On the set of (b,c)-invertible elements

Abstract

Let b and c be two elements in a semigroup S. This paper is devoted to studying the structures of S||(b,c) and H(b,c) in a semigroup S, where S||(b,c) stands for the set of all (b, c)-invertible elements and H(b,c) = {y ? S | bS1 = yS1, S1y = S1c}. Denote the (b, c)-inverse of a ? S||(b,c) by a||(b,c). If S||(b,c) , ?, then H(b,c) = {a||(b,c)| a ? S||(b,c)}. We first find some new equivalent conditions for H(b,c) to be a group and analyze its structure from the viewpoint of generalized inverses. Then a necessary and sufficient condition under which S||(b,c) is a subsemigroup of S with the reverse order law holding for (b, c)-inverses is presented. At last, given a, b, c, d, x, y, z ? S and y ? S||(b,c), we prove that any two of the conditions x ? S||(a,c), z ? S||(b,d) and zy||(b,c)x ? S||(a,d) imply the rest one.