Stojaković, Zoran; Dudek, Wiesław A. Generalized Weisner designs and quasigroups. (English) Zbl 1084.20519 Novi Sad J. Math. 28, No. 2, 143-153 (1998). An \(i\)-Weisner \(n\)-quasigroup is defined as an \(n\)-ary quasigroup \((Q;A)\) satisfying: \[ A(A(x_1^n),A(x_2^{n-1},x_1),\dots,A(x_n,x_1^{n-1}))=x_i,\quad 0<i\leq n. \] The identity generalizes the Schröder law (\(xy\cdot yx=x\)) and Stein’s third law (\(xy\cdot yx=y\)) for binary quasigroups. A necessary and sufficient condition for the existence of an \(i\)-W-\(n\) quasigroup is given. It is shown that every \(i\)-W-\(n\) quasigroup of order \(v\) defines \(n\) orthogonal \((n-10)\)-quasigroups of order \(v\). A \(v^3\times 6\) orthogonal array is defined for \(i\)-W-\(n\) quasigroups (\(i=2,3\)) of order \(v\). Some conjugates of \(i\)-W-\(n\) quasigroups are proven to be \(j\)-W-\(n\) quasigroups. A result on spectra of \(i\)-W-\(n\) quasigroups is also given. Reviewer: Aleksandar Krapež (Beograd) MSC: 20N15 \(n\)-ary systems \((n\ge 3)\) 05B30 Other designs, configurations 05B15 Orthogonal arrays, Latin squares, Room squares Keywords:Weisner designs; \(n\)-quasigroups; orthogonal arrays; spectra PDFBibTeX XMLCite \textit{Z. Stojaković} and \textit{W. A. Dudek}, Novi Sad J. Math. 28, No. 2, 143--153 (1998; Zbl 1084.20519) Full Text: EuDML