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 monoid and a, b, c ? S. Firstly, give several characterizations for idempotency of a||(b,c) as follows: exists if only cab = cb, cS cbS, Sb Scb both 1||(b,c) exist 1||(b,c), which establish relationship between 1||(b,c). They imply that merely depends on but independent when idempotent. Particularly, b c, more ensure a||b by inner outer inverses are given. Finally, a||bn any n N+ revealed.