Symmetric Properties of (b,c)-Inverses

Abstract

Let b and c be two elements in a semigroup S. The (b,c)-inverse is an important outer inverse because it unifies many common generalized inverses. This paper is devoted to presenting some symmetric properties of (b,c)-inverses and (c,b)-inverses. We first find that S contains a (b,c)-invertible element if and only if it contains a (c,b)-invertible element. Then, for four given elements a,b,c,d in S, we prove that a is (b,c)-invertible and d is (c,b)-invertible if and only if abd is invertible along c and dca is invertible along b. Inspired by this result, the (b,c)-invertibility is characterized by one-sided invertible elements. Furthermore, we show that a is inner (b,c)-invertible and d is inner (c,b)-invertible if and only if c is inner (a,d)-invertible and b is inner (d,a)-invertible.