ST and TS as Product and Sum

The set of $\mathsf{ST}$-valid inferences is neither the intersection, nor the union of the sets of $\mathsf{K}_3$- and $\mathsf{LP}$-valid inferences, but despite the proximity to both systems, an extensional characterization of $\mathsf{ST}$ in terms of a natural set-theoretic operation on the sets of $\mathsf{K}_3$- and $\mathsf{LP}$-valid inferences is still wanting. In this paper, we show that it is their relational product. Similarly, we prove that the set of $\mathsf{TS}$-valid inferences can be identified using a dual notion, namely as the relational sum of the sets of $\mathsf{LP}$- and $\mathsf{K}_3$-valid inferences. We discuss links between these results and the interpolation property of classical logic. We also use those results to revisit the duality between $\mathsf{ST}$ and $\mathsf{TS}$. We present a combined notion of duality on which $\mathsf{ST}$ and $\mathsf{TS}$ are dual in exactly the same sense in which $\mathsf{LP}$ and $\mathsf{K}_3$ are dual to each other.