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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     author = {Parascovia Syrbu and Dina Ceban},
     title = {On $\pi$-quasigroups of type~$T_1$},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {36--43},
     publisher = {mathdoc},
     number = {2},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2014_2_a4/}
}
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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/