The Vietoris Monad and Weak Distributive Laws

Garner, Richard

doi:10.1007/s10485-019-09582-w

Cited by 18 publications

(79 citation statements)

References 28 publications

(33 reference statements)

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“…Böhm [11] and Street [37] have studied various weaker notions of distributive law; here we shall use the one that consists in dropping the axiom involving η T in Definition 1, following the approach of Garner [15].…”

Section: Definitionmentioning

confidence: 99%

“…There are suitable weaker notions of liftings and extensions which also bijectively correspond to weak distributive laws as proved in [11,15].…”

Section: Definitionmentioning

confidence: 99%

“…The image alongP of a S-algebra (X, a) will be a set Y together with a structure map α a that makes it a S-algebra in turn. Garner [15,Proposition 13] gives us the recipe to build Y and α a appropriately. Y is obtained by splitting the following idempotent in Set:…”

Section: Proof Of Theorem 24mentioning

confidence: 99%

“…Nevertheless, some sort of weakening of the notion of distributive law-e.g., distributive laws of functors over monads [26]-proved to be ubiquitous in computer science: they are GSOS specifications [38], they are sound coinductive up-to techniques [7] and complete abstract domains [5]. In this paper we will exploit weak distributive laws in the sense of [15] that have been recently shown successful in composing the monads for nondeterminism and probability [17].…”

Section: Introductionmentioning

confidence: 99%

“…We proceed as follows. Rather than composing P f , we found it convenient to compose the full, not necessarily finite, powerset monad P with S. In this way we can reuse several results in [12] that provide necessary and sufficient conditions on the semiring S for the existence of a canonical weak [15] distributive law δ : SP → PS. Our first contribution (Theorem 21) consists in showing that such δ has a convenient alternative characterisation, whenever the underlying semiring is a positive semifield, a condition that is met, e.g., by the semirings of Booleans and non-negative reals.…”

Section: Introductionmentioning

confidence: 99%

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Combining Semilattices and Semimodules

Bonchi

Santamaria

2021

Lecture Notes in Computer Science

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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 .

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Section: Definitionmentioning

confidence: 99%

“…There are suitable weaker notions of liftings and extensions which also bijectively correspond to weak distributive laws as proved in [11,15].…”

Section: Definitionmentioning

confidence: 99%

Section: Proof Of Theorem 24mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Combining Semilattices and Semimodules

Bonchi

Santamaria

2021

Lecture Notes in Computer Science

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Algebraic Presentation of Semifree Monads

Rosset

Hansen

Endrullis

2022

Coalgebraic Methods in Computer Science

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Preservation and Reflection of Bisimilarity via Invertible Steps

Turkenburg¹,

Kupke²,

Rot³

et al. 2023

Lecture Notes in Computer Science

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In the theory of coalgebras, distributive laws give a general perspective on determinisation and other automata constructions. This perspective has recently been extended to include so-called weak distributive laws, covering several constructions on state-based systems that are not captured by regular distributive laws, such as the construction of a belief-state transformer from a probabilistic automaton, and ultrafilter extensions of Kripke frames.In this paper we first observe that weak distributive laws give rise to the more general notion of what we call an invertible step: a pair of natural transformations that allows to move coalgebras along an adjunction. Our main result is that part of the construction induced by an invertible step preserves and reflects bisimilarity. This covers results that have previously been shown by hand for the instances of ultrafilter extensions and belief-state transformers.

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The Vietoris Monad and Weak Distributive Laws

Cited by 18 publications

References 28 publications

Combining Semilattices and Semimodules

Combining Semilattices and Semimodules

Algebraic Presentation of Semifree Monads

Preservation and Reflection of Bisimilarity via Invertible Steps

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