On De Graaf spaces of pseudoquotients

TL;DR

This paper generalizes the concept of pseudoquotient spaces to non-commutative semigroups satisfying Ore conditions, establishing a topology where the extended group acts as homeomorphisms.

Contribution

It extends the construction of pseudoquotient spaces to non-commutative settings with Ore conditions, and studies their topological and group action properties.

Findings

The space $B(X,S)$ can be topologized naturally.

The group $G$ acts as homeomorphisms on $B(X,S)$.

$X$ embeds into $B(X,S)$ as a subset.

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 $S$ 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 $\mathcal{B}(X,S)$.