.png){kind=icon}
.png){kind=icon}
Return
The reverse representation problem
.png){kind=icon}
PRE
AI
DOI：10.1007/s00233-026-10625-7.png)
.png){kind=icon}
Original request for help.png){kind=icon}
Share.png){kind=icon}
Save
Abstract
En 中文
Cayley's theorem tells us that all groups G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{G}$$\end{document} occur as subgroups of the group of permutations over some set X. In this paper we consider a 'sort-of' converse to this question: given a set X and some transformation group S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{S}$$\end{document} over X, what are the possible group structures on X that result in groups represented by S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{S}$$\end{document}? We solve this problem in the more general setting of faithful semigroups and observe that the solutions to this problem, which we term unrepresentations, have an inherent group structure. We study this phenomenon in depth before finishing with an analysis of the special case of unrepresentations of Clifford semigroups.
Keywords:
Representation
Cayley's Theorem
Clifford Semigroup
Heap
Transformation