Automorphisms of semigroups of k-linked upfamilies

Abstract

A family \( \mathcal{A} \) of non-empty subsets of a set X is called an upfamily, if, for each set \( A\in \mathcal{A} \); any set B ⊃ A belongs to \( \mathcal{A} \). An upfamily \( \mathrm{\mathcal{L}} \) is called k-linked, if \( \cap \mathrm{\mathcal{F}}\ne \varnothing \) for any subfamily \( \mathrm{\mathcal{F}}\subset \mathrm{\mathcal{L}} \) of cardinality \( \left|\mathrm{\mathcal{F}}\right|\le k \). The extension N_k(X) consists of all k-linked upfamilies on X. Any associative binary operation ∗ : X × X → X can be extended to an associative binary operation ∗ : N_k(X) × N_k(X) → N_k(X). Here, we study automorphisms of the extensions of groups and finite monogenic semigroups. We also describe the automorphism groups of extensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups.