An application of groupoids of fractions to inverse semigroups
An inverse semigroup \(S\) is said to be \(E\)-unitary if for each \(s\in S\), \(e\leq s\) implies \(s^2=s\), where \(\leq\) is the natural partial order on \(S\) and \(e^2=e\). A McAlister triple is a triple \((G,X,Y)\), where \(G\) is a group, \(X\) is a poset, \(Y\) is a subset of \(X\) and \(G\) acts on \(X\) by order automorphisms such that the following three axioms are satisfied: (1) \(Y\) is a meet semilattice and an order ideal of \(X\); (2) \(GY=X\); (3) \(gY\cap Y\neq\emptyset\) for each \(g\in G\). If \((G,X,Y)\) is a McAlister triple then \(P(G,X,Y)=\{(y,g)\in Y\times G\mid g^{-1}y\in Y\}\) endowed with a multiplication defined by \((y,g)(z,h)=(y\wedge gz,gh)\) is an \(E\)-unitary inverse semigroup called a \(P\)-semigroup, see \textit{J. M. Howie} [Fundamentals of semigroup theory, Clarendon Press, Oxford (1995; Zbl 0835.20077)]. \textit{D. B. McAlister} [Trans. Am. Math. Soc. 196, 351-370 (1974; Zbl 0297.20072)] has proved the \(P\)-theorem: every \(E\)-unitary inverse semigroup is isomorphic to a \(P\)-semigroup. In this paper a new proof for the \(P\)-theorem as an application of category theory to the theory of inverse semigroups is given. In order to do that the following result is established: Proposition. Let \(\mathcal C\) be a cancellative division category and let \(\mathcal G\) be its groupoid of fractions. Let \(G\) be the endomorphism group of the identity 1 in \(\mathcal G\); let \(Y=\{{\mathcal C}a\mid a\in{\mathcal C}\) and \({\mathbf d}(a)=1\}\) and \(X=\{{\mathcal C}g\mid g\in{\mathcal C}\) and \({\mathbf d}(g)=1\}\). Define an action of \(G\) on \(X\) by \(gA=Ag^{-1}\). Then \((G,X,Y)\) is a McAlister triple.
scientific article; zbMATH DE number 1404460