The Existence of E-Free Objects in E-Varieties of Regular Semigroups

Yeh, Yeong-Nan

doi:10.1142/s0218196792000281

Cited by 36 publications

(41 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [42], Yeh also showed that various analogues of properties normally possessed by varieties, in the usual sense, carry over to e-varieties in those situations (but not generally). We have, for instance, the following, the first part of which is [42,Corollary 4.7] Finally, the concept of the Mal'cev product of e-varieties has proven very fruitful in the contexts of varieties of inverse and, especially, completely regular semigroups. A recent result of the second author may be regarded as representing e-varieties of the form C ∞ U as certain such products.…”

Section: Results 14 An E-variety Of Regular Semigroups Possesses An mentioning

confidence: 99%

“…Y.T. Yeh [42] extended the definition to cover e-varieties in general and determined those e-varieties of regular semigroups which possess e-free semigroups on all nonempty sets, as follows.…”

Section: Proposition 13 For Any E-variety U Of Regular Semigroups mentioning

confidence: 99%

“…By expanding on an idea of Yeh [42], Auinger [3,4] has considered locally inverse semigroups as 'binary semigroups' -semigroups with an additional binary operation (derived from its 'sandwich sets' -for more details see §10). The elements of e-free locally inverse semigroups may then be considered as 'words' of this type.…”

Section: That Starts At a Thus There Is An Induced Bijection C/t → Smentioning

confidence: 99%

“…Following [42] and [3,4], the rule x ∧ y = S(y y, xx ), where y ∈ V (y) and x ∈ V (x), yields a well defined binary operation on any locally inverse semigroup T .…”

Section: General Propertiesmentioning

confidence: 99%

See 3 more Smart Citations

Untitled

1997

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

Section: Results 14 An E-variety Of Regular Semigroups Possesses An mentioning

confidence: 99%

Section: Proposition 13 For Any E-variety U Of Regular Semigroups mentioning

confidence: 99%

Section: That Starts At a Thus There Is An Induced Bijection C/t → Smentioning

confidence: 99%

“…Following [42] and [3,4], the rule x ∧ y = S(y y, xx ), where y ∈ V (y) and x ∈ V (x), yields a well defined binary operation on any locally inverse semigroup T .…”

Section: General Propertiesmentioning

confidence: 99%

See 2 more Smart Citations

Untitled

1997

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…For the nonorthodox case, Y. T. Yeh [26] has shown that all bifree objects exist in an e-variety "V if and only if all members of 7/" are either ¿'-solid or locally inverse. In the latter case, the methods of [26] suggest that we introduce another binary operation A on a locally inverse semigroup S and that we treat 5 as an algebra of type (2,2).…”

Section: Introductionmentioning

confidence: 99%

A System of Bi-Identities for Locally Inverse Semigroups

Auinger¹

1995

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

Abstract. A class of regular semigroups closed under taking direct products, regular subsemigroups, and homomorphic images is an existence-variety (or evariety) of regular semigroups. Each e-variety of locally inverse semigroups can be characterized by a set of bi-identities. These are identities of terms of type (2, 2) in two sorts of variables X and X'. In this paper we obtain a basis of bi-identities for the e-variety of locally inverse semigroups and for certain sub-e-varieties.

show abstract

The existence of free products in existence varieties of regular semigroups

Auinger¹

1995

Algebra Universalis

View full text Add to dashboard Cite

The Existence of E-Free Objects in E-Varieties of Regular Semigroups

Cited by 36 publications

References 0 publications

Untitled

Untitled

A System of Bi-Identities for Locally Inverse Semigroups

The existence of free products in existence varieties of regular semigroups

Contact Info

Product

Resources

About