On existence varieties of locally inverse semigroups

Auinger, Karl; Doyle, Joseph; Jones, Peter R.

doi:10.1017/s0305004100072042

Cited by 6 publications

(10 citation statements)

References 31 publications

(73 reference statements)

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“…The results of this section, other than Result 10.1 and Theorem 10.5(iii), are new. When both U and V are locally inverse, so that U V also has that property [5], the outcome of our construction is equivalent to that of Auinger and Polák. However, their arguments are syntactic, based on 'words' in free binary semigroups (to which allusion was made in §8.1), and do not extend to the non-locally inverse situation.…”

Section: An Alternative Product Of Regular Semigroupsmentioning

confidence: 82%

“…Recently, K. Auinger and L. Polák [5] have defined a product S T of regular semigroups S and T that yields a regular semigroup whenever T is locally inverse. There is an associated product U V of e-varieties U and V, defined whenever V ⊆ LI.…”

Section: An Alternative Product Of Regular Semigroupsmentioning

confidence: 99%

“…It was shown in [5] that S T = {(s, t) ∈ S × T : tt s = s for all t ∈ V (t)}. Define a product on the set S T by (x, u)(y, v) = ( u(v∧u)u x u y, uv), where u ∈ V (u).…”

Section: General Propertiesmentioning

confidence: 99%

“…Define a product on the set S T by (x, u)(y, v) = ( u(v∧u)u x u y, uv), where u ∈ V (u). According to [5], the product is independent of the choice of inverse for u. Readers uncomfortable with the ∧ operation may prefer the cumbersome notation vv (u uvv ) u u to v ∧ u; here indicates taking an arbitrary inverse of the indicated element.…”

Section: General Propertiesmentioning

confidence: 99%

“…The final section examines briefly an alternative semidirect product of e-varieties, studied by Polák [32] and Auinger and Polák [5], generalizing certain semidirect products of varieties of inverse semigroups. Their product relies on a variant multiplication; it is well defined as long as the second factor is locally inverse.…”

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confidence: 99%

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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: An Alternative Product Of Regular Semigroupsmentioning

confidence: 82%

Section: An Alternative Product Of Regular Semigroupsmentioning

confidence: 99%

“…It was shown in [5] that S T = {(s, t) ∈ S × T : tt s = s for all t ∈ V (t)}. Define a product on the set S T by (x, u)(y, v) = ( u(v∧u)u x u y, uv), where u ∈ V (u).…”

Section: General Propertiesmentioning

confidence: 99%

Section: General Propertiesmentioning

confidence: 99%

mentioning

confidence: 99%

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