The bifree regularE-solid semigroups

Szendrei, Mária B.

doi:10.1007/bf02574082

Cited by 7 publications

(11 citation statements)

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“…In the case K = G, these also coincide with C ∞ U, by Corollary 1.9. In particular, we obtain the decompositions C ∞ CR(H) = CR(H) * G, for any group variety H. Most important, perhaps, is the decomposition ES = CR * G. This particular equation was obtained by Szendrei [35] as a consequence of her description of the e-free E-solid semigroups; in the sequel we shall, to the contrary, derive such a description and, more generally, a description of the e-free semigroups in any semidirect product U * K, as above, from these equations. The key to obtaining these equations is to apply Result 2.4 on the e-locality of CR(H), in conjunction with a 'regular' version of the 'Derived Semigroupoid Theorem' of Tilson [37, Theorem B.1].…”

Section: Decompositions Involving Groupsmentioning

confidence: 81%

“…Some well known sub-e-varieties of LI have similar decompositions. In §5, we prove that ES also has a simple decomposition, as CR * G (first proved by M. Szendrei [35]), where CR is the (e-) variety of completely regular semigroups and G is the (e-) variety of groups. This decomposition is a very special case of the equality of the semidirect product U * K with the Mal'cev product UmK for certain ('e-local') E-solid e-varieties U and for any group variety K; and of the description of U * G, for U completely regular, as the class C ∞ U of all E-solid semigroups whose self-conjugate cores belong to U (ES being C ∞ CR).…”

mentioning

confidence: 89%

“…Similarly, we may regard S as being generated as a regular semigroup by T × X = (T × X) ∪ (T × X) . However, a nice technical device (introduced by Szendrei [35] and transparent, in terms of the Cayley graph of T ) is to identify this set with the set T ×X, by identifying (t, x) with (tx, x ), for all t ∈ T, x ∈ X. Thus we shall regard T ×X as a subset of S, with (tx, x ) ∈ V ((t, x)) for all t ∈ T, x ∈ X.…”

Section: E-free E-solid Semigroupsmentioning

confidence: 99%

“…Since the e-free completely regular semigroups are well described [14], we obtain a relatively simple solution to the 'word problem' in EF ES X . In fact, this semigroup has recently been described by Szendrei [35]; however, our technique also determines the e-free semigroup on X in any e-variety C ∞ CR(H), where CR(H) is the e-variety comprising all completely regular semigroups, the subgroups of which belong to the group variety H; and also determines the e-free semigroups in certain e-varieties of orthodox semigroups.…”

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confidence: 99%

“…A graphical description of EF ES X is given in §7.3 that is similar to that of Szendrei [35] (and ultimately derives from work of Margolis and Pin [22] on inverse semigroups). It has the decided advantage, over the purely semigroup-theoretic description, of providing a set of canonical forms for the elements of the e-free semigroup (rather than just solving the 'word problem').…”

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confidence: 99%

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Semidirect products of regular semigroups

Jones

Trotter

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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Section: Decompositions Involving Groupsmentioning

confidence: 81%

mentioning

confidence: 89%

Section: E-free E-solid Semigroupsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Semidirect products of regular semigroups

Jones

Trotter

1997

Trans. Amer. Math. Soc.

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E-solid e-varieties are not finitely based

Auinger

Szendrei²

1998

Israel J. Math.

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Inductive groupoids and cross-connections of regular semigroups

Muhammed

Volkov

2018

Acta Math. Hungar.

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There are two major structure theorems for an arbitrary regular semigroup using categories, both due to Nambooripad. The first construction using inductive groupoids departs from the biordered set structure of a given regular semigroup. This approach belongs to the realm of the celebrated Ehresmann-Schein-Nambooripad Theorem and its subsequent generalisations. The second construction is a generalisation of Grillet's work on cross-connected partially ordered sets, arising from the principal ideals of the given semigroup. In this article, we establish a direct equivalence between these two seemingly different constructions. We show how the cross-connection representation of a regular semigroup may be constructed directly from the inductive groupoid of the semigroup, and vice versa.

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The bifree regularE-solid semigroups

Cited by 7 publications

References 13 publications

Semidirect products of regular semigroups

Semidirect products of regular semigroups

E-solid e-varieties are not finitely based

Inductive groupoids and cross-connections of regular semigroups

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