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 is an epimorphism if and only if , furthermore, a map is an epimorphism if and only if .[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
editZig-zag
editZig-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;
in which and , 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 .
Dominion
editDominion:[5][6] Let U be a submonoid of a monoid (or a subsemigroup of a semigroup) S. The dominion is the set of all elements such that, for all homomorphisms coinciding on U, .
We call a subsemigroup U of a semigroup U closed if , and dense if .[2][12]
Isbell's zigzag theorem
editIsbell's zigzag theorem:[13]
If U is a submonoid of a monoid S then if and only if either 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
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 . Then if and only if in the tensor product .
Application
edit- Let U be a commutative subsemigroup of a semigroup S. Then is commutative.[10]
- Every epimorphism 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 is an epimorphism in the category of all rings and ring homomorphisms by proving that any pair of ring homomorphisms which agree on are fact equal.
A proof sketch for example of non-surjective epimorphism in the category of rings by using zig-zag
|
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We show that: Let to be ring homomorphisms, and , . When for all , then for all .
as required. |
See also
editReferences
editCitations
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.
Further reading
edit- Alam, Noor; Higgins, Peter M.; Khan, Noor Mohammad (2020). "Epimorphisms, dominions and -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
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.