A short proof of Isbell's zigzag theorem (Q1175101)

A subsemigroup \(U\) of a semigroup \(S\) dominates an element \(d\in S\) if for every semigroup \(T\) and all morphisms \(\phi_ 1: S\to T\), \(\varphi_ 2: S\to T\), \(\varphi_ 1\mid U = \varphi_ 2\mid U\) implies \(\varphi_ 1(d) = \varphi_ 2(d)\). The set of all elements in \(S\) dominated by \(U\) is called the dominion of \(U\) in \(S\). In the paper a short proof of Isbell's Zigzag Theorem which characterizes semigroup dominions is given.