The lattices of varieties and pseudovarieties of band monoids

Wismath, Shelly L.

doi:10.1007/bf02573192

Cited by 46 publications

(29 citation statements)

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“…Wismath [34] showed that the lattice of pseudovarieties of band monoids consists exactly of the trivial pseudovariety, BSM, SLSM and the intersections with M of the pseudovarieties obtained from SL using the operators s l , s r and intersection. So it corresponds to the solid line portion of Figure 1 (excluding any intersection with the non solid lines).…”

Section: The Structure Of the Lattice Of Pseudo6arieties Of Bandsmentioning

confidence: 99%

The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue

Trotter

Weil

1997

Algebra Universalis

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We use classical results on the lattice L(B) of varieties of band (idempotent) semigroups to obtain information on the structure of the lattice Ps(DA) of subpseudovarieties of DA, -where DA is the largest pseudovariety of finite semigroups in which all regular semigroups are band semigroups. We bring forward a lattice congruence on Ps(DA), whose quotient is isomorphic to L(B), and whose classes are intervals with effectively computable least and greatest members. Also we characterize the pro-identities satisfied by the members of an important family of subpseudovarieties of DA. Finally, letting V k be the pseudovariety generated by the k-generated elements of DA (k ] 1), we use all our results to compute the position of the congruence class of V k in L(B). IntroductionThe lattice of pseudovarieties of finite semigroups has been the object of much attention over the past few decades, with motivations drawn not only from universal algebra, but also from theoretical computer science. The link between theoretical computer science and semigroup theory goes back to Schü tzenberger's work in the late 1950s, and the role played there by pseudovarieties of finite semigroups was detailed by Eilenberg in the 1970s [7]. For more recent developments, the reader is referred to the books by Pin [21] and Almeida [1].Independently of this research, the lattice L(CR) of varieties of completely regular semigroups also received considerable attention, through the work of Polák [25,26,27], Reilly [28], Pastijn and Trotter [18], and others. Until recently however, the interaction between the techniques and results developed in the study of these two lattices was only minimal. This situation has changed in the last couple of years, for instance in Reilly [29], Auinger, Hall, Reilly and Zhang [5], and Reilly and Zhang [30].This paper is a contribution to the effort of using results on varieties of completely regular semigroups to study pseudovarieties of finite semigroups. Here we choose a restricted framework, namely that provided by the lattice of band Presented by B. M Schein.

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Section: The Structure Of the Lattice Of Pseudo6arieties Of Bandsmentioning

confidence: 99%

The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue

Trotter

Weil

1997

Algebra Universalis

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“…However, these articles are devoted mainly to the examination of identities of monoids. At the same time, although the first results about the lattice MON was found a long time ago (see [5,15,19]), only a little information was recently known about the lattice MON.…”

Section: Introductionmentioning

confidence: 99%

On the ascending and descending chain conditions in the lattice of monoid varieties

Gusev¹

2019

Sib. Elektron. Mat. Izv.

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In this work we consider monoids as algebras with an associative binary operation and the nullary operation that fixes the identity element. We found an example of two varieties of monoids with finite subvariety lattices such that their join covers one of them and has a continuum cardinality subvariety lattice that violates the ascending chain condition and the descending chain condition.2010 Mathematics Subject Classification. Primary 20M07, secondary 08B15.

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“…(Furthermore, we saw in the introduction that U is generated by a finite set of finite band monoids, which actually narrows down the choice of U to T , SL, B k , B k and B k ∨ B k , k 2, see [34].) Therefore, for the same choice of k, we have…”

Section: The Case Of Band Varietiesmentioning

confidence: 97%

On free spectra of finite completely regular semigroups and monoids

Dolinka

2009

Journal of Pure and Applied Algebra

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a b s t r a c tWe show that a finite completely regular semigroup has a sub-log-exponential free spectrum if and only if it is locally orthodox and has nilpotent subgroups. As a corollary, it follows that the Seif Conjecture holds true for completely regular monoids. In the process, we derive solutions of word problems of free objects in a sequence of varieties of locally orthodox completely regular semigroups from solutions of word problems in relatively free bands.

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The lattices of varieties and pseudovarieties of band monoids

Abstract: + Nationai i,brary of Canada &Sliotheque nabonale du Canada Col iect~ons D e v e l o p r n e~~n c h Directton du developpernent des collectrons Canadtan Theses o n Service des theses canadl nnes

Cited by 46 publications

References 8 publications

The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue

The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue

On the ascending and descending chain conditions in the lattice of monoid varieties

On free spectra of finite completely regular semigroups and monoids

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