Zigzags and their applications
Peter M. Higgins · 1992
Abstract We begin this chapter with a new and short proof of the Zigzag Theorem of John Isbell, which characterizes semigroup dominions by means of certain sequences of equations known as zigzags. A variety of applications concerning epimorphisms and amalgams will illustrate the usefulness of this result. First we look at the connection between epimorphisms and dominions. Observe that to say that a homomorphism α:S →T is epi (pre-cancellative) is the same as saying that whenever homomorphisms β, γ: T → V are such that β | Sα= γ |Sα, then β = γ. If this is the case, we say that Sα isdense in T, or that Sα isepimorphically embedded in T via the inclusioni: Sα →T. Therefore the study of epimorphisms is equivalent to the study of epimorphic embeddings.