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E-varieties and E-pseudovarieties of regular semigroups

In this thesis, analogues of certain universal algebra results are established for classes of regular semigroups. In [2], Hall introduced a concept of variety for regular semigroups, which he defined as follows. An e-variety is a class of regular semigroups closed under the formation of homomorphic images, regular subsemigroups and direct products. An analogue of the notion of free object (termed bifree object) was developed for classes of orthodox semigroups by Kadourek and Szendrei [3], and generalized to classes of regular semigroups by Yeh [6]. In Chapter 3, we establish that analogues of several universal algebra theorems concerning free objects in varieties carry over to bifree objects in e-varieties. For example, we show that bifree objects (when they exist) in the e-variety generated by a single regular semigroup, 5 , can be embedded in powers of 5 . We prove Birkoff-type theorems for classes of E-solid (alternatively, locally inverse) regular semigroups, and we explore the links between bifree objects, fully invariant congruences and lattices of e-varieties. An analogue of variety (termed pseudovariety) appropriate for finite algebras was introduced by Eilenberg and Schiitzenberger [l]. In Chapter 4, we introduce a notion of pseudovariety (termed e-pseudovariety) appropriate for regular semigroups. An e-pseudovariety is defined to be a class of finite regular semigroups closed under the formation of homomorphic images, regular subsemigroups and finite direct products. After verifying that not every e-pseudovariety consists of the finite members of an e-variety, or of the regular members of a semigroup pseudovariety, we consider questions concerning identities and e-pseudovarieties. For example, we show that each e-pseudovariety is ultimately denned by a sequence (and not necessarily denned by a set) of regular unary semigroup identities. However, it is also shown that sequences of such identities do not necessarily define e-pseudovarieties. We proceed to show that a class of E-solid finite regular semigroups is an e-pseudovariety if and only if it is ultimately defined by a sequence of semigroup identities. (A similar result is given for locally inverse finite regular