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Quadratic quasigroup varieties

1.
On quasigroup varieties closed under isotopy, Publ. Inst. Math. Beograd 39 (1986), 89-95.

2.
Quadratic functional equations on quasigroups (with A. Krapez), preprint (1986).

3.
Quadratic varieties of quasigroups, preprint (1986).



SUMMARY:


This is my thesis (Belgrade 1986, in Serbo-Croatian), of which only a small part [1] has been published. A theory is developed for solving functional equations which are quadratic in the sense that every variable occurs twice, and in which unknown functions are quasigroups. ([A] is a survey of what this is all about.) The theory is geometrical and it involves association of a cubic graph to each functional equation of the form described, then factorization of such graphs into connected sum of irreducible ones, then embedding irreducibles in surfaces of smallest possible genus, then solution of functional equations corresponding to the irreducible graphs, then the solution of the original equation.

Now to every quadratic quasigroup identity one can associate a functional equation of the above form. Applying the machinery developed for functional equations, I was able to deduce essential results about quasigroup varieties defined by quadratic equations and will here just state a couple of typical results. It has been observed that some quasigroup identities force every quasigroup satisfying them to be isotopic to a group; some identities even force all quasigroups satisfying them to be isotopic to abelian groups (cf. [DK]). I give the following easy to check criteria: 1. The variety defined by a quadratic equality E consists entirely of group isotopes if and only if the graph associated to E contains a subgraph homeomorphic to the complete graph K4. 2. The variety defined by E consists entirely of isotopes of abelian groups if and only if the graph associated to it contains a copy of K4 and neither of the identities $E^{\epsilon\zeta} \; (\epsilon,\zeta=\pm1)$ is a trivial group identity. Here $E^{\epsilon\zeta}$ denotes the group identity obtained from E by interpreting the multiplication $x\cdot y$ in it as $x^\epsilon\cdot y^\zeta$.



REFERENCES:


[DK]
J. Dénes and A. D. Keedwell, Latin Squares and their Applications, Academic Press: 1974.

[A]
J. Aczél, Quasigroups, nets, and nomograms, Adv. Math. 1 (1965), 383-450.


next up previous
Next: Early papers Up: SUMMARY OF RESULTS Previous: Axiomatization for semigroups embeddable

2000-03-16