UNIVERSITATIS APULENSIS No 15 / 2008 G - N - QUASIGROUPS

In this paper we present criteria for an n-quasigroup to be isotopic to an n-group. We call a such n-quasigroup G−n-quasigroup. Applications to functional equations on quasigroups are presented in a subsequent paper. 2000 Mathematics Subject Classification: 20N15. Some important n-quasigroup classes are the following. An n-quasigroup (A,α) of the form α(x1 ) = n ∑ i=1 fi(xi)+a, where (A,+) is a group, f1, . . . , fn are some automorphisms of (A,+), a is some fixed element ofA is called linear nquasigroup (over group (A,+)). A linear quasigroup over an abelian group is called T − n-quasigroup. An n-quasigroup with identity α(α(x 11 ), . . . , α(x nn n1 )) = α(α(x n1 11 ), . . . , α(x nn 1n ) is called medial n-quasigroup. All these quasigroups are isotopic to n-groups. This motivates the purpose of our work to find criteria for an n-quasigroup to be isotopic to an n-group.