# zbMATH — the first resource for mathematics

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.
##### MSC:
 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
Full Text: