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

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