| Title:
|
Semisymmetrization and Mendelsohn quasigroups (English) |
| Author:
|
Smith, Jonathan D. H. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
553-566 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The semisymmetrization of an arbitrary quasigroup builds a semisymmetric quasigroup structure on the cube of the underlying set of the quasigroup. It serves to reduce homotopies to homomorphisms. An alternative semisymmetrization on the square of the underlying set was recently introduced by A. Krapež and Z. Petrić. Their construction in fact yields a Mendelsohn quasigroup, which is idempotent as well as semisymmetric. We describe it as the Mendelsohnization of the original quasigroup. For quasigroups isotopic to an abelian group, the relation between the semisymmetrization and the Mendelsohnization is studied. It is shown that the semisymmetrization is the total space for an action of the Mendelsohnization on the abelian group. The Mendelsohnization of an abelian group isotope is then identified as the idempotent replica of its semisymmetrization, with fibers isomorphic to the abelian group. (English) |
| Keyword:
|
semisymmetric |
| Keyword:
|
quasigroup |
| Keyword:
|
Mendelsohn triple system |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 07332729 |
| idMR:
|
MR4230960 |
| DOI:
|
10.14712/1213-7243.2021.001 |
| . |
| Date available:
|
2021-02-25T12:48:07Z |
| Last updated:
|
2023-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148665 |
| . |
| Reference:
|
[1] Chernousov V., Elduque A., Knus M.-A., Tignol J.-P.: Algebraic groups of type $D_4$, triality, and composition algebras.Doc. Math. 18 (2013), 413–468. MR 3084556 |
| Reference:
|
[2] Colbourn C. J., Rosa A.: Triple Systems.Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Zbl 1030.05017, MR 1843379 |
| Reference:
|
[3] Curtis R. T.: A classification of Howard Eve's `equihoops'.preprint, Bowdoin College, Brunswick, ME, 1979. |
| Reference:
|
[4] Donovan D. M., Griggs T. S, McCourt T. S., Opršal J., Stanovský D.: Distributive and anti-distributive Mendelsohn triple systems.Canad. Math. Bull. 59 (2016), no. 1, 36–49. MR 3451896, 10.4153/CMB-2015-053-2 |
| Reference:
|
[5] Drápal A.: On multiplication groups of relatively free quasigroups isotopic to Abelian groups.Czechoslovak Math. J. 55 (2005), no. 1, 61–86. MR 2121656, 10.1007/s10587-005-0004-2 |
| Reference:
|
[6] Goračinova-Ilieva L., Markovski S.: Construction of Mendelsohn designs by using quasigroups of $(2,q)$-varieties.Comment. Math. Univ. Carolin. 57 (2016), no. 4, 501–514. MR 3583302 |
| Reference:
|
[7] Holshouser A., Klein B., Reiter H.: The commutative equihoop and the card game SET.Pi Mu Epsilon J. 14 (2015), no. 3, 175-–190. MR 3445104 |
| Reference:
|
[8] Im B., Ko H.-J., Smith J. D. H.: Semisymmetrizations of abelian group isotopes.Taiwanese J. Math. 11 (2007), no. 5, 1529–1534. MR 2368669, 10.11650/twjm/1500404884 |
| Reference:
|
[9] Im B., Nowak A. W., Smith J. D. H.: Algebraic properties of quantum quasigroups.J. Pure Appl. Algebra 225 (2021), no. 3, 106539, 35 pages. MR 4137718, 10.1016/j.jpaa.2020.106539 |
| Reference:
|
[10] Jacobson N.: Lie Algebras.Interscience Tracts in Pure and Applied Mathematics, 10, Interscience Publishers (a division of John Wiley & Sons), New York, 1962. Zbl 0333.17009, MR 0143793 |
| Reference:
|
[11] Ježek J., Kepka T.: Quasigroups, isotopic to a group.Comment. Math. Univ. Carolinae 16 (1975), 59–-76. MR 0367103 |
| Reference:
|
[12] Krapež A., Petrić Z.: A note on semisymmetry.Quasigroups Related Systems 25 (2017), no. 2, 269–278. MR 3738007 |
| Reference:
|
[13] MacLane S.: Categories for the Working Mathematician.Graduate Texts in Mathematics, 5, Springer, New York, 1971. Zbl 0705.18001, MR 0354798 |
| Reference:
|
[14] Mal'cev A. I.: Multiplication of classes of algebraic systems.Sibirsk. Mat. Ž. 8 (1967), 346–365 (Russian); translated in Siberian Math. J. 8 (1967), 54–-267; The metamathematics of algebraic systems. Collected papers: 1936–1967; translated by B. F. Wells, III., Studies in Logic and the Foundations of Mathematics, 66, North-Holland Publishing, Amsterdam, 1971, pages 422-–446. MR 0213276 |
| Reference:
|
[15] Mendelsohn N. S.: A natural generalization of Steiner triple systems.Computers in number theory, Proc. Sci. Res. Council Atlas Sympos., No. 2, Oxford, 1969, Academic Press, London, 1971, pages 323–338. MR 0321755 |
| Reference:
|
[16] Nowak A.: Distributive Mendelsohn triple systems and the Eisenstein integers.available at arXiv: 1908.04966 [math.CO] (2019), 30 pages. MR 4158514 |
| Reference:
|
[17] Okubo S.: Introduction to Octonion and Other Non-Associative Algebras in Physics.Montroll Memorial Lecture Series in Mathematical Physics, 2, Cambridge University Press, Cambridge, 1995. MR 1356224 |
| Reference:
|
[18] Okubo S., Osborn J. M.: Algebras with nondegenerate associative symmetric bilinear forms permitting composition.Comm. Algebra 9 (1981), no. 12, 1233–1261. MR 0618901, 10.1080/00927878108822644 |
| Reference:
|
[19] Paige L. J.: A class of simple Moufang loops.Proc. Amer. Math. Soc. 7 (1956), 471–482. Zbl 0070.25302, MR 0079596, 10.1090/S0002-9939-1956-0079596-1 |
| Reference:
|
[20] Petersson H. P.: Eine Identität fünften Grades, der gewisse Isotope von Kompositions-Algebren genügen.Math. Z. 109 (1969), 217–238 (German). MR 0242910, 10.1007/BF01111407 |
| Reference:
|
[21] Romanowska A. B., Smith J. D. H.: Modal Theory: An Algebraic Approach to Order, Geomtery, and Convexity.Research and Exposition in Mathematics, 9, Heldermann, Berlin, 1985. MR 0788695 |
| Reference:
|
[22] Romanowska A. B., Smith J. D. H.: Modes.World Scientific Publishing Co., River Edge, 2002. Zbl 1060.08009, MR 1932199 |
| Reference:
|
[23] Shcherbacov V.: Elements of Quasigroup Theory and Applications.Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, 2017. MR 3644366 |
| Reference:
|
[24] Smith J. D. H.: Mal'cev Varieties.Lecture Notes in Mathematics, 554, Springer, Berlin, 1976. Zbl 0344.08002, MR 0432511, 10.1007/BFb0095447 |
| Reference:
|
[25] Smith J. D. H.: Homotopy and semisymmetry of quasigroups.Algebra Universalis 38 (1997), no. 2, 175–184. MR 1608968, 10.1007/s000120050046 |
| Reference:
|
[26] Smith J. D. H.: An Introduction to Quasigroups and Their Representations.Studies in Advanced Mathematics, Chapman and Hall/CRC, Boca Raton, 2007. Zbl 1122.20035, MR 2268350 |
| Reference:
|
[27] Smith J. D. H.: Four lectures on quasigroup representations.Quasigroups Related Systems 15 (2007), no. 1, 109–140. MR 2379128 |
| Reference:
|
[28] Smith J. D. H.: Evans' normal form theorem revisited.Internat. J. Algebra Comput. 17 (2007), no. 8, 1577–1592. MR 2378053 |
| Reference:
|
[29] Smith J. D. H.: Quasigroup homotopies, semisymmetrization, and reversible automata.Internat. J. Algebra Comput. 18 (2008), no. 7, 1203–1221. MR 2468744, 10.1142/S0218196708004846 |
| Reference:
|
[30] Smith J. D. H., Romanowska A. B.: Post-Modern Algebra.Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, New York, 1999. Zbl 0946.00001, MR 1673047 |
| Reference:
|
[31] Smith J. D. H., Vojtěchovský P.: Okubo quasigroups.preprint, 2019. |
| Reference:
|
[32] Soublin J.-P.: Médiations.C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A115-–A117 (French). MR 0200374 |
| Reference:
|
[33] Stein S. K.: On the foundations of quasigroups.Trans. Amer. Math. Soc. 85 (1957), 228-–256. MR 0094404, 10.1090/S0002-9947-1957-0094404-6 |
| Reference:
|
[34] Zorn M.: Alternativkörper und quadratische Systeme.Abh. Math. Sem. Univ. Hamburg 9 (1933), 395–402 (German). MR 3069613, 10.1007/BF02940661 |
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