Number 60

2025

Publication date: 01.12.2025

Description

Financed by the Faculty of Mathematics and Computer Science of the Jagiellonian University

Licence: CC BY

Editorial team

Editor-in-Chief Marek Zaionc

Managing Editor Ewa Capinska

Issue content

Selection method for inquisitive modal logic

Stipe Marić, Tin Perkov

Reports on Mathematical Logic, Number 60, 2025, pp. 3-22

https://doi.org/10.4467/20842589RM.25.001.22715

The selection method is one of the methods to prove that various modal logics have the finite model property. For a given formula that is satisfiable in some model, we select a finite tree-like submodel, while preserving the satisfiability of the observed formula. In this paper, we adapt the selection method for the inquisitive modal logic InqML_⊞. We first define a tree-like model in the inquisitive setting and show that each satisfiable formula is satisfiable in a tree-like model. Then, using the notions of n-bisimulation and characteristic formulas, we show that InqML⊞ has the finite tree model property, i.e., each satisfiable formula is satisfiable in a finite tree-like model. Furthermore, we show analogous results for the inquisitive modal logic InqML_⇒, and as a consequence we obtain the decidability of InqML⇒.

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Divisibility of ultralters II: The big picture

Boris Šobot

Reports on Mathematical Logic, Number 60, 2025, pp. 23-46

https://doi.org/10.4467/20842589RM.25.001.22716

A divisibility relation on ultrafilters is defined as follows: F | G if and only if every set in F upward closed for divisibility also belongs to G. After describing the first ω levels of this quasiorder, in this paper we generalize the process of determining the basic divisors of an ultrafilter. First we describe these basic divisors, obtained as (equivalence classes of) powers of prime ultrafilters. Using methods of nonstandard analysis we define the pattern of an ultrafilter: the collection of its basic divisors as well as the multiplicity of each of them. All such patterns have a certain closure property in an appropriate topology. We isolate the family of sets belonging to every ultrafilter with a given pattern. We show that every pattern with the closure property is realized by an ultrafilter. Finally, we apply patterns to obtain an equivalent condition for an ultrafilter to be self-divisible.

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Positive regions of computable binary relations

Huishan Wu

Reports on Mathematical Logic, Number 60, 2025, pp. 47-65

https://doi.org/10.4467/20842589RM.25.001.22717

Positive region plays a fundamental role in rough set-based attribute reduction. We study positive regions of decision systems and of binary relations in rough set theory within the framework of reverse mathematics and computability theory. First, we propose the notion of infinite decision systems and prove that the existence of positive regions of decision systems is equivalent to arithmetic comprehension over the weak base theory RCA₀. We also show that the complexity of positive regions of computable decision systems lies exactly in π⁰₂of the arithmetic hierarchy. Next, we study positive regions of equivalence relations and binary relations. We show that the existence of each of the two positive regions is equivalent to arithmetic comprehension over RCA₀; however, the exact complexity of positive regions of computable equivalence relations lies in π⁰₁ and the exact complexity of positive regions of computable binary relations lies in ∑⁰₂ of the arithmetic hierarchy.

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Słowa kluczowe: Inquisitive modal logic, selection method, tree model property, divisibility, Stone-Čech compactification, ultrafilter, mathematics, computability theory, rough set theory, positive region, decision system

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