On Elements whose (b, c)-Inverse is Idempotent in a Monoid

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 is an element of 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 is an element of N+ is revealed.