Möbius monoids and their connection to inverse monoids.
type: Item
instance of: scholarly article
title: Möbius monoids and their connection to inverse monoids.
zbMATH Open document ID: 1344.20079
author: Emil Daniel Schwab
published in: Semigroup Forum
publication date: 2015-10-12T00:00:00Z
review text: A monoid \(M\) satisfying the following conditions is called a \textit{Möbius monoid}: 1) \(M\) is decomposition-finite (i.e., for any \(a\in M\) there is a finite number of pairs \((b,c)\in M\times M\) such that \(a=bc\)); 2) \(ab=1\Rightarrow a=b=1\); 3) \(ab=b\Rightarrow a=1\). \(\mathfrak M\) denotes a non-trivial right cancellative left rigid (i.e., \(Sa\cap Sb\neq\emptyset\) implies \(Sa\cap Sb=Sc\) for some \(c\in\mathfrak M\)) Möbius monoid. Set \(\mathfrak M_0=\mathfrak M\) if \(\mathfrak Ma\cap\mathfrak Mb\neq\emptyset\) for any \(a,b\in\mathfrak M\) and \(\mathfrak M_0=\mathfrak M\cup\{0\}\) otherwise, where \(0\) is an extra adjoined zero. Set also \(S(\mathfrak M_0)=\mathfrak M\times\mathfrak M\) if \(\mathfrak M_0=\mathfrak M\) and \(S(\mathfrak M_0)=\mathfrak M\times\mathfrak M\cup\{(0,0)\}\) otherwise. Defining an operation suitably, \(S(\mathfrak M_0)\) becomes a bisimple (0-bisimple) and combinatorial (i.e., the Green \(\mathcal H\) relation is the identity relation) inverse monoid. Moreover, any bisimple (0-bisimple) combinatorial inverse monoid \(S\) for which \((E^*(S),\leq)\) is locally finite arises from a right cancellative and left rigid Möbius monoid \(\mathfrak M\). A construction and subsequent examples are given how to get from a Möbius monoid a proper Möbius category. On the other hand, the author defines a \textit{quasi-division category} and constructs an inverse monoid on it which he calls \textit{Leech-Lawson monoid} and derives some properties of these monoids.
reviewed by: Peeter Normak
Mathematics Subject Classification ID: 20M18
Mathematics Subject Classification ID: 18B40
zbMATH DE Number: 6493119
zbMATH Keywords: inverse semigroups
zbMATH Keywords: Möbius monoids
zbMATH Keywords: inverse monoids
zbMATH Keywords: Möbius categories
zbMATH Keywords: Möbius functions
zbMATH Keywords: bicyclic semigroups
zbMATH Keywords: polycyclic monoids
MaRDI profile type: Publication
full work available at URL: s00233-014-9666-0
OpenAlex ID: W2027440577
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DOI: 10.1007/S00233-014-9666-0
description: scientific article; zbMATH DE number 6493119