arXiv subjects

Paula Menchón

Publications and source records attributed to Paula Menchón.

13 recordsLinked to original sources

arXiv2026

A Modal Expansion of Kleene Algebras via Twist Structures

Kleene triples, introduced by Jalali, provide a representation of arbitrary Kleene algebras in terms of twist-products. We introduce modal Kleene algebras and modal Kleene triples, and establish a categorical equivalence between the corresponding categories, extending Jalali's duality to the modal setting. We study the centered case and show that Kleene algebras with implication can be naturally described within this framework by viewing implication as a family of modal operators. Finally, we develop a topological duality for modal Kleene triples.

math.LO↗

arXiv2026

Twist-structures isomorphic to modal Nelson lattices

In this paper, we introduce a new variety of Heyting algebras with two unary modal operators that are not interdefinable but satisfy the weakest condition necessary to define modal operators on Nelson lattices. To achieve this, we utilize the representation of Nelson lattices as twist structures over Heyting algebras and establish a categorical equivalence. Finally, we develop a topological duality for this new variety and apply it to derive a topological duality for modal Nelson lattices.

math.LO↗

arXiv2026

On Nondefinability of Interior-Connectedness via the Contact Relation

This short paper is a small contribution to the field of Boolean contact algebras. We analyze the nondefinability of the property of interior-connectedness, and we prove certain minimality conditions for algebras and spaces that can be used in demonstrating that the aforementioned property cannot be expressed by means of contact within regular closed algebras.

math.GN↗

arXiv2026

Towards a logic of affordances

We aim to construct a formal theory of affordances seen as ternary relations. Beginning with a characterization of affordances proposed by James J. Gibson, and utilizing the tools provided by Zdzisław Pawlak's information systems and rough sets, we construct a mathematically precise definition of both crisp and rough affordances. Then, we analyze modal and approximation operators that enable reasoning about affordances in both scenarios.

math.LO↗

arXiv2026

Hemi-Nelson algebras

The aim of this paper is to generalize the link between Heyting algebras and Nelson algebras, established independently by Fidel and Vakarelov at the end of the 1970s, in the framework of bounded distributive hemi-implicative lattices. For this purpose, we introduce the variety of hemi-Nelson algebras. Moreover, we characterize the lattice of congruences of a hemi-Nelson algebra in terms of certain implicative filters. We also esta\-blish an equivalence between the algebraic category of bounded distributive hemi-implicative lattices and the one of centered hemi-Nelson algebras.

math.LO↗

arXiv2025

A mixed logic with binary operators

In previous work "Betweenness algebras" we introduced and examined the class of betweenness algebras. In the current paper we study a larger class of algebras with binary operators of possibility and sufficiency, the weak mixed algebras. Furthermore, we develop a system of logic with two binary modalities, sound and complete with respect to the class of frames closely related to the aforementioned algebras, and we prove an embedding theorem which solves an open problem from "Betweenness algebras".

math.LO↗

arXiv2025

Extended Contact Algebras: Algebraic analysis and duality theory

The ternary extended contact relation was introduced in (Ivanova, 2020) as a more expressive counterpart of the standard binary contact relation. The class of Boolean algebras expanded with the relation was named Extended Contact Algebras (ECAs). In this work, we take an algebraic perspective on ECAs, interpreting the ternary relation as a form of entailment. We introduce Pseudo-Inference Algebras, purely algebraic tructures where the ternary relation is replaced by a monotone ternary operator, capturing the logical character of extended contact. We show that the subclass of relational Pseudo-Inference Algebras corresponds precisely to ECAs and generates a subvariety of strict PSI-Algebras, which forms a discriminator variety. Furthermore, we extend Stone duality to this ternary context, introducing descriptive PSI-frames and establishing three interrelated dualities that differ in their morphisms while sharing the same class of topological objects. The framework developed in the paper provides a nified relational semantics for Boolean algebras equipped with monotone ternary operators, connecting spatial and logical notions within a categorical and topological setting.

math.LO↗

arXiv2025

Conditional algebras

Drawing on the classic paper by Chellas "Basic conditional logic" (1975), we propose a general algebraic framework for studying a binary operation of conditional that models universal features of the "if..., then..." connective as strictly related to the unary modal necessity operator. To this end, we introduce a variety of conditional algebras, and we develop its duality and canonical extensions theory.

math.LO↗

arXiv2024

Subresiduated Nelson Algebras

In this paper we generalize the well known relation between Heyting algebras and Nelson algebras in the framework of subresiduated lattices. In order to make it possible, we introduce the variety of subresiduated Nelson algebras. The main tool for its study is the construction provided by Vakarelov. Using it, we characterize the lattice of congruences of a subresiduated Nelson algebra through some of its implicative filters. We use this characterization to describe simple and subdirectly irreducible algebras, as well as principal congruences. Moreover, we prove that the variety of subresiduated Nelson algebras has equationally definable principal congruences and also the congruence extension property. Additionally, we present an equational base for the variety generated by the totally ordered subresiduated Nelson algebras. Finally, we show that there exists an equivalence between the algebraic category of subresiduated lattices and the algebraic category of centedred subresiduated Nelson algebras.

math.LO↗

arXiv2024

Rotations of Gödel algebras with modal operators

The present paper is devoted to study the effect of connected and disconnected rotations of Gödel algebras with operators grounded on directly indecomposable structures. The structures resulting from this construction we will present are nilpotent minimum (with or without negation fixpoint, depending on whether the rotation is connected or disconnected) with special modal operators defined on a directly indecomposable algebra. In this paper we will present a (quasi-)equational definition of these latter structures. Our main results show that directly indecomposable nilpotent minimum algebras (with or without negation fixpoint) with modal operators are fully characterized as connected and disconnected rotations of directly indecomposable Gödel algebras endowed with modal operators.

math.GM↗

arXiv2022

From contact relations to modal operators, and back

One of the standard axioms for Boolean Contact Algebras says that if a region x is in contact with the join of y and z, then x is in contact with at least one of the two regions. Our intention is to examine a stronger version of this axiom according to which if x is in contact with the supremum of some family S of regions, then there is a y in S that is in contact with x. We study a modal possibility operator which is definable in complete algebras in the presence of the aforementioned axiom, and we prove that the class of complete algebras satisfying the axiom is closely related to the class of modal KTB-algebras. We also demonstrate that in the class of complete extensional contact algebras the axiom is equivalent to the statement: every region is isolated. Finally, we present an interpretation of the modal operator in the class of the so-called resolution contact algebras.

math.LO↗

arXiv2021

Algebras and relational frames for Gödel modal logic and some of its extensions

Gödel modal logics can be seen as extenions of intutionistic modal logics with the prelinearity axiom. In this paper we focus on the algebraic and relational semantics for Gödel modal logics that leverages on the duality between finite Gödel algebras and finite forests, i.e. finite posets whose principal downsets are totally ordered. We consider different subvarieties of the basic variety of Gödel algebras with two modal operators (GAOs for short) and their corresponding classes of forest frames, either with one or two accessibility relations. These relational structures can be considered as prelinear versions of the usual relational semantics of intuitionistic modal logic. More precisely we consider two main extensions of finite Gödel algebras with operators: the one obtained by adding Dunn axioms, typically studied in the fragment of positive classical (and intuitionistic) logic, and the one determined by adding Fischer Servi axioms. We present Jónsson-Tarski like representation theorems for the different types of finite GAOs considered in the paper.

math.LO↗

arXiv2021

A topological duality for monotone expansions of semilattices

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

math.LO↗