On semigroups with lower semimodular lattice of subsemigroups.
type: Item
instance of: scholarly article
title: On semigroups with lower semimodular lattice of subsemigroups.
zbMATH Open document ID: 1221.20043
author: Peter R. Jones
published in: Journal of Algebra
publication date: 2011-01-14T00:00:00Z
full work available at URL: https://epublications.marquette.edu/cgi/viewcontent.cgi
review text: The question of which semigroups have a lower semimodular lattice of subsemigroups has been open since the early 1960s, when the corresponding question was answered for modularity and for upper semimodularity. The author provides a characterization of such semigroups in the language of principal factors. Since it is easily seen that semigroups for which Green's relation \(\mathcal J\) is trivial have this property, a description in such terms is natural. A semigroup \(S\) has a lower semimodular lattice of subsemigroups if and only if {\parindent=8mm \begin{itemize} \item[(I)]each non-null principal factor of \(S\) is either: \begin{itemize} \item[(a)]a group with lower semimodular subgroup lattice or a singular band; such a semigroup with zero adjoined; or, up to isomorphism, the five-element combinatorial Brandt semigroup \(B_2\); or \item[(b)]a \(\mathcal D\)-trivial idempotent-free [0-]simple semigroup; \end{itemize} \item[(II)]for any nontrivial subgroup of \(S\), with identity \(e\), say, if \(ea\in H_e\) for some \(a\in S\), then either \(e\in\langle a\rangle\) or \(e=ea\); \item[(III)]for each element \(x\) of \(S\) that does not belong to a nontrivial subgroup of \(S\), \begin{itemize} \item[(a)]if \(x=xab\) for some \(a,b\in S\), then either \(x\in\langle a,b\rangle\) or \(x=xa\); and dually, \item[(b)]if the associated principal factor is null and \(x=bxa\) for some \(a,b\in S\), then either \(x\in\langle a,b\rangle\) or \(x=xa\), \item[(c)]if the associated principal factor is of type I(b) and \(x=a_0xa_1x\cdots xa_n\) for some \(a_0,\dots,a_n\in S^1\) and \(n\geq 1\), then either \(x\in\langle a_0,\dots,a_n\rangle\) or \(n=1\) and \(x=xa_1\). \end{itemize} \end{itemize}} In the case of periodic semigroups -- a case that turns out to include all eventually regular semigroups -- the characterization becomes quite explicit and yields interesting consequences. In the general case, it remains an open question whether there exists a simple, but not completely…
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 06C10
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 5836825
zbMATH Keywords: lower semimodular subsemigroup lattices
zbMATH Keywords: varieties of semigroups
zbMATH Keywords: principal factors
zbMATH Keywords: \(\mathcal J\)-trivial semigroups
zbMATH Keywords: \(\mathcal D\)-trivial semigroups
zbMATH Keywords: Green relations
zbMATH Keywords: lattices of subsemigroups
MaRDI profile type: Publication
OpenAlex ID: W1978029640
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DOI: 10.1016/J.JALGEBRA.2010.07.046
description: scientific article; zbMATH DE number 5836825