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||bn for any n ? N+ is revealed.