding to additional arithmetical operations on real numbers, such as multiplication or division.