A class of unary semigroups admitting a Rees matrix representation |
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Authors: | B Billhardt E Giraldes P Marques-Smith P Mendes Martins |
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Institution: | 1. FB 10 Mathematik und Naturwissenschaften, Universit?t Kassel, 34127, Kassel, Germany 2. CM-UTAD, U.T.A.D., 5000, Vila Real, Portugal 3. Centro de Matemática, Universidade do Minho, 4710-057, Braga, Portugal
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Abstract: | In Billhardt et al. (Semigroup Forum 79:101–118, 2009) 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, first presented in Blyth et al. (Glasgow Math. J. 36:163–171, 1994). 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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