Some types of ideals in residuated lattices

It is clear to characterize logic algebras, ideals and filters play important role, therefore in this research, we survey the structure of various ideals and find some types of ideals such as positive implicative ideals, strong ideals and MV-ideals in residuated lattices. We also introduce the conce...

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Bibliographic Details
Published inJournal of intelligent & fuzzy systems Vol. 35; no. 5; pp. 5647 - 5655
Main Author Shoar, S. Khosravi
Format Journal Article
LanguageEnglish
Published Amsterdam IOS Press BV 01.01.2018
Subjects
Algebra
Boolean algebra
Lattices (mathematics)
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Summary:It is clear to characterize logic algebras, ideals and filters play important role, therefore in this research, we survey the structure of various ideals and find some types of ideals such as positive implicative ideals, strong ideals and MV-ideals in residuated lattices. We also introduce the concept of quasi ideals and show that any ideal is a quasi ideal, but the converse does not hold in general. We clarify the relations between these ideals in residuated lattices, strong residuated lattices and BL-algebras. For instance, we prove that in strong residuated lattices the concept of quasi ideals and ideals are the same and in BL-algebras the concepts of quasi ideals, ideals, strong ideals and MV-ideals coincide, whereas they are different in residuated lattices and MTL-algebras. We detect the relation between these ideals with MV-algebras, strong algebras and Boolean algebras.
ISSN:1064-1246
1875-8967
DOI:10.3233/JIFS-171197