Deconstructing Lawvere with distributive laws

PROs, PROPs and Lawvere categories are related notions adapted to the study of algebraic structures borne by an object in a category: PROs are monoidal, PROPs are symmetric monoidal and Lawvere categories are cartesian. This paper connects the three notions using Lack's technique for composing PRO(P)s via distributive laws. We show that Lawvere categories can be seen as the composite PROP CCm;T, where T expresses the algebraic structure in linear form and CCm express the ability of copying and discarding them. In turn the PROP T can be decomposed in terms of PROs as P;S where P expresses the ability of permuting variables and S is the PRO encoding the syntactic structure without permutations.