Quasivarieties and varieties of ordered algebras: regularity and exactness

Kurz, Alexander; Velebil, Jiřı́

doi:10.1017/s096012951500050x

Cited by 22 publications

(48 citation statements)

References 31 publications

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“…The similarity of these correspondences in our applications suggests that there should be a (possibly enriched) notion of exact category that covers our examples; cf. Kurz and Velebil's [15] 2-categorical view of ordered algebras. This would allow to move more work to the generic theory.…”

Section: Discussionmentioning

confidence: 99%

Equational Axiomatization of Algebras with Structure

Milius

Urbat

2019

Lecture Notes in Computer Science

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This paper proposes a new category theoretic account of equationally axiomatizable classes of algebras. Our approach is wellsuited for the treatment of algebras equipped with additional computationally relevant structure, such as ordered algebras, continuous algebras, quantitative algebras, nominal algebras, or profinite algebras. Our main contributions are a generic HSP theorem and a sound and complete equational logic, which are shown to encompass numerous flavors of equational axiomizations studied in the literature.

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Section: Discussionmentioning

confidence: 99%

Equational Axiomatization of Algebras with Structure

Milius

Urbat

2019

Lecture Notes in Computer Science

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“…An ordered category is thus a special form of 2category, and thus the well-developed theory of 2-categories (see, e.g., [29]) can be applied. To site just a couple of examples where the 2-categorical machinery works very well for particular order-enriched categories we mention [30,31,32], which involves a translation of a 2-categorical notion to a condition on a monad known as the Kock-Zöberlein condition, and [33] in the area of ordered universal algebra. However, as noted generally already in [34], the standard 2-categorical constructions yield the 'wrong' results in certain ordered categories arising in computer science.…”

Section: Order Extensionsmentioning

confidence: 99%

Mahavier completeness and classifying diagrams

Maehara

Weiss

2017

Topology and its Applications

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Generalised inverse limits of compacta were introduced by Ingram and Mahavier in 2006. The main difference between ordinary inverse limits and their generalised cousins is that the former concerns diagrams of singlevalued functions while the latter permits multivalued functions. However, generalised inverse limits are not merely limits in the Kleisli category of a hyperspace monad, a fact that independently motivated each of the authors of this article to come up with the same formalism which restores the link with category theory through the concept of Mahavier limit of an order diagram in an order extension of a category B. Mahavier limits of diagrams in B coincide with ordinary limits in B, and so Mahavier limits are an extension of ordinary limits along the functor that views an ordinary diagram as a diagram in the extension. Within that context it is natural to consider Mahavier completeness, namely when all small diagrams admit Mahavier limits, as well as classifying diagrams, namely the existence of a right adjoint to the mentioned functor on diagrams. In this work we show that these two conditions are equivalent, and we study some of the properties of classifying diagrams and of the adjunction.

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“…Conversely, if F ⊣ U : A → Set is an ordinary variety and the only order on algebras in A making all operations monotone is the trivial discrete order (as it is the case in BA), then F C ⊣ DU : A → Pos is an ordered variety, see [45]. The proposition above guarantees that for a functor L : BA → BA, it does not matter whether we consider it as an ordinary functor on the variety BA, or whether we consider it as a locally monotone functor on the ordered variety BA.…”

Section: If Any Of the Above Conditions Is Satisfied Then Alsomentioning

confidence: 99%

Positive fragments of coalgebraic logics

Balan¹,

Kurz²,

Velebil³

2015

Logical Methods in Computer Science

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Abstract. Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond.In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors.To facilitate this analysis we prove a number of category theoretic results on functors on the categories Set of sets and Pos of posets:Every functor Set → Pos has a Pos-enriched left Kan extension Pos → Pos. Functors arising in this way are said to have a presentation in discrete arities. In the case that Set → Pos is actually Set-valued, we call the corresponding left Kan extension Pos → Pos its posetification.A set functor preserves weak pullbacks if and only if its posetification preserves exact squares. A Pos-functor with a presentation in discrete arities preserves surjections.The inclusion Set → Pos is dense. A functor Pos → Pos has a presentation in discrete arities if and only if it preserves coinserters of 'truncated nerves of posets'. A functor Pos → Pos is a posetification if and only if it preserves coinserters of truncated nerves of posets and discrete posets.A locally monotone endofunctor of an ordered variety has a presentation by monotone operations and equations if and only if it preserves Pos-enriched sifted colimits. ACM CCS: [Theory of computation]:Logic.

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Quasivarieties and varieties of ordered algebras: regularity and exactness

Cited by 22 publications

References 31 publications

Equational Axiomatization of Algebras with Structure

Equational Axiomatization of Algebras with Structure

Mahavier completeness and classifying diagrams

Positive fragments of coalgebraic logics

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