Some classes of linear quasigroups

• Autorzy: Nĕmec P.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

linear structures; arithmetic; linear functions

PL

konstrukcje liniowe; arytmetyka; funkcje liniowe

• Wydawca: Wydawnictwo Uniwersytetu Humanistyczno-Przyrodniczego im. Jana Długosza w Częstochowie
• Czasopismo: Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
• Rocznik: 1998
• Tom: Vol. 4
• Strony: 22--33
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Nĕmec P.

Bibliografia

• [1] V.D. Belousov, Osnovy teorii kvazigrup i lup, Nauka, Moskva, 1967.
• [2] R.H. Bruck, A Survey of Binary Systems, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958.
• [3] T.Kepka, Structure of triabelian quasigroups, Comment. Math. Univ. Carolinae 17 (1976), 229-240.
• [4] T. Kepka, Hamiltonian guasimodules and trimedial quasigroups, Acta Univ. Carolinae Math. Phys. 26,1 (1985), 11-20.
• [5] P. Nĕmec, Arithmetical forms of quasigroups, Comment. Math. Univ. Carolinae 29 (1988), 295-302.
• [6] P. Nĕmec, Commutative Moufang loops corresponding to linear quasigroups, Comment. Math. Univ. Carolinae 29 (1988), 303-308.
• [7] J.-P.Soublin, Etude algébrique de la notion de moyenne, J. Math. Pures Appl. 50 (1971), 53-264.
• [8] K. Toyoda, On axioms of linear functions, Proc. Imp. Acad. Tokyo 17 (1941), 221-227.