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 $U$ be a subsemigroup of $S$ containing $U$ , the inclusion map $U\hookrightarrow S$ is an epimorphism if and only if ${\rm {{Dom}_{S}(U)=S}}$ , furthermore, a map $\alpha \colon S\to T$ is an epimorphism if and only if ${\rm {{Dom}_{T}({\rm {{im}\;\alpha )=T}}}}$ . 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 Isbell (1966) for semigroups, and continuing by Philip (1974), completing Isbell's original proof. The pure algebraic proofs were given by Howie (1976) and Storrer (1976).

Statement

Zig-zag

Zig-zag: If $U$ is a submonoid of a monoid (or a subsemigroup of a semigroup) $S$ , then a system of equalities;

${\begin{aligned}d&=x_{1}u_{1},&u_{1}&=v_{1}y_{1}\\x_{i-1}v_{i-1}&=x_{i}u_{i},&u_{i}y_{i-1}&=v_{i}y_{i}\;(i=2,\dots ,m)\\x_{m}v_{m}&=u_{m+1},&u_{m+1}y_{m}&=d\end{aligned}}$

in which $u_{1},\dots ,u_{m+1},v_{1},\dots ,v_{m}\in U$ and $x_{1},\dots ,x_{m},y_{1},\dots ,y_{m}\in S$ , 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 $(u_{1},v_{1},u_{2},v_{2},\dots ,u_{m},v_{m},u_{m+1})$ .

Dominion

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

We call a subsemigroup $U$ of a semigroup $S$ closed if ${\rm {{Dom}_{S}(U)=U}}$ , and dense if ${\rm {{Dom}_{S}(U)=S}}$ .

Isbell's zigzag theorem

Isbell's zigzag theorem:

If $U$ is a submonoid of a monoid $S$ then $d\in {\rm {{Dom}_{S}(U)}}$ if and only if either $d\in U$ 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

$d=x_{1}u_{1}=x_{1}v_{1}y_{1}=x_{2}u_{2}y_{1}=x_{2}v_{2}y_{2}=\cdots =x_{m}v_{m}y_{m}=u_{m+1}y_{m}$

This statement also holds for semigroups.

For monoids, this theorem can be written more concisely:

Let $S$ be a monoid, let $U$ be a submonoid of $S$ , and let $d\in S$ . Then $d\in \mathrm {Dom} _{S}(U)$ if and only if $d\otimes 1=1\otimes d$ in the tensor product $S\otimes _{U}S$ .

Application

Let $U$ be a commutative subsemigroup of a semigroup $S$ . Then ${\rm {{Dom}_{S}(U)}}$ is commutative.
Every epimorphism $\alpha \colon S\to T$ from a finite commutative semigroup $S$ to another semigroup $T$ is surjective.
Inverse semigroups are absolutely closed.
Example of non-surjective epimorphism in the category of rings: The inclusion $i:(\mathbb {Z} ,\cdot )\hookrightarrow (\mathbb {Q} ,\cdot )$ is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of ring homomorphisms ${\displaystyle \beta ,\gamma$ to any ring $R$ which agree on $\mathbb {Z}$ 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 $\beta ,\gamma$ to be ring homomorphisms, and $n,m\in \mathbb {Z}$ , $n\neq 0$ . When $\beta (m)=\gamma (m)$ for all $m\in \mathbb {Z}$ , then $\beta \left({\frac {m}{n}}\right)=\gamma \left({\frac {m}{n}}\right)$ for all ${\frac {m}{n}}\in \mathbb {Q}$ .

${\begin{aligned}\beta \left({\frac {m}{n}}\right)&=\beta \left({\frac {1}{n}}\cdot m\right)=\beta \left({\frac {1}{n}}\right)\cdot \beta (m)\\&=\beta \left({\frac {1}{n}}\right)\cdot \gamma (m)=\beta \left({\frac {1}{n}}\right)\cdot \gamma \left(mn\cdot {\frac {1}{n}}\right)\\&=\beta \left({\frac {1}{n}}\right)\cdot \gamma (mn)\cdot \gamma \left({\frac {1}{n}}\right)=\beta \left({\frac {1}{n}}\right)\cdot \beta (mn)\cdot \gamma \left({\frac {1}{n}}\right)\\&=\beta \left({\frac {1}{n}}\cdot mn\right)\cdot \gamma \left({\frac {1}{n}}\right)=\beta (m)\cdot \gamma \left({\frac {1}{n}}\right)=\gamma (m)\cdot \gamma \left({\frac {1}{n}}\right)\\&=\gamma \left(m\cdot {\frac {1}{n}}\right)=\gamma \left({\frac {m}{n}}\right),\end{aligned}}$

as required.

References

Citations

↑ (Isbell 1966)
1 2 3 4 (Howie 1996)
1 2 3 4 (Higgins 1988)
1 2 3 4 (Higgins 1990)
1 2 3 (Hoffman 2008)
1 2 (Storrer 1976)
1 2 3 (Howie & Isbell 1967, Theorem 2.3.)
↑ (Hall 1982)
1 2 (Higgins 1986)
1 2 3 4 (Higgins 2016)
↑ (Mitchell 1972)
↑ (Higgins 1983)
↑ (Howie 1996, Theorem 1.2.)
↑ (Higgins 1985)
↑ (Stenström 1971)
↑ (Renshaw 2002)

Bibliography

Higgins, P. M. (1981). "Epis are onto for generalized inverse semigroups". Semigroup Forum. 23: 255–260. doi:10.1007/BF02676649. S2CID 122139547. Archived from the original on 2022-11-30. Retrieved 2023-08-20.
Higgins, P. M. (1983). "The determination of all varieties consisting of absolutely closed semigroups". Proceedings of the American Mathematical Society. 87 (3): 419–421. doi:10.1090/S0002-9939-1983-0684630-8.
Higgins, Peter M. (1986). "Completely semisimple semigroups and epimorphisms". Proceedings of the American Mathematical Society. 96 (3): 387–390. doi:10.1090/S0002-9939-1986-0822424-3. S2CID 123529614.
Higgins, Peter M. (1988). "Epimorphisms and amalgams". Colloquium Mathematicum. 56: 1–17. doi:10.4064/cm-56-1-1-17. Archived from the original on 2023-07-29. Retrieved 2023-07-29.
Higgins, Peter M. (1990). "A short proof of Isbell's zigzag theorem". Pacific Journal of Mathematics. 144 (1): 47–50. doi:10.2140/pjm.1990.144.47.
Higgins, Peter M. (2016). "Ramsey's theorem in algebraic semigroup". First International Tainan-Moscow Algebra Workshop: Proceedings of the International Conference held at National Cheng Kung University Tainan, Taiwan, Republic of China, July 23–August 22, 1994. Walter de Gruyter GmbH & Co KG. ISBN 9783110883220. Archived from the original on August 13, 2023. Retrieved August 13, 2023.
Hall, T. E. (1982). "Epimorphisms and dominions". Semigroup Forum. 24: 271–284. doi:10.1007/BF02572773. S2CID 120129600. Archived from the original on 2023-08-11. Retrieved 2023-08-10.
Hall, T. E.; Jones, P. R. (1983). "Epis are onto for finite regular semigroups". Proceedings of the Edinburgh Mathematical Society. 26 (2): 151–162. doi:10.1017/S0013091500016850. S2CID 120509107.
Isbell, John R. (1966). "Epimorphisms and Dominions". Proceedings of the Conference on Categorical Algebra. pp. 232–246. doi:10.1007/978-3-642-99902-4_9. ISBN 978-3-642-99904-8. Archived from the original on 2023-07-26. Retrieved 2023-07-26.
Howie, J.M; Isbell, J.R (1967). "Epimorphisms and dominions. II". Journal of Algebra. 6: 7–21. doi:10.1016/0021-8693(67)90010-5.
Howie, John M. (1976). An introduction to semigroup theory. L.M.S. Monographs; 7. Academic Press. ISBN 9780123569509.
Howie, John M. (1996). "Isbell's zigzag theorem and its consequences". Semigroup Theory and its Applications. pp. 81–92. doi:10.1017/CBO9780511661877.007. ISBN 9780521576697. Archived from the original on 2023-08-05. Retrieved 2023-08-05.
Isbell, John R. (1969). "Epimorphisms and Dominions. IV". Journal of the London Mathematical Society: 265–273. doi:10.1112/jlms/s2-1.1.265.
Hoffman, Piotr (2008). "A Proof of Isbell's Zigzag Theorem". Journal of the Australian Mathematical Society. 84 (2): 229–232. doi:10.1017/S1446788708000384. S2CID 55107808.
Mitchell, Barry (1972). "The Dominion of Isbell". Transactions of the American Mathematical Society. 167: 319–331. doi:10.1090/S0002-9947-1972-0294441-0. JSTOR 1996142.
Renshaw, James (2002). "On Free Products of Semigroups and a New Proof of Isbell's Zigzag Theorem". Journal of Algebra. 251 (1): 12–15. doi:10.1006/jabr.2002.9143.
Stenström, Bo (1971). "Flatness and localization over monoids". Mathematische Nachrichten. 48 (1–6): 315–334. doi:10.1002/mana.19710480124.
Storrer, H. (1976). "An algebraic proof of Isbell' s zigzag theorem". Semigroup Forum. 12: 83–88. doi:10.1007/BF02195912. S2CID 121208494. Archived from the original on 2022-07-17. Retrieved 2023-07-31.

Footnote

↑ These pure algebraic proofs were based on the tensor product characterization of the dominant elements for monoid by Stenström (1971).
↑ See Hoffman or Mitchell for commutative diagram.
↑ Some results were corrected in Isbell (1969).

External links

Nicol, Andrew W. "WHAT IS... THE ZIGZAG THEOREM?" (PDF). S2CID 51745066.

[1] (Isbell 1966)

[Howie1996-2] 1 2 3 4 (Howie 1996)

[Higgins1988-3] 1 2 3 4 (Higgins 1988)

[Higgins1990-4] 1 2 3 4 (Higgins 1990)

[Hoffman-5] 1 2 3 (Hoffman 2008)

[Storrer1976-6] 1 2 (Storrer 1976)

[HowieIsbell1967-8] 1 2 3 (Howie & Isbell 1967, Theorem 2.3.)

[9] (Hall 1982)

[Higgins1986-10] 1 2 (Higgins 1986)

[Higgins2016-11] 1 2 3 4 (Higgins 2016)

[12] (Mitchell 1972)

[14] (Higgins 1983)

[15] (Howie 1996, Theorem 1.2.)

[16] (Higgins 1985)

[17] (Stenström 1971)

[18] (Renshaw 2002)

[7] These pure algebraic proofs were based on the tensor product characterization of the dominant elements for monoid by Stenström (1971).

[13] See Hoffman or Mitchell for commutative diagram.

[19] Some results were corrected in Isbell (1969).

Isbell's zigzag theorem

Statement

Zig-zag

Dominion

Isbell's zigzag theorem

Application

See also

References

Citations

Bibliography

Further reading

Footnote

External links