On conjugate sets of quasigroups

It is known that the set of conjugates (the conjugate set) of a binary quasigroup can contain 1, 2, 3 or 6 elements. We establish a connection between different pairs of conjugates and describe all six possible conjugate sets, with regard to the equality (“assembling”) of conjugates. Four identities which correspond to the equality of a quasigroup to its conjugates are pointed out. Every conjugate set is characterized with the help of these identities. The conditions of the equality of a $T$-quasigroup to conjugates are established and some examples of $T$-quasigroups with distinct conjugate sets are given.