Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1912.12901

On selected developments in the theory of natural dualities

Miroslav Haviar
View original source ↗Read source document ↗Export metadata ↓

Abstract

This is a survey on selected developments in the theory of natural dualities where the author had the opportunity to make with his foreign colleagues several breakthroughs and move the theory forward. It is aimed as author's reflection on his works on the natural dualities in Oxford and Melbourne over the period of twenty years 1993-2012 (before his attention with the colleagues in universal algebra and lattice theory has been fully focused on the theory of canonical extensions and the theory of bilattices). It is also meant as a remainder that the main problems of the theory of natural dualities, Dualisability Problem and Decidability Problem for Dualisability, remain still open. Theory of natural dualities is a general theory for quasi-varieties of algebras that generalizes `classical' dualities such as Stone duality for Boolean algebras, Pontryagin duality for abelian groups, Priestley duality for distributive lattices, and Hofmann-Mislove-Stralka duality for semilattices. We present a brief background of the theory and then illustrate its applications on our study of Entailment Problem, Problem of Endodualisability versus Endoprimality and then a famous Full versus Strong Problem with related developments.

Explore related subjects

math.CTmath.RA

Keep this discovery

BibTeXRIS

Miroslav Haviar. 2019-12-30. On selected developments in the theory of natural dualities. https://arxiv.org/abs/1912.12901

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A model structure for cartesian 2-fibrations

Cartesian 2-fibrations provide a way to understand indexed categories, but their classical ``straightening'' construction requires several layers of weak coherence data. This paper develops a homotopical framework that replaces much of this bookkeeping with a fully strict model. By using marked 2-categories to record the cartesian morphisms and 2-cells, we construct a model structure whose fibrant objects are precisely the cartesian 2-fibrations over a fixed 2-category $\mathcal{C}$. We then show that the marked Grothendieck construction identifies these 2-fibrations, up to weak equivalence, with strict 2-functors from $\mathcal{C}$ into $2\mathrm{Cat}$. As an additional contribution, we construct localizations of 2-categories that simultaneously invert selected morphisms and 2-cells.

math.CT↗
arXiv2026

A Natural Fuzzy Order on Fuzzy Numbers

This paper introduces a natural fuzzy order on fuzzy numbers that extends the natural orders on real numbers and interval numbers. We investigate its completeness properties and show that the space of uniformly bounded fuzzy numbers is conically complete and conically cocomplete, and that it is complete if and only if the underlying continuous t-norm is the G\"odel t-norm. Moreover, it is proved that this space constitutes a \([0,1]\)-enriched domain if and only if the underlying continuous t-norm satisfies the (S) condition. These results provide a foundation for ordering fuzzy numbers.

math.CT↗
arXiv2026

Noetherian forms of free non-symmetric operads

In this paper, we study certain categories of labeled finite rooted ordered trees over a fixed set of labels where each label is equipped with an arity: a fixed number of children that the vertex with the given label must have. Equivalently, these are expression trees for operations in a free non-symmetric operad. A morphism between these trees matches a pruning of one tree (a prefix) with an entire subtree of another (a suffix). We characterize such categories, up to isomorphism, in terms of suitable exactness properties. It turns out that these categories exhibit strong algebraic behavior, in the sense that every such category, when appended with a strict initial object, has a particularly nice noetherian form.

math.CT↗