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.[1] Dominion is a concept in semigroup theory, within the study of the properties of epimorphisms. For example, let U is a subsemigroup of S containing U, the inclusion map is an epimorphism if and only if , furthermore, a map is an epimorphism if and only if .[2] 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.[3] Proofs of this theorem are topological in nature, beginning with Isbell (1966) for semigroups, and continuing by Philip (1974), completing Isbell's original proof.[3][4][5] The pure algebraic proofs were given by Howie (1976) and Storrer (1976).[3][4][note 1]

Statement

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Zig-zag

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The dashed line is the spine of the zig-zag.

Zig-zag:[7][2][8][9][10][note 2] If U is a submonoid of a monoid (or a subsemigroup of a semigroup) S, then a system of equalities;

 

in which   and  , is called a zig-zag of length m in S over U with value d. By the spine of the zig-zag we mean the ordered (2m + 1)-tuple  .

Dominion

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Dominion:[5][6] Let U be a submonoid of a monoid (or a subsemigroup of a semigroup) S. The dominion   is the set of all elements   such that, for all homomorphisms   coinciding on U,  .

We call a subsemigroup U of a semigroup U closed if  , and dense if  .[2][12]

Isbell's zigzag theorem

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Isbell's zigzag theorem:[13]

If U is a submonoid of a monoid S then   if and only if either   or there exists a zig-zag in S over U with value d that is, there is a sequence of factorizations of d of the form

 

This statement also holds for semigroups.[7][14][9][4][10]

For monoids, this theorem can be written more concisely:[15][2][16]

Let S be a monoid, let U be a submonoid of S, and let  . Then   if and only if   in the tensor product  .

Application

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  • Let U be a commutative subsemigroup of a semigroup S. Then   is commutative.[10]
  • Every epimorphism   from a finite commutative semigroup S to another semigroup T is surjective.[10]
  • Inverse semigroups are absolutely closed.[7]
  • Example of non-surjective epimorphism in the category of rings:[3] The inclusion   is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of ring homomorphisms   which agree on   are fact equal.
A proof sketch for example of non-surjective epimorphism in the category of rings by using zig-zag

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

 

as required.

See also

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References

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Citations

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Bibliography

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Further reading

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Footnote

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  1. ^ These pure algebraic proofs were based on the tensor product characterization of the dominant elements for monoid by Stenström (1971).[6][4]
  2. ^ See Hoffman[5] or Mitchell[11] for commutative diagram.
  3. ^ Some results were corrected in Isbell (1969).
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