Symmetric Properties of (b, c)-Inverses

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.