Some algebraic laws for spans

This paper investigates some key algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions, analyzing the monoidal structures induced over both spans and cospans by cartesian product and disjoint union of sets. Our results find analogous counterparts in (and are partly inspired by) the theory of relational algebras, thus our paper also sheds some light on the relationship between (co)spans and the categories of (multi)relations and of equivalence relations. And, since (co)spans yield an intuitive presentation of dynamical systems with input and output interfaces, our results introduce an expressive, two-fold algebra that can serve as a specification formalism for rewriting systems and for composing software modules