Associativity is a well-known property of some binary operations (or the corresponding groupoids). Instead of the binary classification into associative and non-associative operations, more refined approaches for quantifying the degree of associativity or non-associativity of a binary operation have been proposed in the literature. One such method is the so-called associative spectrum, which was introduced by Csákány and Waldhauser in 2000. The associative spectrum of a groupoid \(G\) is an integer sequence whose \(n\)-th term equals the number of distinct term operations induced on \(G\) by the bracketings of \(n\) variables.

In this talk, we provide a brief introduction to the associative spectrum, and give an overview on our work on the associative spectra of graph algebras and quasigroups. Graph algebras were introduced by Shallon in 1979 and provide a useful representation of directed graphs as algebras with a binary operation that is not necessarily associative. A quasigroup is a groupoid satisfying the condition that for all elements \(a\) and \(b\), there exist unique elements \(x\) and \(y\) such that \(xa = b\) and \(ay = b\). This is a generalization of groups; associativity and an identity element are not required.

This talk is based on joint work with Tamás Waldhauser (University of Szeged).