Ladders and canonical varieties of completely regular semigroups
type: Item
instance of: scholarly article
title: Ladders and canonical varieties of completely regular semigroups
zbMATH Open document ID: 1413.20039
author: Mario Petrich
published in: Periodica Mathematica Hungarica
publication date: 2018-07-23T00:00:00Z
review text: The lattice \({\mathcal L} (\mathcal{CR})\) of varieties of completely regular semigroups contains as an ideal the well-studied lattice \({\mathcal{L}} (\mathcal{B})\) of bands. Two associated complete congruences are defined on \({\mathcal L} (\mathcal{CR})\): \(\mathcal{U} \mathrel{\mathbb{B}^{\wedge}} \mathcal{V}\) if \(\mathcal{U} \cap \mathcal{B} = \mathcal{V} \cap \mathcal{B}\); and \(\mathcal{U} \mathrel{\mathbb{B}^{\vee}} \mathcal{V}\) if \(\mathcal{U} \vee \mathcal{B} = \mathcal{V} \vee \mathcal{B}\). The meet-semilattice \(\Delta\) of upper ends \(\mathcal{V}^{\mathbb{B}^\wedge}\) of the \(\mathbb{B}^{\wedge}\)-classes can be expected to carry significant information. With a few exceptions, the members of \(\Delta\) are intersections of the `canonical' varieties the author introduced in [J. Aust. Math. Soc. 83, No. 1, 87--104 (2007; Zbl 1142.20035)] and so are quite well understood. The first main theorem computes the upper ends \(\mathcal{V}\vee \mathcal{B}\) of the \(\mathbb{B}^{\vee}\)-classes of the members of \(\Delta\). As a useful tool, the Polák ladders are calculated for most of these varieties. Various properties of the resulting meet-semilattice of upper ends are also found.
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 6909513
zbMATH Keywords: completely regular semigroup
zbMATH Keywords: ladder
zbMATH Keywords: canonical variety
reviewed by: Peter R. Jones
MaRDI profile type: Publication
full work available at URL: s10998-017-0215-y
OpenAlex ID: W2757871199
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DOI: 10.1007/S10998-017-0215-Y
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description: scientific article; zbMATH DE number 6909513