Let b and c be two elements in a semigroup S. This paper is devoted to studying the structures of S ||(b,c) and H (b,c) in a semigroup S, where S ||(b,c) stands for the set of all (b, c)-invertible elements and H (b,c) = {y ∈ S | bS 1 = yS 1 , S 1 y = S 1 c}. Denote the (b, c)-inverse of a ∈ S ||(b,c) by a ||(b,c). If S ||(b,c) ∅, then H (b,c) = {a ||(b,c) | a ∈ S ||(b,c) }. We first find some new equivalent conditions for H (b,c) to be a group and analyze its structure from the viewpoint of generalized inverses. Then a necessary and sufficient condition under which S ||(b,c) is a subsemigroup of S with the reverse order law holding for (b, c)-inverses is presented. At last, given a, b, c, d, x, y, z ∈ S and y ∈ S ||(b,c) , we prove that any two of the conditions x ∈ S ||(a,c) , z ∈ S ||(b,d) and zy ||(b,c) x ∈ S ||(a,d) imply the rest one.