Preideals in EQ-algebras

EQ -algebras were introduced by Novak ( 2006 ) as an algebraic structure of truth values for fuzzy-type theory (FFT). Novák and De Baets ( 2009 ) introduced various kinds of EQ -algebras such as good, residuated, and IEQ -algebras. In this paper, we define the notion of (pre)ideal in bounded EQ -alg...

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Bibliographic Details
Published inSoft computing (Berlin, Germany) Vol. 25; no. 20; pp. 12703 - 12715
Main Authors Akhlaghinia, N., Borzooei, R. A., Kologani, M. Aaly
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2021
Subjects
Artificial Intelligence
Computational Intelligence
Control
Engineering
Foundations
Mathematical Logic and Foundations
Mechatronics
Robotics
Generated preideal
(Pre)ideal
Distributive lattice
Heyting algebra
Bounded
Complete lattice
algebra
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Summary:EQ -algebras were introduced by Novak ( 2006 ) as an algebraic structure of truth values for fuzzy-type theory (FFT). Novák and De Baets ( 2009 ) introduced various kinds of EQ -algebras such as good, residuated, and IEQ -algebras. In this paper, we define the notion of (pre)ideal in bounded EQ -algebras ( BEQ -algebras) and investigate some properties. Then, we introduce a congruence relation on good BEQ -algebras by using ideals, and then, we solve an open problem in Paad ( 2019 ). Moreover, we show that in IEQ -algebras, there is a one-to-one correspondence between congruence relations and the set of ideals. In the following, we characterize the generated preideal in BEQ -algebras, and by using this, we prove that the family of all preideals of a BEQ -algebra is a complete lattice. Then, we show that the family of all preideals of a prelinear IEQ -algebras is a distributive lattice and becomes a Heyting algebra. Finally, we show that we can construct an MV -algebra from the family of all preideals of a prelinear IEQ -algebra.
ISSN:1432-7643
1433-7479
DOI:10.1007/s00500-021-06071-y