Several co-associative laws and pre-B-algebras

Abstract

In this paper, we introduce several co-associative laws and the notion of a pre-B-algebra. We show that every B-algebra is both a pre-B-algebra and a perpendicular to-algebra. We apply the notions of a post groupoid and a pre-semigroup of a groupoid to the set (N, +) of all nonnegative integers, and we prove that the groupoid (N, +) cannot be a post groupoid of a B-algebra or an edge d-algebra.