I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article mention is made of the "trace product" - it is defined as:
if $a,b \in S$, then $ab$ is a trace product if $ab \in \mathcal{R}_a \cap \mathcal{L}_b$.
Here $\mathcal{R}_a$ and $\mathcal{L_b}$ are the relevant classes from Green's relations. So I traced (no pun intended) this product back to two articles, one by Rees and one by Miller and Clifford (Regular D-classes in semigroups). So here the explanation is given of a semigroup $D^0$, constructed from a $D$ class and called the trace of $D$. The definition of the binary operation of this semigroup then includes this trace product. So this semigroup is used to prove that $D$ is partially isomorphic to a regular matrix semigroup.
I am not so familiar with the abstract algebra of semigroups - does anyone know if this trace product mentioned here bears any relation to the trace of a matrix? or perhaps trace of a matrix product? as it is usually referred to in the context of matrix theory. Perhaps in a semigroup of matrices (say square matrices of the same size) you get some classes as per Green's definitions that can be characterized in terms of the trace of a matrix, and it is related to this trace product (it's a long shot I know...)?
NB: You can read it online if you register - and the section on the trace construction is on page 276/277.