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![The structure of the partial automorphism monoid PAut(Γ0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {PAut}}(\Gamma _0)$$\end{document}](publication/344389464/figure/fig3/AS:11431281126874435@1678850029964/The-structure-of-the-partial-automorphism-monoid-PAutG0documentclass12ptminimal.png)
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A partial automorphism of a finite graph is an isomorphism between its vertex-induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of standard tools of inverse semigroup theory, namely Gree...
Citations
... For other related papers on regular sets and regular orbits, see [1,11,16,20,22,24]. For some interesting generalizations see [17,18]. ...
... We have applied another approach, the results of which can be checked easily using GAP. The group G = M 12 || ψ M 12 different from M (2) 12 is generated by (1,19,17,23,2,4,8,9) (6,16,21,22)(3, 12) (5,24,15,7,18,10,20,13), and (1,23,8,19,9,6,4,21,16,22,2) (3,14,12,7,11,20,13,5,15,24,10) It is contained in the transitive group T generated by (1,16,23,19,9,21,2,4) (9,10,15,13) and (1,19,15,8,20,23,24,9,14,11,5,10,22,13,2)(3, 6, 4) (7,16,12,17,18). ...
... We have applied another approach, the results of which can be checked easily using GAP. The group G = M 12 || ψ M 12 different from M (2) 12 is generated by (1,19,17,23,2,4,8,9) (6,16,21,22)(3, 12) (5,24,15,7,18,10,20,13), and (1,23,8,19,9,6,4,21,16,22,2) (3,14,12,7,11,20,13,5,15,24,10) It is contained in the transitive group T generated by (1,16,23,19,9,21,2,4) (9,10,15,13) and (1,19,15,8,20,23,24,9,14,11,5,10,22,13,2)(3, 6, 4) (7,16,12,17,18). ...
A permutation group G on a set $$\Omega $$ Ω is called orbit closed if every permutation of $$\Omega $$ Ω preserving the orbits of G in its action on the power set $$P(\Omega )$$ P ( Ω ) belongs to G . It is called a relation group if there exists a family $$R \subseteq P(\Omega )$$ R ⊆ P ( Ω ) such that G is the group of all permutations preserving R . We prove that if a finite orbit closed permutation group G is simple, or is a subgroup of a simple group, then it is a relation group. This result justifies our general conjecture that with a few exceptions every finite orbit closed group is a relation group. To obtain the result, we prove that most of the finite simple permutation groups are relation groups. We also obtain a complete description of those finite simple permutation groups that have regular sets, and prove that (with one exception) if a finite simple permutation group G is a relation group, then every subgroup of G is a relation group.
A finite semigroup S is called structurally uniform if any two subsemigroup of S are isomorphic whose heights in the partially ordered set of all subsemigroups of S are equal. Note that this class contains the class of all finite semigroups for which the inverse monoid of local automorphisms is congruence permutable. In this paper, we present a classification of finite structurally uniform nilsemigroups.
