Dually normal relations on sets

Abstract

In this paper, the concept of dual normal relations on sets is introduced and generalized. Intrinsic characterizations of them are obtained.

Finitely bi-quasieegular relations

“…For that we need the concept of finite extension of a relation. That notion and belonging notation we borrow from articles [1], [2] and [7]. For any set X, let X (<ω) = {F ⊆ X : F is finite and nonempty}.…”

“…In this article, since regular and finitely regular relations had important applications in lattice theory, following the concepts of finitely conjugative relations ( [1], Jiang Guanghao and Xu Luoshan), finitely dual normal relations ( [2], Jiang Guanghao and Xu Luoshan) and finitely quasi-conjugative relations ( [5], D. A. Romano and M. Vinčić), we introduce and analyze the notions of finitely bi-quasiregular relations after citing some previous results of the second author on bi-quasiregular relations ( [6]). …”

“…(See, for example, articles [4], [5] and [6].) Notions and notations which are not explicitly exposed but are used in this article, can be found in book [3] and articles [1], [2], [4], [5], [6] and [7].…”

Weak Finitely Dual Quasi-Conjugative Relations

“…The fundamental works of K. A. Zareckii [9], B. M. Schein [7] and others on regular relations motivated several mathematicians to investigate similar classes of relations, obtained by putting α −1 , α c or (α c ) −1 in place of one or both α's on the right side of the regularity equation α = α • β • α (where β is some relation). The following class of elements in the semigroup B(X) have been investigated: normal relation in [1] by G. Jiang, L. Xu, J. Cai and G. Han;, dually normal relation in [2] by G. Jiang and L. Xu; quasi-conjugative and bi-normal relations [4,5,6,8] by this author and M. Vinčić. For example:…”