On De Graaf spaces of pseudoquotients

Abstract

A space of pseudoquotients, $\mathcal{B}(X,S)$, is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f) \sim (y,g)$ if $gx=fy$. In this note, we consider a generalization of this construction where the assumption of commutativity of $S$ is replaced by Ore type conditions. As in the commutative case, $X$ can be identified with a subset of $\mathcal{B}(X,S)$, and $S$ can be extended to a group, $G$, of bijections on $\mathcal{B}(X,S)$. We introduce a natural topology on $\mathcal{B}(X,S)$ and show that all elements of $G$ are homeomorphisms on $\mathcal{B}(X,S)$.