The commuting inverse along an element in semigroups
Long Wang, Dijana Mosić, Hua Yao · Communications in Algebra · 2021
In this article, we consider commuting properties of the generalized inverse along one element and two elements. The conditions for the existence of the solution to the equation axd=d=dxa under the Green’s relation x≤Hd are given. It is proven that the solution is the inverse of a along d, and that it commutes with a in this case. Moreover, the concept of complete (b, c)-inverses, a special class of (b, c)-inverses, is proposed. Some characterizations of complete (b, c)-inverses are given. As applications, complete (b, c)-inverses are borrowed to characterize certain types of semigroups.