| Summary: | In [1] the authors introduced the notion of an associate inverse subsemigroup of a regular semigroup, extending the concept of an associate subgroup of a regular semigroup, ¯rst presented in [3]. The main result of the present paper, Theorem 2.15, establishes that a regular semigroup S with an associate inverse subsemigroup S* satisfies three simple identities if and only if it is isomorphic to a generalised Rees matrix semigroup M(T; A;B; P), where T is a Clifford semigroup, A and B are bands, with common associate inverse subsemigroup E(T) satisfying the referred identities, and P is a sandwich matrix satisfying some natural conditions. If T is a group and A, B are left and right zero semigroups, respectively, then the structure described provides a usual Rees matrix semigroup with normalised sandwich matrix, thus generalising the Rees matrix representation for completely simple semigroups.
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