Associativity of products of existence varieties of regular semigroups

Reilly, Norman R.; Zhang, Shuhua

doi:10.1016/s0022-4049(97)00135-7

Cited by 2 publications

(6 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(ii) -Sf ^ c r * r^^ i j r g 4?^, and (iii) [7] Semidirect product of e-varieties 91 RESULT Since every semidirect product of a left zero semigroup by another semigroup is isomorphic to their direct product, we get the following result, see [15]. RESULT 1.6.…”

Section: Results 11 Iffy and V Are E-varieties Such That Fy Orf Cmentioning

confidence: 99%

See 1 more Smart Citation

Associativity of the regular semidirect product of existence varieties

Billhardt¹,

Szendrei²

2000

J Aust Math Soc A

View full text Add to dashboard Cite

The associativity of the regular semidirect product of existence varieties introduced by Jones and Trotter was proved under certain conditions by Reilly and Zhang. Here we establish associativity in many new cases. Moreover, we prove that the regular semidirect product is right distributive with respect to the join operation. In particular, both associativity and right distributivity yield within the varieties of completely simple semigroups. Analogous results are obtained for e-pseudovarieties of finite regular semigroups.2000 Mathematics subject classification: primary 20M17,20M07.

show abstract

Section: Results 11 Iffy and V Are E-varieties Such That Fy Orf Cmentioning

confidence: 99%

“…These classical products are well known to be associative (see [4,13,14]). The equality (^ * r Y) * r TV = <% * r (Y * r W) has been proved recently by Reilly and Zhang [15] in cases, where at least two of the e-varieties %, Y, W are group varieties and some additional conditions are fulfilled.…”

Section: Introductionmentioning

confidence: 87%

Associativity of the regular semidirect product of existence varieties

Billhardt¹,

Szendrei²

2000

J Aust Math Soc A

View full text Add to dashboard Cite

show abstract

“…That Corollary 3.2 does not hold in general follows rather indirectly from an example of Reilly and Zhang [10,Example 5.4]. It is shown in [10] that there is an (e-) variety W of completely simple semigroups with the property that W * G = WmG.…”

Section: Proposition 22 Let S Be a Regular Semigroup And Suppose Thmentioning

confidence: 99%

“…It is shown in [10] that there is an (e-) variety W of completely simple semigroups with the property that W * G = WmG. But according to [6,Proposition 5.1], U * K = UmK for any e-local e-variety U of E-solid regular semigroups and any variety K of groups.…”

Section: Proposition 22 Let S Be a Regular Semigroup And Suppose Thmentioning

confidence: 99%

See 1 more Smart Citation

Rees Matrix Covers and Semidirect Products of Regular Semigroups

Jones

1999

Journal of Algebra

View full text Add to dashboard Cite

G. Trotter and the author introduced a "regular" semidirect product U * V of e-varieties U and V. Among several specific situations investigated there was the case V = RZ, the e-variety of right zero semigroups. Applying a covering theorem of McAlister, it was shown there that in several important cases (for instance for the e-variety of inverse semigroups), U * RZ is precisely the e-variety LU of "locally U" semigroups.The main result of the current paper characterizes membership of a regular semigroup S in U * RZ in a number of ways; one in terms of an associated category S E and another in terms of S regularly dividing a regular Rees matrix semigroup over a member of U. The categorical condition leads directly to a characterization of the equality U * RZ = LU in terms of a graphical condition on U, slightly weaker than "e-locality." Among consequences of known results on e-locality, the conjecture CR * RZ = LCR (with CR denoting the e-variety of completely regular semigroups), is therefore verified. The connection with matrix semigroups then leads to a range of Rees matrix covering theorems that, while slightly weaker than McAlister's, apply to a broader range of examples. K. Auinger and P. G. Trotter (Pseudovarieties, regular semigroups and semidirect products, J. London Math. Soc. (2) 58 (1998), 284-296) have used our results to describe the pseudovarieties generated by several important classes of (finite) regular semigroups.An e-variety of regular semigroups is a class of regular semigroups that is closed under products, quotients, and regular subsemigroups. In a recent paper [6], Trotter and the author introduced a product U * V of e-varieties U V, well-defined if (and only if) either U or V consists of completely simple semigroups, as follows: U * V is the e-variety generated by the semigroups of regular elements of the wreath products (or the semidirect prod-287

show abstract

Associativity of products of existence varieties of regular semigroups

Cited by 2 publications

References 29 publications

Associativity of the regular semidirect product of existence varieties

Associativity of the regular semidirect product of existence varieties

Rees Matrix Covers and Semidirect Products of Regular Semigroups

Contact Info

Product

Resources

About