### Automorphism groups of superextensions of finite monogenic semigroups

#### Abstract

A family \(\mathcal L\) of subsets of a set \(X\) is called

*linked*if \(A\cap B\ne\emptyset\) for any \(A,B\in\mathcal L\). A linked family \(\mathcal M\) of subsets of \(X\) is*maximal linked*if \(\mathcal M\) coincides with each linked family \(\mathcal L\) on \(X\) that contains \(\mathcal M\). The*superextension*\(\lambda(X)\) of \(X\) consists of all maximal linked families on \(X\). Any associative binary operation \(* : X\times X \to X\) can be extended to an associative binary operation \(*: \lambda(X)\times\lambda(X)\to\lambda(X)\). In the paper we study automorphisms of the superextensions of finite monogenic semigroups and characteristic ideals in such semigroups. In particular, we describe the automorphism groups of the superextensions of finite monogenic semigroups of cardinality \(\leq 5\).#### Keywords

monogenic semigroup, maximal linked upfamily, superextension, automorphism group

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