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 $U\hookrightarrow S$ 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 ${\rm {{Dom}_{T}({\rm {{im}\;\alpha )=T}}}}$ .^[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 edit

Zig-zag edit

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;

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

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 edit

Dominion:^[5]^[6] 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 $U$ closed if ${\rm {{Dom}_{S}(U)=U}}$ , and dense if ${\rm {{Dom}_{S}(U)=S}}$ .^[2]^[12]

Isbell's zigzag theorem edit

Isbell's zigzag theorem:^[13]

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.^[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 $d\in S$ . 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 edit

Let $U$ be a commutative subsemigroup of a semigroup $S$ . Then ${\rm {{Dom}_{S}(U)}}$ is commutative.^[10]
Every epimorphism $\alpha \colon S\to T$ 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 $i:(\mathbb {Z} ,\cdot )\hookrightarrow (\mathbb {Q} ,\cdot )$ is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of ring homomorphisms $\beta ,\gamma :\mathbb {Q} \to \mathbb {R}$ 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 edit

Citations edit

^ (Isbell 1966)
^ ^a ^b ^c ^d (Howie 1996)
^ ^a ^b ^c ^d (Higgins 1988)
^ ^a ^b ^c ^d (Higgins 1990)
^ ^a ^b ^c (Hoffman 2008)
^ ^a ^b (Storrer 1976)
^ ^a ^b ^c (Howie & Isbell 1967, Theorem 2.3.)
^ (Hall 1982)
^ ^a ^b (Higgins 1986)
^ ^a ^b ^c ^d (Higgins 2016)
^ (Mitchell 1972)
^ (Higgins 1983)
^ (Howie 1996, Theorem 1.2.)
^ (Higgins 1985)
^ (Stenström 1971)
^ (Renshaw 2002)

Bibliography edit

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.^{[note 3]}
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 edit

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

External links edit

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

[1] (Isbell 1966)

[Howie1996-2] (Howie 1996)

[Higgins1988-3] (Higgins 1988)

[Higgins1990-4] (Higgins 1990)

[Hoffman-5] (Hoffman 2008)

[Storrer1976-6] (Storrer 1976)

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

[9] (Hall 1982)

[Higgins1986-10] (Higgins 1986)

[Higgins2016-11] (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).^[6]^[4]

[13] See Hoffman^[5] or Mitchell^[11] for commutative diagram.

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

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