The extension of L-algebras and states

Abstract

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.