Intuitionistic-like unsharp implication and negation defined on a poset

Result code in 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>
Result on the web
<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>

Result language
angličtina
Original language name
Intuitionistic-like unsharp implication and negation defined on a poset
Original language description
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.
Czech name
—
Czech description
—

Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GF24-14386L" target="_blank" >GF24-14386L: Representations of algebraic semantics for substructural logics</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>S - Specificky vyzkum na vysokych skolach

Publication year
2025
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Similar results(10)