On matrices and $K$-relations

We show that the matrix query language $\mathsf{M}\mathsf{A}\mathsf{T}\mathsf{L}\mathsf{A}\mathsf{N}\mathsf{G}$ corresponds to a natural fragment of the positive relational algebra on $K$-relations. The fragment is defined by introducing a composition operator and restricting $K$-relation arities to two. We then proceed to show that $\mathsf{M}\mathsf{A}\mathsf{T}\mathsf{L}\mathsf{A}\mathsf{N}\mathsf{G}$ can express all matrix queries expressible in the positive relational algebra on $K$-relations, when intermediate arities are restricted to three. Thus we offer an analogue, in a model with numerical data, to the situation in classical logic, where the algebra of binary relations is equivalent to first-order logic with three variables.