Regular semigroups: amalgamation and the lattice of existence varieties

Hall, T. E.

doi:10.1007/bf01190413

Cited by 27 publications

(15 citation statements)

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“…This also will lead to solutions of the word problem for the bifree objects in the respective e-varieties. The next two results are essentially contained in the papers of Hall [7,8]. We now consider the bifree objects in orthodox e-varieties of strict regular semigroups.…”

Section: A Regular *-Semigroup Is Orthodox If and Only If It Satisfiementioning

confidence: 99%

“…The semigroup A®B = A xB/I is completely (O-)simple. A Rees matrix representation of A®B by means of the respective representations of A and B can be obtained easily (see [7]). Let S = M°(I, G; P) be a completely 0-simple orthodox *-semigroup.…”

Section: Varieties Of Strict Orthodox ^-Semigroupsmentioning

confidence: 99%

“…By Theorem 3.7 each combinatorial strict orthodox *-semigroup satisfies (7) and (6'). On the other hand, (7) implies that the Clifford semigroups eSe* are inverse semigroups satisfying the law x=x 2 . Consequently, as above, each local submonoid eSe is a semilattice and thus 5 is a combinatorial strict regular semigroup.…”

Section: Varieties Of Strict Orthodox ^-Semigroupsmentioning

confidence: 99%

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Bifree objects in e-varieties of strict orthodox semigroups and the lattice of strict orthodox *-semigroup varieties

Auinger

1993

Glasgow Math. J.

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For regular semigroups, the appropriate analogue of the concept of a variety seems to be that of an e(xistence)-variety, developed by Hall [6,7,8]. A class V of regular semigroups is an e-variety if it is closed under taking direct products, regular subsemigroups and homomorphic images. For orthodox semigroups, this concept has been introduced under the term “bivariety” by Kaďourek and Szendrei [12]. Hall showed that the collection of all e-varieties of regular semigroups forms a complete lattice under inclusion. Further, he proved a Birkhoff-type theorem: each e-variety is determined by a set of identities. For e-varieties of orthodox semigroups a similar result has been proved by Kaďourek and Szendrei. At variance with the case of varieties, prima facie the free objects in general do not exist for e-varieties. For instance, there is no free regular or free orthodox semigroup. This seems to be true for most of the naturally appearing e-varieties (except for cases of e-varieties which coincide with varieties of unary semigroups such as the classes of all inverse and completely regular semigroups, respectively). This is true if the underlying concept of free objects is denned as usual. Kaďourek and Szendrei adopted the definition of a free object according to e-varieties of orthodox semigroups by taking into account generalized inverses in an appropriate way. They called such semigroups bifree objects. These semigroups satisfy the properties one intuitively expects from the “most general members” of a given class of semigroups. In particular, each semigroup in the given class is a homomorphic image of a bifree object, provided the bifree objects exist on sets of any cardinality. Concerning existence, Kaďourek and Szendrei were able to prove that in any class of orthodox semigroups which is closed under taking direct products and regular subsemigroups, all bifree objects exist and are unique up to isomorphism. Further, similar to the case of varieties, there is an order inverting bijection between the fully invariant congruences on the bifree orthodox semigroup on an infinite set and the e-varieties of orthodox semigroups. Recently, Y. T. Yeh [22] has shown that suitable analogues to free objects exist in an e-variety V of regular semigroups if and only if all members of V are either E-solid or locally inverse.

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Section: A Regular *-Semigroup Is Orthodox If and Only If It Satisfiementioning

confidence: 99%

Section: Varieties Of Strict Orthodox ^-Semigroupsmentioning

confidence: 99%

Section: Varieties Of Strict Orthodox ^-Semigroupsmentioning

confidence: 99%

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Bifree objects in e-varieties of strict orthodox semigroups and the lattice of strict orthodox *-semigroup varieties

Auinger

1993

Glasgow Math. J.

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“…In these papers the structure of such semigroups and special cases thereof has been fully described. Some recent results for this class of semigroups can be found in the papers of Hall [8,9]. DEFINITION.…”

Section: Introductionmentioning

confidence: 99%

Free Objects in Joins of Strict Inverse and Completely Simple Semigroup Varieties

Auinger¹

1992

Journal of the London Mathematical Society

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“…The equations follow from [9]. The strict inclusions are immediate from Theorem 4.5, which gives I * RZ = LI = I * RB.…”

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confidence: 96%

Untitled

1997

Trans. Amer. Math. Soc.

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Abstract. Within the usual semidirect product S * T of regular semigroups S and T lies the set Reg (S * T ) of its regular elements. Whenever S or T is completely simple, Reg (S * T ) is a (regular) subsemigroup. It is this 'product' that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes, U and V, the e-variety U * V generated by {Reg (S * T ) : S ∈ U, T ∈ V} is well defined if and only if either U or V is contained within the e-variety CS of completely simple semigroups. General properties of this product, together with decompositions of many important e-varieties, are obtained. For instance, as special cases of general results the e-variety LI of locally inverse semigroups is decomposed as I * RZ, where I is the variety of inverse semigroups and RZ is that of right zero semigroups; and the e-variety ES of E-solid semigroups is decomposed as CR * G, where CR is the variety of completely regular semigroups and G is the variety of groups.In the second half of the paper, a general construction is given for the e-free semigroups (the analogues of free semigroups in this context) in a wide class of semidirect products U * V of the above type, as a semidirect product of e-free semigroups from U and V, "cut down to regular generators". Included as special cases are the e-free semigroups in almost all the known important e-varieties, together with a host of new instances. For example, the e-free locally inverse semigroups, E-solid semigroups, orthodox semigroups and inverse semigroups are included, as are the e-free semigroups in such sub-e-varieties as strict regular semigroups, E-solid semigroups for which the subgroups of its self-conjugate core lie in some given group variety, and certain important varieties of completely regular semigroups. Graphical techniques play an important role, both in obtaining decompositions and in refining the descriptions of the e-free semigroups in some e-varieties.Similar techniques are also applied to describe the e-free semigroups in a different 'semidirect' product of e-varieties, recently introduced by Auinger and Polák. The two products are then compared.The semidirect product has a venerable history in semigroup theory. In conjunction with the wreath product, it has played a central role in the decomposition theory of finite semigroups. In the context of regular semigroups, variations on the usual product have been introduced for inverse semigroups and, recently, for locally inverse semigroups. We take a different approach, by studying the set Reg (S * T ) of regular elements of the usual semidirect product of regular semigroups S and T . In many important cases, these elements form a (regular) subsemigroup of S * T . These cases are most easily interpreted in the framework of e-varieties.

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Regular semigroups: amalgamation and the lattice of existence varieties

Cited by 27 publications

References 25 publications

Bifree objects in e-varieties of strict orthodox semigroups and the lattice of strict orthodox *-semigroup varieties

Bifree objects in e-varieties of strict orthodox semigroups and the lattice of strict orthodox *-semigroup varieties

Free Objects in Joins of Strict Inverse and Completely Simple Semigroup Varieties

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