Distributive elements of the lattice of semigroup varieties
type: Item
instance of: scholarly article
title: Distributive elements of the lattice of semigroup varieties
zbMATH Open document ID: 1223.08002
published in: Algebra and Logic
publication date: 2011-01-24T00:00:00Z
review text: Let \textbf{SEM} be the lattice of all semigroup varieties. A pair of identities \(wx=xw=w\), where the letter \(x\) does not occur in the word \(w\), is written as \(w=0\). Let \({\mathcal{D}}\) be the set of the following varieties of semigroups: \({\mathcal{SEM}}\) = variety of all semigroups; \({\mathcal{SL}}=\text{var}\{x=x^2, xy=yx\}\); \({\mathcal{M}}_1=\text{var}\{x^2y=xyx=yx^2=0\}\); \({\mathcal{M}}^n_1=\text{var}\{x^2y=xyx=yx^2=x_1x_2\dots x_n=0\}\); \({\mathcal{M}}_2=\text{var}\{x^2=xyx=0\}\); \({\mathcal{M}}^n_2=\text{var}\{x^2=xyx=x_1x_2\dots x_n=0\}\) \((n\in \mathbb{N})\). The authors prove that the set of all distributive elements of the lattice \textbf{SEM} is equal to the set \({\mathcal D}\cup\{{\mathcal M}_1\vee {\mathcal S}{\mathcal L}, \;{\mathcal M}^n_1\vee {\mathcal S}{\mathcal L}, \;{\mathcal M}_2\vee {\mathcal S}{\mathcal L}, \;{\mathcal M}^n_2\vee {\mathcal S}{\mathcal L}\}\).
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 5840717
zbMATH Keywords: semigroup variety
zbMATH Keywords: lattice of varieties
zbMATH Keywords: distributive element
author: Boris Vernikov
author: Vyacheslav Yu. Shaprynskiǐ
reviewed by: Danica Jakubiková-Studenovská
MaRDI profile type: Publication
full work available at URL: s10469-010-9090-9
OpenAlex ID: W2078632424
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DOI: 10.1007/S10469-010-9090-9
description: scientific article; zbMATH DE number 5840717