Actions and expansions of ordered groupoids
An ordered groupoid is a small category consisting of invertible morphisms with an order on the set of all morphisms satisfying three axioms connecting composition of morphisms with the order. These axioms enable us to define a new partial associative binary operation -- a pseudoproduct. The Birget-Rhodes expansion \(G^{BR}\) of an ordered groupoid \(G\) is defined and it is shown that it coincides with the prefix expansion of \(G\). An ordered premorphism between ordered groupoids is defined -- it is a mapping preserving order, invertibility and which is subdecreasing with respect to composition of morphisms and pseudoproduct. Any ordered premorphism from \(G\) into \(H\) induces a functor from \(G^{BR}\) into \(H\). A symmetric groupoid \(\mathbb G(X)\) for a set \(X\) is formed by all subsets of \(X\) and partial bijections between them. A partial action of an ordered groupoid \(G\) on \(X\) is an ordered premorphism \(\theta :G@>>>\mathbb G(X)\), and if \(\theta\) is a functor then \( \theta\) is a global action of \(G\) on \(X\). It is proved that for every partial action \(\theta\) of \(G\) on \(X\) there exist an injective mapping from \(X\) into \(Y\) and a global action \(\Theta\) of \(G\) on \(Y\) such that \( \theta\) is a suitable `restriction' of \(\Theta\) (this process is called a globalisation of a partial action). The connection between the Margolis-Meakin expansion over an ordered groupoid \(G\) and the Birget-Rhodes expansion of \(G\) is presented.
scientific article; zbMATH DE number 2173219