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An Exploration of Ideals and Filters In Triangle Algebras
Version 1
: Received: 30 July 2024 / Approved: 30 July 2024 / Online: 15 August 2024 (14:09:46 CEST)
How to cite: Noumen, E.; Yinga, F. T.; KOGUEP, B. B. N.; Cornelis, C. An Exploration of Ideals and Filters In Triangle Algebras. Preprints 2024, 2024081173. https://doi.org/10.20944/preprints202408.1173.v1 Noumen, E.; Yinga, F. T.; KOGUEP, B. B. N.; Cornelis, C. An Exploration of Ideals and Filters In Triangle Algebras. Preprints 2024, 2024081173. https://doi.org/10.20944/preprints202408.1173.v1
Abstract
In the study of algebraic structures related to logical systems, Ideals and filters have different
meanings and they are algebraic notions related to logical provable formulas. Unlike the classical
Boolean lattice theory, ideals and filters are not dual notions in residuated lattices. An interesting
subclass of residuated lattices is the class of triangle algebras, which is an equational representation
of interval-valued residuated lattices that provides an algebraic framework for using closed intervals
as truth values in fuzzy logic. The main aim of this article is to introduce and study the concept of
ideals in triangle algebras and investigate the connection between ideals and filters. We first point out
that the construction procedure for the filter generated by a subset of a triangle algebra established
by Zahiri et al. is incorrect, and we proceed to give an alternative characterization.
Keywords
filters; ideals; congruences; residuated lattices; triangle algebras; interval-valued fuzzy 10 logic.
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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