Free Completely Distributive Lattices

Markowsky, George

doi:10.2307/2043137

Cited by 5 publications

(5 citation statements)

References 3 publications

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“…A more concrete construction of free objects in CABA can be given utilizing the theory of canonical extensions. It is well known that free objects over any set exist in the category of complete and completely distributive lattices (see Markowski [Mar79] and Dwinger [Dwi81, Thm. 4.2]).…”

Section: Thomason Dualitymentioning

confidence: 99%

Duality for powerset coalgebras

Bezhanishvili¹,

Carai²,

Morandi³

2022

Logical Methods in Computer Science

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Let CABA be the category of complete and atomic boolean algebras and complete boolean homomorphisms, and let CSL be the category of complete meet-semilattices and complete meet-homomorphisms. We show that the forgetful functor from CABA to CSL has a left adjoint. This allows us to describe an endofunctor H on CABA such that the category Alg(H) of algebras for H is dually equivalent to the category Coalg(P) of coalgebras for the powerset endofunctor P on Set. As a consequence, we derive Thomason duality from Tarski duality, thus paralleling how J\'onsson-Tarski duality is derived from Stone duality.

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Section: Thomason Dualitymentioning

confidence: 99%

Duality for powerset coalgebras

Bezhanishvili¹,

Carai²,

Morandi³

2022

Logical Methods in Computer Science

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“…In [28] it is shown that the category Ω-CD is monadic over Set, hence a(n infinitary) variety [21]. In particular, the free completely distributive Ω-category over a set A is Í(dA) [2,38], similarly to the ordered case [22].…”

Section: Completely Distributive Quantale Enriched Categoriesmentioning

confidence: 99%

An equational approach to enriched distributivity

Bălan¹,

Kurz²

2021

Preprint

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The familiar adjunction between ordered sets and completely distributive lattices can be extended to generalised metric spaces, that is, categories enriched over a quantale (a lattice of "truth values"), via an appropriate distributive law between the "down-set" monad and the "up-set" monad on the category of quantale-enriched categories. If the underlying lattice of the quantale is completely distributive, and if powers distribute over non-empty joins in the quantale, then this distributive law can be concretely formulated in terms of operations, equations and choice functions, similar to the familiar distributive law of lattices.

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“…Algebras for the monad A are completely distributive lattices [27]. The sets of sets in A(X) can be seen as DNF formulae over elements of X, where the outer powerset is disjunctive and the inner one is conjunctive.…”

Section: Examplesmentioning

confidence: 99%

Learning Automata with Side-Effects

Heerdt

Sammartino

Silva

2020

Lecture Notes in Computer Science

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Automata learning has been successfully applied in the verification of hardware and software. The size of the automaton model learned is a bottleneck for scalability, and hence optimizations that enable learning of compact representations are important. This paper exploits monads, both as a mathematical structure and a programming construct, to design and prove correct a wide class of such optimizations. Monads enable the development of a new learning algorithm and correctness proofs, building upon a general framework for automata learning based on category theory. The new algorithm is parametric on a monad, which provides a rich algebraic structure to capture non-determinism and other side-effects. We show that this allows us to uniformly capture existing algorithms, develop new ones, and add optimizations.

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Free Completely Distributive Lattices

Cited by 5 publications

References 3 publications

Duality for powerset coalgebras

Duality for powerset coalgebras

An equational approach to enriched distributivity

Learning Automata with Side-Effects

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