Combining Semilattices and Semimodules

Abstract

We describe the canonical weak distributive law $$\delta :\mathcal S\mathcal P\rightarrow \mathcal P\mathcal S$$ δ : S P → P S of the powerset monad $$\mathcal P$$ P over the S-left-semimodule monad $$\mathcal S$$ S , for a class of semirings S. We show that the composition of $$\mathcal P$$ P with $$\mathcal S$$ S by means of such $$\delta $$ δ yields almost the monad of convex subsets previously introduced by Jacobs: the only difference consists in the absence in Jacobs’s monad of the empty convex set. We provide a handy characterisation of the canonical weak lifting of $$\mathcal P$$ P to $$\mathbb {EM}(\mathcal S)$$ EM ( S ) as well as an algebraic theory for the resulting composed monad. Finally, we restrict the composed monad to finitely generated convex subsets and we show that it is presented by an algebraic theory combining semimodules and semilattices with bottom, which are the algebras for the finite powerset monad $$\mathcal P_f$$ P f .

Processes Parametrised by an Algebraic Theory

“…For example, processes with multiset branching given by the theory of commutative monoids produces nondeterministic processes with a simplistic notion of resources. Another example is nondeterministic weighted processes with branching captured by the monad arising from the weak distributive law between the free semimodule and powerset monads [46]. Yet another instantiation arises from the theory of monoids (presenting the list monad), which produces processes related to breadth-first search algorithms.…”

Weakening and Iterating Laws using String Diagrams

“…Remark 3.15 Deleting the (η − ) axiom is one of the many ways of weakening the notion of distributive law. In recent years, this approach has been very fruitful for computer science applications [14,15,13,6]. One may ask, symmetrically, what happens by deleting the (η + ) axiom instead.…”

“…Simplifying earlier work from [31,2,3], Garner proposed to delete the one axiom that was obstructing, in many concrete cases, the existence of a law. Several papers followed on this revived notion of weak distributive law [14,15,6,27]. In general, finding examples of weak distributive laws remains difficult, and very few techniques are known to produce them.…”

Algebraic Presentation of Semifree Monads

“…To overcome this, Garner [11] defines a notion of weak distributive law which drops the problematic axiom. The usefulness of this concept has been demonstrated as several monads have been proven to result from weak distributive laws: the Vietoris monad [11], the convex powerset monad [13] and the monad of finitely generated convex subsets [7].…”