IMCS/Publications/BASM/Issues/BASM n.1 (68), 2012/

Conjugate sets of loops and quasigroups. DC-quasigroups

Authors: Beliavscaia Galina, T. V. Popovich


It is known that the set of conjugates (the conjugate set) of a binary quasigroup can contain 1, 2, 3 or 6 elements. We investigate loops, $IP$-quasigroups and $T$-quasigroups with distinct conjugate sets described earlier. We study in more detail the quasigroups all conjugates of which are pairwise distinct (shortly, $DC$-quasigroups). The criterion of a $DC$-quasigroup (a $DC$-$IP$-quasigroup, a $DC$-$T$-quasigroup) is given, the existence of $DC$-$T$-quasi\-groups for any order $n\geq 5$, $n\neq 6$, is proved and some examples of $DC$-quasigroups are given.

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