  In this paper, we first prove that if a is both left (b, c)-invertible and
left (c, b)-invertible, then a is both (b, c)-invertible and (c, b)-invertible
in a *-monoid, which generalized the recent result about the inverse along an
element by Wang and Mosic, under the conditions (ab)* = ab and (ac)* = ac. In
addition, we consider that ba is (c, b)- invertible, and at the same time ca is
(b, c)-invertible under the same conditions, which extend the related results
about Moore-Penrose inverses by Chen et al. to (b, c)-inverses. As
applications, we obtain that under condition (a2)* = a2, a is an EP element if
and only if a is one-sided core invertible if and only if a is group
invertible.
