- 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 **K*_{4}*. 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 **K*_{4}* and neither of the identities
*
* is a trivial
group identity.* Here
denotes the group
identity obtained from *E* by interpreting the multiplication
in it as
.

*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.