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Rafał Gruszczyński

Publications and source records attributed to Rafał Gruszczyński.

10 recordsLinked to original sources
arXiv2026

Contact relations on bounded distributive lattices and related structures

We study a discrete representation of contact relations on bounded distributive lattices and some of their reducts. In the course of the paper, we apply ideas and techniques developed by Ivo Düntsch, Dimiter Vakarelov, and Michael Winter, and we show---among others---that contact relations on a distributive join semi-lattice $L$ with the bottom element correspond to reflexive and symmetric relations which are closed in the product of the Stone space of $L$.

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↗
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

Hybrid logic for strict betweenness

The paper is devoted to modal properties of the ternary strict betweenness relation as used in the development of various systems of geometry. We show that such a relation is non-definable in a basic similarity type with a binary operator of possibility, and we put forward two systems of hybrid logic, one of them complete with respect to the class of dense linear betweenness frames without endpoints, and the other with respect to its subclass composed of Dedekind complete frames.

math.LO↗
arXiv2023

Mathematical methods in region-based theories of space: the case of Whitehead points

Regions-based theories of space aim -- among others -- to define points in a geometrically appealing way. The most famous definition of this kind is probably due to Whitehead. However, to conclude that the objects defined are points indeed, one should show that they are points of a geometrical or a topological space constructed in a specific way. This paper intends to show how the development of mathematical tools allows showing that Whitehead's method of extensive abstraction provides a construction of objects that are fundamental building blocks of specific topological spaces.

math.LO↗
arXiv2023

Grzegorczyk and Whitehead points: the story continues

One of the main goals of region-based theories of space is to formulate a geometrically appealing definition of points. The paper is devoted to the analysis of two such seminal definitions: Alfred N. Whitehead's (1929) and Andrzej Grzegorczyk's (1960). Relying on the work of Loredana Biacino's and Ginagiacomo Gerla's (1996), we improve their results, solve some open problems concerning the mutual relationship between Whitehead and Grzegorczyk points, and put forward open problems for future investigation.

math.LO↗
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↗