The extension of L-algebras and states

In this paper, we introduce EL-algebra, a new algebraic structure extended from L-algebras, where the top element is not assumed and each EL-algebra contains an L-algebra. First, we study the dual atom and the branch on EL-algebras, from which we find that the implication of any two elements in an E...

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Bibliographic Details
Published inFuzzy sets and systems Vol. 455; pp. 35 - 52
Main Authors Mao, Lingling, Xin, Xiaolong, Zhang, Shunli
Format Journal Article
LanguageEnglish
Published Elsevier B.V 15.03.2023
Subjects
Atom
Bosbach state
EL-algebra
L-algebra
Riečan state
Bosbach state
EL-algebra
Riečan state
Atom
L-algebra
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Summary:In this paper, we introduce EL-algebra, a new algebraic structure extended from L-algebras, where the top element is not assumed and each EL-algebra contains an L-algebra. First, we study the dual atom and the branch on EL-algebras, from which we find that the implication of any two elements in an EL-algebra is always in branch V(1), which is an L-algebra. Based on this result, some relationships among EL-algebras, L-algebras, BCK-algebras, and l-groups are presented. Moreover, we introduce and investigate prime EL-algebras, ideals and congruence relations on EL-algebras. Then, the notions of Bosbach state and state-morphism on EL-algebras are presented, and some of their properties are investigated. Finally, using two different kinds of orthogonal relations on EL-algebras, ⊤ and ⊥, we define two classes of Riečan states on EL-algebras. In addition, some relationships between the Bosbach state and these two Riečan states are studied.
ISSN:0165-0114
1872-6801
DOI:10.1016/j.fss.2022.10.017