arXiv subjects

Filip Jankovec

Publications and source records attributed to Filip Jankovec.

5 recordsLinked to original sources

arXiv2026

The pairwise Stone space of an S4 De Morgan algebra

The purpose of this study is to investigate the bitopological duality theory of De Morgan algebras equipped with a closure operator, known as S4 De Morgan algebras. We first introduce certain expansions of pairwise Stone spaces, which we call pairwise S4 De Morgan Stone spaces (henceforth, PS4D-spaces). These consist of a pairwise Stone space $X$ equipped with a twist continuous involution $g\colon X\to X$, as well as a binary relation $R\subseteq X\times X$ that is reflexive and transitive. We first demonstrate that the bitopological spectrum $S_0(A)$ of prime filters of an S4 De Morgan algebra $A$ gives rise to a PS4D-space. A topological representation is then obtained by exhibiting an isomorphism from $A$ to the S4 De Morgan algebra $A_0(S_0(A))$ of $(\tau_1,\delta_2)$-biclopen subsets of $S_0(A)$ whose operation of De Morgan involution is defined through $g$ and whose closure operator is defined through $R$. We then provide an algebraic realization theorem by showing that every PS4D-space $X$ is bihomeomorphic and relationally isomorphic to the bitopological spectrum $S_0(A_0(X))$ of prime filters of $A_0(X)$. With the introduction of suitable bicontinuous frame morphisms, we show that the category $\mathbf{S4D}$ of S4 De Morgan algebras is dually equivalent to the category $\mathbf{PStone_{S4D}}$ of PS4D-spaces. As an application, we provide bitopological characterizations of filters and ideals in general De Morgan algebras under our established duality as well as bitopological soundness and completeness results for an S4-type modal extension of the calculus FDE of first-degree entailment.

math.LO↗

arXiv2026

Pointed Modal Abelian Logic, Algebraically

In this article, we investigate the pointed modal logic of reals. We first establish its relational (Kripke) semantics with bounded valuations in the Abelian l-group of real numbers with the distinguished negative constant -1. To study this logic algebraically, we introduce the variety of negatively pointed modal Abelian l-groups, in particular we focus on the strongly pointed members thereof. Constructing complex algebras and canonical frames, we establish a Truth Lemma connecting the relational and algebraic frameworks. Since finitary axiomatizations cannot fully capture the Kripke validities of the reals we introduce further algebraic constraints, in particular including an infinitary Archimedean-style rule. Finally, we prove a corresponding `infinitary algebraic completeness' result for pointed modal Abelian logic with respect to the variety of pointed modal Abelian l-groups.

cs.LO↗

arXiv2025

Satisfiability in {\L}ukasiewicz logic and its unbounded relative

Unbounded {\L}ukasiewicz logic is a substructural logic that combines features of infinite-valued {\L}ukasiewicz logic with those of abelian logic. The logic is finitely strongly complete w.r.t.~the additive $\ell$-group on the reals expanded with a distinguished element $-1$. We show that the existential theory of this structure is NP-complete. This provides a complexity upper bound for the set of theorems and the finite consequence relation of unbounded {\L}ukasiewicz logic. The result is obtained by reducing the problem to the existential theory of the MV-algebra on the reals, the standard semantics of {\L}ukasiewicz logic. This provides a new connection between both logics. The result entails a translation of the existential theory of the standard MV-algebra into itself.

math.LO↗

arXiv2025

Subvarieties of pointed Abelian l-groups

This paper provides a complete classification of all subvarieties of pointed Abelian lattice-ordered groups (l-groups), as well as all subquasivarieties that are generated by their totally ordered members. We present two complementary approaches to achieve this classification. First, using purely l-group-theoretic methods, we analyze the structure of lexicographic products and values to identify all join-irreducible members of the lattice of subvarieties of positively pointed Abelian l-groups. We provide a novel equational basis for each of these subvarieties, leading to a complete description of the entire subvariety lattice. As a direct application, our l-group-theoretic classification yields an alternative, self-contained proof of Komori's classification of subvarieties of MV-algebras. Second, we explore the connection to MV-algebras via an extended version of Mundici's functor. We prove that this functor preserves universal classes, a result of independent model-theoretic interest. This allows us to lift the classification of universal classes of totally ordered MV-algebras, due to Gispert, to a complete classification of universal classes of totally ordered pointed Abelian l-groups. As a direct consequence, we obtain a complete structural description of the lattice of subquasivarieties that are generated by their totally ordered members.

math.LO↗

arXiv2024

Superabelian logics

This paper presents a unified algebraic study of a family of logics related to Abelian logic (Ab), the logic of Abelian lattice-ordered groups. We treat Ab as the base system and refer to its expansions as superabelian logics. The paper focuses on two main families of expansions. First, we investigate the rich landscape of infinitary extensions of Ab, providing an axiomatization for the infinitary logic of real numbers and showing that there exist $2^{2^\omega}$ distinct logics in this family. Second, we introduce pointed Abelian logic (pAb), the logic of pointed Abelian lattice-ordered groups, by adding a new constant to the language. This framework includes {\L}ukasiewicz unbound logic. We provide axiomatizations for its finitary and infinitary versions as extensions of pAb and establish their precise relationship with standard {\L}ukasiewicz logic via a formal translation. Finally, the methods developed for this analysis are generalized to axiomatize the logics of other prominent pointed groups.

math.LO↗