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Polynomial \(k\)-ary operations, matrices, and \(k\)-mappings. (English) Zbl 1213.20068
The connection between the product of two matrices of order \(k\times k\) over a field and the product of the \(k\)-mappings corresponding to the \(k\)-operations defined by these matrices, is established. In this respect, the author considers a \(k\)-permutation \(\overline\theta\) with components that are polynomial \(k\)-operations defined by a non-singular matrix \(A\). Then the inverse matrix \(A^{-1}\) is defined by the components of the bijective \(k\)-mapping inverse to the \(k\)-permutation \(\overline\theta\). Unlike the binary case, for \(k\geq 3\) it is proved that the components of the \(k\)-permutation inverse to a \(k\)-permutation, all components of which are polynomial \(k\)-quasigroups, are not necessarily \(k\)-quasigroups, although they are invertible at least in two places. Finally, some transformations of orthogonal systems of polynomial \(k\)-operations over a field are investigated.
20N15 \(n\)-ary systems \((n\ge 3)\)
08A40 Operations and polynomials in algebraic structures, primal algebras
20N05 Loops, quasigroups
05B15 Orthogonal arrays, Latin squares, Room squares
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