Automorphism groups of semigroups of upfamilies

Abstract

A family [Formula: see text] of nonempty subsets of a set [Formula: see text] is called an upfamily if for each set [Formula: see text] any set [Formula: see text] belongs to [Formula: see text]. The extension [Formula: see text] of [Formula: see text] consists of all upfamilies on [Formula: see text]. Any associative binary operation [Formula: see text] can be extended to an associative binary operation [Formula: see text]. In the paper, we study automorphisms of extensions of groups, finite monogenic semigroups, null semigroups, right zero semigroups and left zero semigroups. Also, we describe the automorphism groups of extensions of some semigroups of small cardinalities.