Isbell's zigzag theorem

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Short description: Theorem of dominion in abstract algebra

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 [math]\displaystyle{ U \hookrightarrow S }[/math] is an epimorphism if and only if [math]\displaystyle{ \rm{Dom}_S (U) = S }[/math], furthermore, a map [math]\displaystyle{ \alpha \colon S \to T }[/math] is an epimorphism if and only if [math]\displaystyle{ \rm{Dom}_T (\rm{im} \; \alpha) = T }[/math].[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

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;

[math]\displaystyle{ \begin{align} d &= x_1 u_1, &u_1 &= v_1 y_1 \\ x_{i - 1} v_{i - 1} &= x_i u_i, &u_i y_{i - 1} &= v_i y_i \; (i = 2, \dots, m) \\ x_{m} v_{m} &= u_{m+1}, &u_{m + 1} y_{m} &= d \end{align} }[/math]

in which [math]\displaystyle{ u_1, \dots , u_{m + 1}, v_1, \dots , v_{m} \in U }[/math] and [math]\displaystyle{ x_1, \dots , x_{m}, y_1, \dots , y_{m} \in S }[/math], 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 [math]\displaystyle{ (u_1,v_1,u_2,v_2,\dots,u_{m},v_{m}, u_{m+1}) }[/math].

Dominion

Dominion:[5][6] Let U be a submonoid of a monoid (or a subsemigroup of a semigroup) S. The dominion [math]\displaystyle{ \rm{Dom}_S (U) }[/math] is the set of all elements [math]\displaystyle{ s \in S }[/math] such that, for all homomorphisms [math]\displaystyle{ f, g : S \to T }[/math] coinciding on U, [math]\displaystyle{ f(s) = g(s) }[/math].

We call a subsemigroup U of a semigroup U closed if [math]\displaystyle{ \rm{Dom}_S (U) = U }[/math], and dense if [math]\displaystyle{ \rm{Dom}_S (U) = S }[/math].[2][12]

Isbell's zigzag theorem

Isbell's zigzag theorem:[13]

If U is a submonoid of a monoid S then [math]\displaystyle{ d \in \rm{Dom}_S (U) }[/math] if and only if either [math]\displaystyle{ d \in U }[/math] 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

[math]\displaystyle{ d = x_1 u_1=x_1 v_1 y_1 = x_2 u_2 y_1 = x_2 v_2 y_2 = \cdots = x_{m} v_{m} y_{m} = u_{m+1} y_{m} }[/math]

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 [math]\displaystyle{ d \in S }[/math]. Then [math]\displaystyle{ d \in \mathrm{Dom}_{S} (U) }[/math] if and only if [math]\displaystyle{ d \otimes 1 = 1 \otimes d }[/math] in the tensor product [math]\displaystyle{ S \otimes_{U} S }[/math].

Application

  • Let U be a commutative subsemigroup of a semigroup S. Then [math]\displaystyle{ \rm{Dom}_S (U) }[/math] is commutative.[10]
  • Every epimorphism [math]\displaystyle{ \alpha \colon S \to T }[/math] 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 [math]\displaystyle{ i: (\mathbb{Z},\cdot)\hookrightarrow (\mathbb{Q},\cdot) }[/math] is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of ring homomorphisms [math]\displaystyle{ \beta, \gamma: \mathbb{Q} \to \mathbb{R} }[/math] which agree on [math]\displaystyle{ \mathbb{Z} }[/math] 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 [math]\displaystyle{ \beta, \gamma }[/math] to be ring homomorphisms, and [math]\displaystyle{ n,m \in \mathbb{Z} }[/math], [math]\displaystyle{ n \neq 0 }[/math]. When [math]\displaystyle{ \beta(m) = \gamma(m) }[/math] for all [math]\displaystyle{ m \in \mathbb{Z} }[/math], then [math]\displaystyle{ \beta\left(\frac{m}{n}\right) = \gamma\left(\frac{m}{n}\right) }[/math] for all [math]\displaystyle{ \frac{m}{n} \in \mathbb{Q} }[/math].

[math]\displaystyle{ \begin{align} \beta\left(\frac{m}{n}\right) &=\beta\left(\frac{1}{n} \cdot m \right) = \beta\left(\frac{1}{n}\right)\cdot \beta(m)\\ &= \beta\left(\frac{1}{n}\right)\cdot \gamma(m) = \beta\left(\frac{1}{n}\right)\cdot \gamma \left(mn \cdot \frac{1}{n} \right)\\ &= \beta\left(\frac{1}{n}\right)\cdot \gamma(mn) \cdot \gamma\left(\frac{1}{n}\right) = \beta\left(\frac{1}{n}\right) \cdot \beta(mn)\cdot \gamma\left(\frac{1}{n}\right)\\ &= \beta\left(\frac{1}{n} \cdot mn \right)\cdot \gamma \left(\frac{1}{n} \right) = \beta(m )\cdot \gamma\left(\frac{1}{n}\right) = \gamma(m) \cdot \gamma\left(\frac{1}{n}\right)\\ &= \gamma \left(m \cdot \frac{1}{n}\right) = \gamma \left(\frac{m}{n}\right), \end{align} }[/math]

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)
  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)
  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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