Intuitionistic-like unsharp implication and negation defined on a poset

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61989592%3A15310%2F25%3A73633263" target="_blank" >RIV/61989592:15310/25:73633263 - isvavai.cz</a>

Výsledek na webu

<a href="https://mbpapers.math.cas.cz/full/150/4/mb150_4_2.pdf" target="_blank" >https://mbpapers.math.cas.cz/full/150/4/mb150_4_2.pdf</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.21136/MB.2024.0179-23" target="_blank" >10.21136/MB.2024.0179-23</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Intuitionistic-like unsharp implication and negation defined on a poset

Popis výsledku v původním jazyce

The aim of the present paper is to show that the concepts of the intuitionistic implication and negation formalized by means of a Heyting algebra can be generalized in such a way that these concepts are formalized by means of a bounded poset. In this case it is not assumed that the poset is relatively pseudocomplemented. The considered logical connectives negation, implication or even conjunction are not operations in this poset but so-called operators since they assign to given entries not necessarily an element of the poset as a result but a subset of mutually incomparable elements. We show that these operators for negation and implication can be characterized by several simple conditions formulated in the language of posets together with the operator of taking the lower cone. Moreover, our implication and conjunction form an adjoint pair. We call these connectives “unsharp” or “inexact” in accordance with the existing literature. We also introduce the concept of a deductive system of a bounded poset with implication and prove that it induces an equivalence relation satisfying a certain substitution property with respect to implication. Moreover, the restriction of this equivalence to the base set is uniquely determined by its kernel, i.e., the class containing the top element.

Název v anglickém jazyce

Intuitionistic-like unsharp implication and negation defined on a poset

Popis výsledku anglicky

The aim of the present paper is to show that the concepts of the intuitionistic implication and negation formalized by means of a Heyting algebra can be generalized in such a way that these concepts are formalized by means of a bounded poset. In this case it is not assumed that the poset is relatively pseudocomplemented. The considered logical connectives negation, implication or even conjunction are not operations in this poset but so-called operators since they assign to given entries not necessarily an element of the poset as a result but a subset of mutually incomparable elements. We show that these operators for negation and implication can be characterized by several simple conditions formulated in the language of posets together with the operator of taking the lower cone. Moreover, our implication and conjunction form an adjoint pair. We call these connectives “unsharp” or “inexact” in accordance with the existing literature. We also introduce the concept of a deductive system of a bounded poset with implication and prove that it induces an equivalence relation satisfying a certain substitution property with respect to implication. Moreover, the restriction of this equivalence to the base set is uniquely determined by its kernel, i.e., the class containing the top element.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF24-14386L" target="_blank" >GF24-14386L: Reprezentace algebraických sémantik pro substrukturální logiky</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Mathematica Bohemica

ISSN

0862-7959

e-ISSN

2464-7136

Svazek periodika

150

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

CZ - Česká republika

Počet stran výsledku

16

Strana od-do

497-512

Kód UT WoS článku

001674156100002

EID výsledku v databázi Scopus

2-s2.0-105021818711