Geodesic
On $\pi$-quasigroups of type~$T_1$
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2014), pp. 36-43.

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Quasigroups satisfying the identity $x(x\cdot xy)=y$ are called $\pi$-quasigroups of type $T_1$. The spectrum of the defining identity is precisely $q=0$ or $1\pmod3$, except for $q=6$. Necessary conditions when a finite $\pi$-quasigroup of type $T_1$ has the order $q=0\pmod3$, are given. In particular, it is proved that a finite $\pi$-quasigroup of type $T_1$ such that the order of its inner mapping group is not divisible by three has a left unit. Necessary and sufficient conditions when the identity $x(x\cdot xy)=y$ is invariant under the isotopy of quasigroups (loops) are found. 
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Parascovia Syrbu; Dina Ceban. On $\pi$-quasigroups of type~$T_1$. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2014), pp. 36-43. http://geodesic.mathdoc.fr/item/BASM_2014_2_a4/

[1] Belousov V., “Parastrophic-orthogonal quasigroups”, Quasigroups and Related Systems, 14 (2005), 3–51 | MR

[2] Belousov V., Foundations of the theory of quasigroups and loops, Nauka, Moscow, 1967 (in Russian) | MR | Zbl

[3] Belousov V., “Balanced identities in quasigroups”, Mat. Sb., 70(112):1 (1966), 55–97 (in Russian) | MR | Zbl

[4] Belousov V., Gwaramija A., “On Stein quasigroups”, Soob. Gruz. SSR, 44 (1966), 537–544 (in Russian) | MR | Zbl

[5] Bennett F. E., “The spectra of a variety of quasigroups and related combinatorial designs”, Discrete Math., 77 (1989), 29–50 | DOI | MR | Zbl

[6] Bennett F. E., “Quasigroups”, Handbook of Combinatorial Designs, eds. C. J. Colbourn, J. H. Dinitz, CRC Press, 1996 | MR

[7] Kepka T., Nemec P., “$T$-quasigroups, I”, Acta univ. Carolinae. Math. Phys., 1:12 (1971), 31–39 | MR

[8] Kinyon M. K., Phillips J. D., “Commutants of Bol loops of odd order”, Proc. Amer. Math. Soc., 132 (2004), 617–619 | DOI | MR | Zbl

[9] Pflugfelder H. O., Quasigroups and loops: introduction, Sigma Series in Pure Math, 1990 | MR

[10] Syrbu P., “On $\pi $-quasigroups isotopic to abelian groups”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2009, no. 3(61), 109–117 | MR | Zbl