Moje počty / The Math I do

I am an algebraist with combinatorial interests. Most of the mathematics I have done is concerned with loop theory, associativity triples, latin bitrades, groups and left distributive systems.

Loop theory

A latin square is a square array such that every symbol appears in in each column and each row exactly once. If this array is considered as an operational table of a binary operation, then this operation yields a quasigroup. Each row and each column expresses a permutation of symbols. Permutations induced by rows are called left translations, and permutations induced by columns are called right translations. A binary operation yields a quasigroup if and only if all left and right translations are permutations of the underlying set.
A quasigroup is called a loop if it contains a unit element, i.e., if both the left and right translations contain the identity mapping.
Papers I wrote about loops are concerned with multiplication groups, laws (i.e., identities) that hold in loops in general, conjugacy closed and left conjugacy closed loops, and cyclic extensions of a loops.

Associativity triples

Consider a binary operation. A triple (a,b,c) is called associative if a(bc) = (ab)c. It is natural to ask how many such triples may exist, either for binary operations in general, or for quasigroups in particular. This may be further specialized to, say, quasigroups isotopic to groups.
At an early stage of my career I was concerned with the problem posed by Dénes and Keedwell in their famous book on Latin squares: ''What is the least number of nonassociative triples that may be possessed by a quasigroup that is not a group?'' It turns out that, up to some exceptions, quasigroups with few nonassociative triples have to have operational tables that are as close as possible to operational tables of groups. That led me to the study of latin bitrades (see below). More recent efforts are concerned with quasigroups that a(Q) the number of associative triples it possesses. This is sometimes called the associative index of Q. Now, for n>1 denote by a(n) the least associative index among quasigroups of order n. It is known that a(2)=4, a(3)=9, a(4)=16, a(5)=15, a(6)=16, a(7)=17, a(8)=16, a(9)=9, a(10)=???.