Generalized Weisner Designs and Quasigroups
نویسندگان
چکیده
An n-ary quasigroup (Q, A) such that for some i ∈ {1, . . . , n} the identity A(A(x1 ), A(x n−1 2 , x1), . . . , A(xn, x n−1 1 )) = xi holds is called an i-Weisner n-quasigroup ( i-W-n-quasigroup ). iW-nquasigroups represent a generalization of quasigroups satisfying Schröder law (xy · yx = x) and quasigroups satisfying Stein’s third law (xy · yx = y). Properties of i-W-n-quasigroups which are satisfied for all i are determined. Necessary and sufficient conditions for an n-quasigroup to be an i-W-n-quasigroup are obtained. It is proved that every i-W-nquasigroup of order v defines an orthogonal set of n (n−1)-quasigroups of order v. It is shown that some i-W-n-quasigroups are equivalent to orthogonal arrays. Conjugates of i-W-n-quasigroups are investigated, connections among these conjugates for different values of n, i are established. The existence of several classes of i-W-n-quasigroups is proved. AMS Mathematics Subject Classification (1980): 05B30, 20N15
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