Isbell's zigzag theorem

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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, the inclusion map $U\hookrightarrow S$ is an epimorphism if and only if ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}\mathrm{=}\mathrm{S}$, furthermore, a map $\alpha :S\to T$ is an epimorphism if and only if ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{T}}\mathrm{(}\mathrm{i}\mathrm{m}\mathrm{\alpha }\mathrm{)}\mathrm{=}\mathrm{T}$. Proofs of this theorem are topological in nature, beginning with Isbell (1966) for semigroups, and continuing by Philip (1974), completing Isbell's original proof. A pure algebraic proof was given by Howie (1976) and Storrer (1976).

Statement

Zig-zag

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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;

$\begin{array}{rlrl}d& =x_1{u}_1,& u_1& =v_1{y}_1\\ x_{i-1}{v}_{i-1}& =x_i{u}_i,& u_i{y}_{i-1}& =v_i{y}_i(i=2,\dots ,m)\\ x_m{v}_m& =u_{m+1},& u_{m+1}{y}_m& =d\end{array}$

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 ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}$ 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 ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}\mathrm{=}\mathrm{U}$, and dense if ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}\mathrm{=}\mathrm{S}$.

Isbell's zigzag theorem

Isbell's zigzag theorem:

If U is a submonoid of a monoid S then $d\in {\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}$ 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 $d\in {\mathrm{D}\mathrm{o}\mathrm{m}}_S(U)$ 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 ${\mathrm{D}\mathrm{o}\mathrm{m}}_{\mathrm{S}}\mathrm{(}\mathrm{U}\mathrm{)}$ is commutative.
• Every epimorphism $\alpha :S\to T$ 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 $i:(\mathbb{\mathbb{Z}},\cdot )\hookrightarrow (\mathbb{\mathbb{Q}},\cdot )$ is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of ring homomorphisms $\beta ,\gamma :\mathbb{\mathbb{Q}}\to \mathbb{ℝ}$ which agree on $\mathbb{\mathbb{Z}}$ 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 $n,m\in \mathbb{\mathbb{Z}}$, $n\ne 0$. When $\beta (m)=\gamma (m)$ for all $m\in \mathbb{\mathbb{Z}}$, then $\beta \left(\frac{m}{n}\right)=\gamma \left(\frac{m}{n}\right)$ for all $\frac{m}{n}\in \mathbb{\mathbb{Q}}$. $\begin{array}{rl}\beta \left(\frac{m}{n}\right)& =\beta \left(\frac{1}{n}\right)\cdot \beta (m)=\beta \left(\frac{1}{n}\right)\cdot \gamma (m)\\ & =\beta \left(\frac{1}{n}\right)\cdot \gamma (mn)\cdot \gamma \left(\frac{1}{n}\right)=\beta \left(\frac{1}{n}\right)\cdot \beta (mn)\cdot \gamma \left(\frac{1}{n}\right)\\ & =\beta (m)\cdot \gamma \left(\frac{1}{n}\right)=\gamma (m)\cdot \gamma \left(\frac{1}{n}\right)\\ & =\gamma \left(\frac{m}{n}\right),\end{array}$ as required.

See also

• Epimorphisms

References

Bibliography

• Higgins, P. M. (1981). "Epis are onto for generalized inverse semigroups". Semigroup Forum. 23: 255–260. doi:10.1007/BF02676649.
• 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.
• Higgins, Peter M. (1988). "Epimorphisms and amalgams". Colloquium Mathematicum. 56: 1–17. doi:10.4064/cm-56-1-1-17.
• 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. Search this book on
• Hall, T. E. (1982). "Epimorphisms and dominions". Semigroup Forum. 24: 271–284. doi:10.1007/BF02572773.
• 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.
• 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. Search this book on
• 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. Search this book on
• 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. Search this book on
• 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.
• 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.

Further reading

• Alam, Noor; Higgins, Peter M.; Khan, Noor Mohammad (2020). "Epimorphisms, dominions and ${\mathscr{H}}$-commutative semigroups". Semigroup Forum. 100 (2): 349–363. arXiv:1908.01813. doi:10.1007/s00233-019-10050-z.
• 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.
• 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.
• 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.
• Scheiblich, E. (1976). "On epics and dominions of bands". Semigroup Forum. 13 (1): 103–114. doi:10.1007/BF02194926.
• 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.
• Higgins, Peter M. (1992). Techniques of Semigroup Theory. Oxford University Press. ISBN 9780198535775. Search this book on
• Howie, John M. (1995). "Semigroup amalgams". Fundamentals of Semigroup Theory. Clarendon Press. ISBN 0-19-851194-9. MR 1455373. Zbl 0835.20077. Search this book on
• 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.

External link

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

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