Superextensions of three-element semigroups - Gavrylkiv, V.M. (Carpathian Mathematical Publications, 2017)
A family A of non-empty subsets of a set X is called an upfamily if for each set AA any set BA belongs to A. An upfamily L of subsets of X is said to be linked if AB e for all A,BL. A linked upfamily M of subsets of X is maximal linked if M coincides with each linked upfamily L on X that contains M. The superextension λ(X) consists of all maximal linked upfamilies on X. Any associative binary operation * : X× X → X can be extended to an associative binary operation °: λ(X)×λ(X)→λ(X) by the formula L°M=_aLa*M_a:LL, {M_a}_aLM for maximal linked upfamilies L, Mλ(X). In the paper we describe superextensions of all three-element semigroups up to isomorphism.