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 .
Zig-zag: If is a submonoid of a monoid (or a subsemigroup of a semigroup) , 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: Let be a submonoid of a monoid (or a subsemigroup of a semigroup) . 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:
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 .
We show that: Let to be ring homomorphisms, and , . When for all , then for all .
as required.