## Rocky Mountain Journal of Mathematics

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

#### Article information

Source
Rocky Mountain J. Math., Volume 45, Number 5 (2015), 1445-1455.

Dates
First available in Project Euclid: 26 January 2016

https://projecteuclid.org/euclid.rmjm/1453817248

Digital Object Identifier
doi:10.1216/RMJ-2015-45-5-1445

Mathematical Reviews number (MathSciNet)
MR3452222

Zentralblatt MATH identifier
1348.44008

#### Citation

Katsevich, Anya; Mikusiński, Piotr. On De Graaf spaces of pseudoquotients. Rocky Mountain J. Math. 45 (2015), no. 5, 1445--1455. doi:10.1216/RMJ-2015-45-5-1445. https://projecteuclid.org/euclid.rmjm/1453817248

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