Isbell's zigzag theorem

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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 U↪S is an epimorphism if and only if DomS(U)=S, furthermore, a map α:S→T is an epimorphism if and only if DomT(imα)=T.[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

Zig-zag

File:Zigzag theorem 2.svg
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; d=x1u1,u1=v1y1xi−1vi−1=xiui,uiyi−1=viyi(i=2,…,m)xmvm=um+1,um+1ym=d in which u1,…,um+1,v1,…,vm∈U and x1,…,xm,y1,…,ym∈S, 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 (u1,v1,u2,v2,…,um,vm,um+1).

Dominion

Dominion:[5][6] Let U be a submonoid of a monoid (or a subsemigroup of a semigroup) S. The dominion DomS(U) is the set of all elements s∈S such that, for all homomorphisms f,g:S→T coinciding on U, f(s)=g(s). We call a subsemigroup U of a semigroup U closed if DomS(U)=U, and dense if DomS(U)=S.[2][12]

Isbell's zigzag theorem

Isbell's zigzag theorem:[13] If U is a submonoid of a monoid S then d∈DomS(U) if and only if either d∈U 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 d=x1u1=x1v1y1=x2u2y1=x2v2y2=⋯=xmvmym=um+1ym 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 d∈S. Then d∈DomS(U) if and only if d⊗1=1⊗d in the tensor product S⊗US.

Application

  • Let U be a commutative subsemigroup of a semigroup S. Then DomS(U) is commutative.[10]
  • Every epimorphism α:S→T 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 i:(ℤ,⋅)↪(ℚ,⋅) is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of ring homomorphisms β,γ:ℚ→ℝ which agree on ℤ 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 β,γ to be ring homomorphisms, and n,m∈ℤ, n≠0. When β(m)=γ(m) for all m∈ℤ, then β(mn)=γ(mn) for all mn∈ℚ. β(mn)=β(1n⋅m)=β(1n)⋅β(m)=β(1n)⋅γ(m)=β(1n)⋅γ(mn⋅1n)=β(1n)⋅γ(mn)⋅γ(1n)=β(1n)⋅β(mn)⋅γ(1n)=β(1n⋅mn)⋅γ(1n)=β(m)⋅γ(1n)=γ(m)⋅γ(1n)=γ(m⋅1n)=γ(mn), as required.

See also

References

Citations

  1. ↑ (Isbell 1966)
  2. ↑ 2.0 2.1 2.2 2.3 (Howie 1996)
  3. ↑ 3.0 3.1 3.2 3.3 (Higgins 1988)
  4. ↑ 4.0 4.1 4.2 4.3 (Higgins 1990)
  5. ↑ 5.0 5.1 5.2 (Hoffman 2008)
  6. ↑ 6.0 6.1 (Storrer 1976)
  7. ↑ 7.0 7.1 7.2 (Howie & Isbell 1967, Theorem 2.3.)
  8. ↑ (Hall 1982)
  9. ↑ 9.0 9.1 (Higgins 1986)
  10. ↑ 10.0 10.1 10.2 10.3 (Higgins 2016)
  11. ↑ (Mitchell 1972)
  12. ↑ (Higgins 1983)
  13. ↑ (Howie 1996, Theorem 1.2.)
  14. ↑ (Higgins 1985)
  15. ↑ (Stenström 1971)
  16. ↑ (Renshaw 2002)

Bibliography

Further reading

Footnote

  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).

External links

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