On the automorphism group of the superextension of a semigroup

Author
V. M. Gavrylkiv
Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
Abstract
A family $\mathcal{A}$ of non-empty subsets of a set $X$ is called an {\it upfamily} if for each set $A\in\mathcal{A}$ any set $B\supset A$ belongs to $\mathcal{A}$. An upfamily $\mathcal L$ of subsets of $X$ is said to be {\it linked} if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $X$ is {\it maximal linked} if $\mathcal M$ coincides with each linked upfamily $\mathcal L$ on $X$ that contains $\mathcal M$. The {\it superextension} $\lambda(X)$ consists of all maximal linked upfamilies on $X$. Any associative binary operation $* \colon X\times X \to X$ can be extended to an associative binary operation $*\colon \lambda(X)\times\lambda(X)\to\lambda(X)$. In the paper we study automorphisms of superextensions of semigroups and describe the automorphism groups of superextensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups. Also we find the automorphism groups of superextensions of all semigroups $S$ of order $|S|\leq3$.
Keywords
semigroup; maximal linked upfamily; superextension; automorphism group
DOI
doi:10.15330/ms.48.1.3-13
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Pages
3-13
Volume
48
Issue
1
Year
2017
Journal
Matematychni Studii
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