Nasir Sohail, Institute of Mathematics, University of Tartu, Estonia
“An open problem concerning dominions of posemigroups”
An element d of a semigroup S is said to be in the dominion of a subsemigroup U of S if for every pair f,g:S —¿T of semigroup homomorphisms, with f(u)=g(u) for all u in U, we have f(d)=g(d). The set of all such elements of S is called the dominion of U in S, denoted by Dom(U,S). If we consider partially ordered semigroups, briefly posemigroups, and posemigroup homomorphisms (which are monotone semigroup homomorphisms) we can similarly define posemigroup dominions; let us denote them by Dom¡U,S¿, where now U is a subposemigroup of a posemigroup S. Note that for posemigroups one can consider both Dom (U,S) and Dom ¡U,S¿. We know that in general U is contained in Dom (U,S) which is again contained in Dom ¡U,S¿.
I know of certain instances where Dom ¡U,S¿ equals Dom (U,S) and I conjecture that this not always the case. My first candidate for the creation of an example is a seven elements semigroup, considered by P.M. Higgins in the early 1980s. In this talk I shall explain how I want to proceed with my example and discuss my obstacles. Also my aim is to receive feedback that may help me to improve or change my strategy.