Rocky Mountain Journal of Mathematics

On De Graaf spaces of pseudoquotients

Anya Katsevich and Piotr Mikusiński

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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

Permanent link to this document
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

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

Keywords
Pseudo quotients Ore condition Ore localization semigroup action

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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References

  • D. Atanasiu and P. Mikusiński, The Fourier transform of Levy measures on a semigroup, Integral Transf. Spec. Funct. 19 (2008), 537–543.
  • ––––, Fourier transform of Radon measures on locally compact groups, Integral Transf. Spec. Funct. 21 (2010), 815–821.
  • D. Atanasiu, P. Mikusiński and D. Nemzer, An algebraic approach to tempered distributions, J. Math. Anal. Appl. 384 (2011), 307–319.
  • D. Atanasiu, P. Mikusiński and A. Siple, Pseudoquotients on commutative Banach algebras, Banach J. Math. Anal. 8 (2014), 60–66.
  • Jan De Graaf and A.F.M. Tom Ter Elst, An algebraic approach to distribution theories, Generalized functions, convergence structures, and their applications, Plenum, New York, 1988.
  • H.E. Heatherly and J.P. Huffman, Noncommutative operational calculus, Electr. J. Diff. Equat. 02 (1999), 11–18.
  • H.J. Kim and P. Mikusiński, On convergence and metrizability of pseudoquotients, Int. J. Mod. Math. 5 (2010), 285–298.
  • J.G.-M. (Jan Mikusiński), Hypernumbers, Part I: Algebra, Stud. Math. 77 (1983), 1–16.
  • J. Mikusiński, Rachunek operatorów, Monogr. Mat. 30, Instytut Matematyczny Polskiej Akademii Nauk, Warszawa, 1953.
  • ––––, Operational calculus, Vol. 1, Pergamon-PWN, Oxford, 1983.
  • ––––, Generalized quotients with applications in analysis, Meth. Appl. Anal. 10 (2003), 377–386.
  • M. Rosenkranz and A. Korporal, A noncommutative Mikusiński calculus, arXiv:1209.1310.
  • Kôsaku Yosida, Operational calculus: A theory of hyperfunctions, Applied Mathematical Sciences, Springer-Verlag, New York, 1984.