On join-complete implication algebras

In this paper, first, we consider an algebra that has a binary operation and a join of arbitrary nonempty subset. A lattice implication algebra is a lattice with a binary operation, which has a join and a meet of finite nonempty subsets. In this work, the notion of join-complete implication algebras L is defined as a join-complete lattice with a binary operation, and some properties of this algebra L are searched. Moreover, we prove that the interval [ a , 1] in L is a lattice implication algebra and show that L satisfies the completely distributive law when it has the smallest element 0. Finally, we state the concept of filter and multipliers of L and provide finite and infinite examples of them. In addition, we research some properties of these concepts in detail.