Rocky Mountain Journal of Mathematics

On De Graaf spaces of pseudoquotients

Anya Katsevich and Piotr Mikusiński

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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

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

First available in Project Euclid: 26 January 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 44A40: Calculus of Mikusiński and other operational calculi
Secondary: 16U20: Ore rings, multiplicative sets, Ore localization

Pseudo quotients Ore condition Ore localization semigroup action


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.

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