On elements whose (b,c)-inverse is idempotent in a monoid

Abstract

In this paper, we investigate the elements whose (b, c)-inverse is idempotent in a monoid. Let S be a monoid and a, b, c ∈ S. Firstly, we give several characterizations for the idempotency of a^||(b, c) as follows: a^||(b, c) exists and is idempotent if and only if cab=cb, cS=cbS, Sb=Scb if and only if both a^||(b, c) and 1^||(b, c) exist and a^||(b, c)=1^||(b, c), which establish the relationship between a^||(b, c) and 1^||(b, c). They imply that a^||(b, c) merely depends on b, c but is independent of a when a^||(b, c) exists and is idempotent. Particularly, when b=c, more characterizations which ensure the idempotency of a^||b by inner and outer inverses are given. Finally, the relationship between a^||b and a^||b^n for any n ∈ N^+ is revealed.