On Existence Varieties of E-Solid or Locally Inverse Semigroups and E-Invariant Congruences

Yeh, Yeong-Nan

doi:10.1006/jabr.1994.1072

Cited by 4 publications

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“…Z£J as well as $Sf are e-varieties (for the former consult [23], the latter can be seen from the mentioned characterization of ^-solid semigroups). Yeh (see [38,39]) defined the notion of ' e-free object' formally in the same fashion as the notion of bifree object (Definition 2), but with 'matched' replaced by 'paired'. Then he was able to show that an e-variety Y of regular semigroups contains (all) e-free objects if and only if Y ^ &^ or f" £ $£f.…”

Section: Preliminariesmentioning

confidence: 99%

On existence varieties of locally inverse semigroups

Auinger¹,

Doyle

Jones

1994

Math. Proc. Camb. Phil. Soc.

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A locally inverse semigroup is a regular semigroup S with the property that eSe is inverse for each idempotent e of S. Motivated by natural examples such as inverse semigroups and completely simple semigroups, these semigroups have been the subject of deep structure-theoretic investigations. The class ℒ ℐ of locally inverse semigroups forms an existence variety (or e-variety): a class of regular semigroups closed under direct products, homomorphic images and regular subsemigroups. We consider the lattice ℒ(ℒℐ) of e-varieties of such semigroups. In particular we investigate the operations of taking meet and join with the e-variety CS of completely simple semigroups. An important consequence of our results is a determination of the join of CS with the e-variety of inverse semigroups – it comprises the E-solid locally inverse semigroups. It is shown, however, that not every e-variety of E-solid locally inverse semigroups is the join of completely simple and inverse e-varieties.

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Section: Preliminariesmentioning

confidence: 99%

On existence varieties of locally inverse semigroups

Auinger¹,

Doyle

Jones

1994

Math. Proc. Camb. Phil. Soc.

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Minimal Noncryptic e-Varieties of Regular Semigroups

Pastijn

Volkov²

1996

Journal of Algebra

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We describe all minimal noncryptic e-varieties of regular semigroups, thus generalising earlier results by Rasin and Reilly that dealt with the completely regular and the inverse cases, respectively. As corollaries, we prove that an e-variety of regular semigroups is cryptic if and only if its intersections with the variety of all completely regular semigroups and the variety of all inverse semigroups are cryptic. We also find an equational characterization of group-bound cryptic varieties; this generalises some recent results by Doyle and Yeh. ᮊ

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Commutativity of operators on the lattice of existence varieties

Reilly

Zhang

1997

Monatshefte f�r Mathematik

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Abstract. In a previous paper the authors introduced seven complete congruences on the lattice L, ee~(.r ) of e-varieties of regular semigroups of the form pe:qlpe~K ",r ~ o q/= ~ o ~U, where ~ is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form c~: q/c~e~ ~.~ c~ q/= ~ c~ ~, where ~ is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence p on Lf~v(~5 ~ and each q/e ~o,(~5~), the p-class of q/is an interval [q/p, q/P] so that there is associated with each such congruence an idempotent operator q/~ q/P on 5rThis paper establishes numerous results concerning the commutativity of operators of this form.

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On Existence Varieties of E-Solid or Locally Inverse Semigroups and E-Invariant Congruences

Cited by 4 publications

References 0 publications

On existence varieties of locally inverse semigroups

On existence varieties of locally inverse semigroups

Minimal Noncryptic e-Varieties of Regular Semigroups

Commutativity of operators on the lattice of existence varieties

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