Möbius monoids and their connection to inverse monoids.
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.
scientific article; zbMATH DE number 6493119
scientific article