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 ${\displaystyle U\hookrightarrow S}$ is an epimorphism if and only if ${\displaystyle {\rm {{Dom}_{S}(U)=S}}}$, furthermore, a map ${\displaystyle \alpha \colon S\to T}$ is an epimorphism if and only if ${\displaystyle {\rm {{Dom}_{T}({\rm {{im}\;\alpha )=T}}}}}$. ^[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

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;

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

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

Dominion

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

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

Isbell's zigzag theorem

Isbell's zigzag theorem: ^[13]

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

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

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

Application

• Let U be a commutative subsemigroup of a semigroup S. Then ${\displaystyle {\rm {{Dom}_{S}(U)}}}$ is commutative. ^[10]
• Every epimorphism ${\displaystyle \alpha \colon S\to T}$ 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 ${\displaystyle i:(\mathbb {Z} ,\cdot )\hookrightarrow (\mathbb {Q} ,\cdot )}$ is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of ring homomorphisms ${\displaystyle \beta ,\gamma$ :\mathbb {Q} \to \mathbb {R} } which agree on ${\displaystyle \mathbb {Z} }$ 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 ${\displaystyle \beta ,\gamma }$ to be ring homomorphisms, and ${\displaystyle n,m\in \mathbb {Z} }$, ${\displaystyle n\neq 0}$. When ${\displaystyle \beta (m)=\gamma (m)}$ for all ${\displaystyle m\in \mathbb {Z} }$, then ${\displaystyle \beta \left({\frac {m}{n}}\right)=\gamma \left({\frac {m}{n}}\right)}$ for all ${\displaystyle {\frac {m}{n}}\in \mathbb {Q} }$. ${\displaystyle {\begin{aligned}\beta \left({\frac {m}{n}}\right)&=\beta \left({\frac {1}{n}}\cdot m\right)=\beta \left({\frac {1}{n}}\right)\cdot \beta (m)\\&=\beta \left({\frac {1}{n}}\right)\cdot \gamma (m)=\beta \left({\frac {1}{n}}\right)\cdot \gamma \left(mn\cdot {\frac {1}{n}}\right)\\&=\beta \left({\frac {1}{n}}\right)\cdot \gamma (mn)\cdot \gamma \left({\frac {1}{n}}\right)=\beta \left({\frac {1}{n}}\right)\cdot \beta (mn)\cdot \gamma \left({\frac {1}{n}}\right)\\&=\beta \left({\frac {1}{n}}\cdot mn\right)\cdot \gamma \left({\frac {1}{n}}\right)=\beta (m)\cdot \gamma \left({\frac {1}{n}}\right)=\gamma (m)\cdot \gamma \left({\frac {1}{n}}\right)\\&=\gamma \left(m\cdot {\frac {1}{n}}\right)=\gamma \left({\frac {m}{n}}\right),\end{aligned}}}$ as required.

See also

• Epimorphisms

References

Citations

1. ( Isbell 1966 )
2. ( Howie 1996 )
3. ( Higgins 1988 )
4. ( Higgins 1990 )
5. (Hoffman 2008)
6. (Storrer 1976)
7. ( Howie & Isbell 1967 , Theorem 2.3.)
8. ( Hall 1982 )
9. ( Higgins 1986 )
10. ( Higgins 2016 )
11. (Mitchell 1972)
12. ( Higgins 1983 )
13. ( Howie 1996 , Theorem 1.2.)
14. ( Higgins 1985 )
15. ( Stenström 1971 )
16. ( 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. ^{[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.

Further reading

• Alam, Noor; Higgins, Peter M.; Khan, Noor Mohammad (2020). "Epimorphisms, dominions and ${\displaystyle {\mathcal {H}}}$-commutative semigroups". Semigroup Forum. 100 (2): 349–363. arXiv: 1908.01813 . doi:10.1007/s00233-019-10050-z. S2CID   202133305.
• Ahanger, Shabir Ahmad; Shah, Aftab Hussain (2020). "Epimorphisms, dominions and varieties of bands". Semigroup Forum. 100 (3): 641–650. doi:10.1007/s00233-019-10047-8. S2CID   253772526.
• Khan, N. M. (1985). "On saturated permutative varieties and consequences of permutation identities". Journal of the Australian Mathematical Society, Series A. 38 (2): 186–197. doi: 10.1017/S1446788700023041 . S2CID   122979127.
• Isbell, John R. (1968). "Epimorphisms and Dominions, III". American Journal of Mathematics. 90 (4): 1025–1030. doi:10.2307/2373286. JSTOR   2373286.
• Isbell, J. R. (1973). "Epimorphisms and dominions, V". Algebra Universalis. 3 (1): 318–320. doi:10.1007/BF02945133. S2CID   125292076.
• Scheiblich, E. (1976). "On epics and dominions of bands". Semigroup Forum. 13 (1): 103–114. doi:10.1007/BF02194926. S2CID   123580458.
• Philip, J.M (1974). "A proof of Isbell's zig-zag theorem". Journal of Algebra. 32 (2): 328–331. doi: 10.1016/0021-8693(74)90141-0 .
• Higgins, Peter M. (1985). "Epimorphisms, dominions and semigroups". Algebra Universalis. 21 (2–3): 225–233. doi:10.1007/BF01188058. S2CID   121142819.
• Higgins, Peter M. (1992). Techniques of Semigroup Theory. Oxford University Press. ISBN   9780198535775.
• Howie, John M. (1995). "Semigroup amalgams". Fundamentals of Semigroup Theory. Clarendon Press. ISBN   0-19-851194-9. MR   1455373. Zbl   0835.20077.
• Campercholi, Miguel (2018). "Dominions and Primitive Positive Functions". The Journal of Symbolic Logic. 83 (1): 40–54. doi:10.1017/jsl.2017.18. hdl: 11336/88474 . JSTOR   26600306. S2CID   19168037.

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

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