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Generalized Weisner designs and quasigroups. (English) Zbl 1084.20519

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.

MSC:

20N15 \(n\)-ary systems \((n\ge 3)\)
05B30 Other designs, configurations
05B15 Orthogonal arrays, Latin squares, Room squares
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