arXiv subjects

Hongliang Lai

Publications and source records attributed to Hongliang Lai.

23 records · Page 2Linked to original sources

arXiv2016

Fixed points of adjoint functors enriched in a quantaloid

Representation theorems are established for fixed points of adjoint functors between categories enriched in a small quantaloid. In a very general setting these results set up a common framework for representation theorems of concept lattices in formal concept analysis (FCA) and rough set theory (RST), which not only extend the realm of formal contexts to multi-typed and multi-valued ones, but also provide a general approach to construct various kinds of representation theorems. Besides incorporating several well-known representation theorems in FCA and RST as well as formulating new ones, it is shown that concept lattices in RST can always be represented as those in FCA through relative pseudo-complements of the given contexts, especially if the contexts are valued in a non-Girard quantaloid.

cs.LO↗

arXiv2016

Monads on Q-Cat and their lax extensions to Q-Dist

For a small quantaloid $\mathcal{Q}$, we consider 2-monads on the 2-category $\mathcal{Q}$-$\bf{Cat}$ and their lax extensions to the 2-category $\mathcal{Q}$-$\bf{Dist}$ of small $\mathcal{Q}$-categories and their distributors, in particular those lax extensions that are flat, in the sense that they map identity distributors to identity distributors. In fact, unlike in the discrete case, a 2-monad on $\mathcal{Q}$-$\bf{Cat}$ may admit only one flat lax extension. Every ordinary monad on the comma category $\bf{Set}/{\rm ob}\mathcal{Q}$ with a lax extension to $\mathcal{Q}$-$\bf{Rel}$ gives rise to such a 2-monad on $\mathcal{Q}$-$\bf{Cat}$, and we describe this process globally as a coreflective embedding. The $\mathcal{Q}$-presheaf and the double $\mathcal{Q}$-presheaf monads are important examples of 2-monads on $\mathcal{Q}$-$\bf{Cat}$ allowing flat lax extensions to $\mathcal{Q}$-$\bf{Dist}$, and so are their submonads, obtained by the restriction to conical (co)presheaves and known as the $\mathcal{Q}$-Hausdorff and double $\mathcal{Q}$-Hausdorff monads, which we define here in full generality, thus generalizing some previous work in the case when $\mathcal{Q}$ is a quantale, or just the "metric" quantale $[0,\infty]$. Their discretization leads naturally to various lax extensions of the relevant $\bf{Set}$-monads used in monoidal topology.

math.CT↗

arXiv2016

Yoneda completeness and flat completeness of ordered fuzzy sets

This paper studies Yoneda completeness and flat completeness of ordered fuzzy sets valued in the quantale obtained by endowing the unit interval with a continuous triangular norm. Both of these notions are natural extension of directed completeness in order theory to the fuzzy setting. Yoneda completeness requires every forward Cauchy net converges (has a Yoneda limit), while flat completeness requires every flat weight (a counterpart of ideals in partially ordered sets) has a supremum. It is proved that flat completeness implies Yoneda completeness, but, the converse implication holds only in the case that the related triangular norm is either isomorphic to the Lukasiewicz t-norm or to the product t-norm.

math.CT↗

arXiv2016

Quantale-valued Approach Spaces via Closure and Convergence

For a quantale $\mathsf{V}$ we introduce $\mathsf{V}$-approach spaces via $\mathsf{V}$-valued point-set-distance functions and, when $\mathsf{V}$ is completely distributive, characterize them in terms of both, so-called closure towers and ultrafilter convergence relations. When $\mathsf{V}$ is the two-element chain $\mathsf{2}$, the extended real half-line $[0,\infty]$, or the quantale ${\bfΔ}$ of distance distribution functions, the general setting produces known and new results on topological spaces, approach spaces, and the only recently considered probabilistic approach spaces, as well as on their functorial interactions with each other.

math.GN↗

arXiv2006

Many-valued complete distributivity

Categories enriched over a commutative unital quantale can be studied as generalized, or many-valued, ordered structures. Because many concepts, such as complete distributivity, in lattice theory can be characterized by existence of certain adjunctions, they can be reformulated in the many-valued setting in terms of categorical postulations. So, it is possible, by aid of categorical machineries, to establish theories of many-valued complete lattices, many-valued completely distributive lattices, and so on. This paper presents a systematical investigation of many-valued complete distributivity, including the topics: (1) subalgebras and quotient algebras of many-valued completely distributive lattices; (2) categories of (left adjoint) functors; and (3) the relationship between many-valued complete distributivity and properties of the quantale of truth values. The results show that enriched category theory is a very useful tool in the study of many-valued versions of order-related mathematical entities.

math.CT↗