Dominions and zigzags for monoids
Let U be a submonoid of a monoid S. Then by the dominion of U in S we mean the set of all elements d of S such that for every pair of (monoid) homomorphisms f,g from S to T that agree on U one has f(d) = g(d). In 1965 J.R. Isbell proved in his famous (zigzag) theorem that d belongs to the dominion of U in S if and only if there exists a specific system of equalities which he termed zigzag. Another version of the theorem was put forward by B. Stenström in 1971, in which zigzags were replaced by certain tensor equalities. One of the most recent proofs of this theorem is due to J. Renshaw that adheres to its formulation due to Stenström. My objective is to acquaint the audience with the techniques used by Renshaw to establish the said proof. This will provide a base for my subsequent talk concerning an ordered analogue of the zigzag theorem.