T X is absolutely closed

Abstract

If S, T are semigroups with S⊂T, then the dominion of S in T, Dom(S,T), is the set of all x e T such that for each semigroup U and for each pair of homomorphisms f,g: T→U with f|S=g|S, then f(x)=g(x). S is absolutely closed if Dom(S,T)=S for all T. That full transformation semigroups are absolutely closed has previously been reported. The intent here is to offer a corrected proof of that theorem.