Isbell's zigzag theorem


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 is a subsemigroup of containing, the inclusion map is an epimorphism if and only if, furthermore, a map is an epimorphism if and only if. 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 for semigroups, and continuing by, completing Isbell's original proof. The pure algebraic proofs were given by and.

Statement

Zig-zag

Zig-zag: If is a submonoid of a monoid, then a system of equalities;
in which and, is called a zig-zag of length in over with value. By the spine of the zig-zag we mean the ordered -tuple.

Dominion

Dominion: Let be a submonoid of a monoid . The dominion is the set of all elements such that, for all homomorphisms coinciding on,.
We call a subsemigroup of a semigroup closed
if, and dense if.

Isbell's zigzag theorem

Isbell's zigzag theorem:
If is a submonoid of a monoid then if and only if either or there exists a zig-zag in over with value that is, there is a sequence of factorizations of of the form
This statement also holds for semigroups.
For monoids, this theorem can be written more concisely:
Let be a monoid, let be a submonoid of, and let. Then if and only if in the tensor product.

Application

We show that: Let to be ring homomorphisms, and,. When for all, then for all.
as required.