Homomorphisms and congruences on regular semigroups with associate inverse subsemigroups
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HOMOMORPHISMS AND CONGRUENCES ON REGULAR SEMIGROUPS WITH ASSOCIATE INVERSE SUBSEMIGROUPS B. Billhardt1 , E. Giraldes2 , P. Marques-Smith3 and P. Mendes Martins4 1
Universit¨ at Kassel, FB 10 Mathematik und Naturwissenschaften D-34127 Kassel, Germany E-mail: [email protected] 2
U.T.A.D., CM-UTAD 5000 Vila Real, Portugal E-mail: [email protected] 3
4
Universidade do Minho, Centro de Matem´ atica 4710-057 Braga, Portugal E-mail: [email protected]
Universidade do Minho, Centro de Matem´ atica 4710-057 Braga, Portugal E-mail: [email protected]
(Received May 3, 2011; Accepted November 11, 2011) [Communicated by M´ aria B. Szendrei]
Abstract An associate inverse subsemigroup of a regular semigroup S is a subsemigroup T of S containing a least associate x∗ of each x ∈ S, in relation to the natural partial order ≤. In [1] the authors describe the structure of regular semigroups with an associate inverse subsemigroup, satisfying two natural conditions. In this paper we describe all ∗-homomorphisms and all ∗-congruences on such semigroups.
1. Introduction In a semigroup S, an element t ∈ S is an associate of s ∈ S if s = sts. Extending the concept of associate subgroup of a semigroup, first presented in [4], the authors introduced in [1] the notion of associate inverse subsemigroup of a regular semigroup S. This is a subsemigroup S ∗ of S containing a least associate x∗ of each Mathematics subject classification number : 20M10. Key words and phrases: regular semigroup, associate inverse subsemigroup, homomorphism. 0031-5303/2013/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
B. BILLHARDT, E. GIRALDES, P. MARQUES-SMITH and P. MENDES MARTINS
x ∈ S, in relation to the natural partial order ≤. Such a semigroup S ∗ is necessarily inverse. The motivation of this study lies in a very simple characterisation of inverse semigroups in terms of the natural partial order. Semigroups with associate inverse subsemigroups occur naturally in the context of inverse transversals (that is, inverse subsemigroups of a semigroup S that contain exactly a single inverse of each s ∈ S). Inverse transversals have been extensively studied by many mathematicians. A review of the theory can be found in [3]. In many important classes of semigroups, these two concepts are not even related (see Section 3 of [1]). This reality provides further motivation for the study of this class of regular semigroups. Semigroups with associate inverse subsemigroups also include other interesting subclasses, such as completely simple semigroups, the structure of which has already been described. The main result of [1] is a description of a class of regular semigroups with associate inverse subsemigroup satisfying two natural conditions. The open problem of knowing whether this class of semigroups is a variety of unary semigroups has been solved by the authors and its solution is presented in [2]. The knowledge of homomorphisms and congruences of semigroups belonging to a particular class C is often essential for
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