Isbell's zigzag theorem

Statement

Isbell's zigzag theorem, a theorem of abstract algebra characterizing the notion of a dominion, was introduced by American mathematician John R. Isbell in 1966. Dominion is a concept in semigroup theory, within the study of the properties of epimorphisms. For example, let $U$ be a subsemigroup of $S$ containing $U$, the inclusion map $U \hookrightarrow S$ is an epimorphism if and only if $\rm{Dom}_S (U) = S$, furthermore, a map $\alpha \colon S \to T$ is an epimorphism if and only if $\rm{Dom}_T (\rm{im} \; \alpha) = T$. The categories of rings and semigroups are examples of categories with non-surjective epimorphism, and the Zig-zag theorem gives necessary and sufficient conditions for determining whether or not a given morphism is epi. Proofs of this theorem are topological in nature, beginning with (Isbell 1966) for semigroups, and continuing by (Philip 1974), completing Isbell's original proof. The pure algebraic proofs were given by (Howie 1976) and (Storrer 1976).