Superextensions of three-element semigroups

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 $B\supset A$ belongs to $\mathcal{A}$. An upfamily $\mathcal L$ of subsets of $X$ is said to be linked if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $X$ is maximal linked if $\mathcal M$ coincides with each linked upfamily $\mathcal L$ on $X$ that contains $\mathcal M$. The superextension $\lambda(X)$ consists of all maximal linked upfamilies on $X$. Any associative binary operation $* : X\times X \to X$ can be extended to an associative binary operation $\circ: \lambda(X)\times\lambda(X)\to\lambda(X)$ by the formula $\mathcal L\circ\mathcal M=\Big\langle\bigcup_{a\in L}a*M_a:L\in\mathcal L,\;\{M_a\}_{a\in L}\subset\mathcal M\Big\rangle$ for maximal linked upfamilies $\mathcal{L}, \mathcal{M}\in\lambda(X)$. In the paper we describe superextensions of all three-element semigroups up to isomorphism.