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Some classes of linear quasigroups

Nĕmec P.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

1998

Vol. 4

22-33

Several important classes of quasigroups can be characterized by means of certain linear constructions (see e.g. [3], [4], [7], [8]). The first to investigate such linear quasigroups seems to be Toyoda [8] as early as in 1941, who showed that a quasigroup Q is medial iff there is an Abelian group Q(+), two automorphisms f,g of Q and an element a ∈ Q such that fg=gf and xy=f(x)+g(y)=+a for all x,y ∈ Q. Further, Belousov [1] (and independently Soublin [7] showed that a quasigroup Q is distributive iff there is a commutative Moufang loop Q(+) and an automorphism f of Q(+) such that 1 - f is an automorphism, f (x) + x € C(Q(+))} and xy=f(x) + (1-f)(y) for all x,y ∈ Q. As a further generalization in this direction, Kepka [3] proved that a quasigroup is trimedial (i.e., each sub quasigroup generated by at most three elements is medial) iff there is a commutative Moufang loop Q(+), two automorphisms f, g of Q(+) and an element a € C(Q(+)) such that fg = gf and xy = f (x) + g(y) + a for all x,y ∈ Q. These results naturally suggest an idea of defining an arithmetical form of a quasigroup Q as a quadruple (Q(+),,f, g,a) such that Q(+) is a commutative Moufang loop, f, g are automorphisms of Q(+), a E Q and xy = (f (x) + g(y)) + a for all x,y ∈ Q. We shall say that Q is a linear quasigroup if it has at least one arithmetical form. All possible arithmetical forms of a linear quasigroup were characterized in [5] and the structure of commutative Moufang loops occurring in different arithmetical forms of a linear quasigroup was investigated in [6]. This contribution is devoted to the description of some particular classes of linear quasigroups.