Lattices do not distribute over powerset

Tellez, Julian Salamanca

doi:10.1007/s00012-020-00680-8

Cited by 3 publications

(2 citation statements)

References 13 publications

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“…First, let us justify the expressions of κ ϕ and κ ψ in equations ( 32) and (33). Recall that the definition of the idempotent derived from a weak distributive law λ : [S, T ] is given in equation (7) and in the third diagram of (12).…”

Section: B1 Expressions Of κ ϕ and κ ψmentioning

confidence: 99%

Weakening and Iterating Laws using String Diagrams

Goy¹

2023

Electronic Notes in Theoretical Informatics and Computer Science

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Distributive laws are a standard way of combining two monads, providing a compositional approach for reasoning about computational effects in semantics. Situations where no such law exists can sometimes be handled by weakening the notion of distributive law, still recovering a composite monad. A celebrated result from Eugenia Cheng shows that combining $n$ monads is possible by iterating more distributive laws, provided they satisfy a coherence condition called the Yang-Baxter equation. Moreover, the order of composition does not matter, leading to a form of associativity. The main contribution of this paper is to generalise the associativity of iterated composition to weak distributive laws in the case of $n = 3$ monads. To this end, we use string-diagrammatic notation, which significantly helps make increasingly complex proofs more readable. We also provide examples of new weak distributive laws arising from iteration.

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Section: B1 Expressions Of κ ϕ and κ ψmentioning

confidence: 99%

Weakening and Iterating Laws using String Diagrams

Goy¹

2023

Electronic Notes in Theoretical Informatics and Computer Science

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“…distributive laws may not always exist. The literature exhibits many such negative results, called no-go theorems [32,21,10,36,29,37]. A possible fix consists in weakening the notion of distributive law to recover transformations that, even if not satisfying all the usual axioms, still enable some weak kind of monad composition.…”

Section: Introductionmentioning

confidence: 99%

Weakening and Iterating Laws using String Diagrams

Goy¹

2022

Preprint

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Distributive laws are a standard way of combining two monads, providing a compositional approach for reasoning about computational effects in semantics. Situations where no such law exists can sometimes be handled by weakening the notion of distributive law, still recovering a composite monad. A celebrated result from Eugenia Cheng shows that combining more monads is possible by iterating more distributive laws, provided they satisfy a coherence condition called the Yang-Baxter equation. Moreover, the order of composition does not matter, leading to a form of associativity. The main contribution of this paper is to generalise the associativity of iterated composition to weak distributive laws. To this end, we use string-diagrammatic notation, which significantly helps make increasingly complex proofs more readable. We also provide examples of new weak distributive laws arising from iteration.

show abstract

Quantale-valued convex structures as lax algebras

Pang

2023

Fuzzy Sets and Systems

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Lattices do not distribute over powerset

Cited by 3 publications

References 13 publications

Weakening and Iterating Laws using String Diagrams

Weakening and Iterating Laws using String Diagrams

Weakening and Iterating Laws using String Diagrams

Quantale-valued convex structures as lax algebras

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