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 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 ${\rm {{Dom}_{T}({\rm {{im}\;\alpha )=T}}}}$. 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

The blue dashed line is the spine of the 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 is 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 ${\rm {{Dom}_{S}(U)=S}}$.

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 $\beta ,\gamma :\mathbb {Q} \to \mathbb {R}$ 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.

See also

• Epimorphisms

References

Bibliography

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• 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.
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• 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.
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• Howie, John M. (1976). An introduction to semigroup theory. L.M.S. Monographs ; 7. Academic Press. ISBN 9780123569509.
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Further reading

• Alam, Noor; Higgins, Peter M.; Khan, Noor Mohammad (2020). "Epimorphisms, dominions and ${\mathcal {H}}$-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. Pure Mathematics and Statistics. 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.

External link

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