Geodesic
On Commutativity and Mediality of Polyagroup Cross Isomorphs
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2005), pp. 141-152 
F. M. Sokhatsky; O. V. Yurevych. On Commutativity and Mediality of Polyagroup Cross Isomorphs. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2005), pp. 141-152. http://geodesic.mathdoc.fr/item/BASM_2005_3_a11/
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     title = {On {Commutativity} and {Mediality} of {Polyagroup} {Cross} {Isomorphs}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The notion of isotopy (cross isomorphism) of $n$-ary operations can be got from the well-known notion of isotopy (isomorphism) by replacing one of its components with a $k$-ary $m$-invertible operation [1, 2]. The idea of consideration of cross isotopy belongs to V. D. Belousov [3], who defined it for binary quasigroups. In the paper necessary and sufficient conditions for commutativity and mediality of a polyagroup cross isomorph (when $n>2k$) are determined. A neutrality criterion of an arbitrary element is stated.

[1] Sokhatsky F. M., “On cross isotopy and cross isomorphism”, Trudy instytuta prykladnoyi matematyky i mekhaniky NAN Ukrainy, 2005, no. 11, 23–33 (in Ukrainian) | MR

[2] Yurevych O. V., “On cross isotopy of polyagrups”, Trudy instytuta prykladnoyi matematyky i mekhaniky NAN Ukrainy, 2005, no. 11, 34–49 (in Ukrainian) | MR

[3] Belousov V. D., “Cross isotopy of quasigroups”, Quasigroups and their systems, Shtiintsa, Chishinau, 1990, 14–20 (in Russian) | MR

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[6] Sokhatsky F. N., “On associativity of multiary operation”, Diskretnaya mathematika, 4:1 (1992), 66–84 (in Russian) | MR

[7] Sokhatsky F. M., “On isotopes group, I”, Ukrainian mathematical journal, 1995, no. 10, 1387–1398 | MR