Quantale-valued preorders: Globalization and cocompleteness

Tao, Yuanye; Lai, Hongliang; Zhang, Dexue

doi:10.1016/j.fss.2012.09.013

Cited by 27 publications

(21 citation statements)

References 16 publications

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“…called respectively the forward globalization and the backward globalization functors [29,36]. Explicitly, the forward globalization of a DQ-category (X, α) is the Q-category (X, G f α) with…”

Section: A Appendix: Q-categories Vs Dq-categoriesmentioning

confidence: 99%

Towards probabilistic partial metric spaces: Diagonals between distance distributions

He¹,

Lai²,

Shen³

2019

Fuzzy Sets and Systems

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The quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories. Motivated by the categorical interpretation of partial metric spaces, we are led to investigate the quantaloid of diagonals between distance distributions, which is expected to establish the categorical foundation of probabilistic partial metric spaces. Observing that the quantale of distance distributions w.r.t. an arbitrary continuous t-norm is non-divisible, we precisely characterize diagonals between distance distributions, and prove that one-step functions are the only distance distributions on which the set of diagonals coincides with the generated down set.for all t ∈ [0, ∞], called a distance distribution, rather than a non-negative real number. As discovered by Chai [4] and investigated by Hofmann-Reis [12], (generalized) probabilistic metric spaces are categories enriched in the quantale ∆ * = (∆, ⊗ * , κ 0,1 ) of distance distributions w.r.t. a continuous t-norm * on the unit interval [0, 1] (see Proposition 3.6 for further explanations) which, in elementary words, are precisely sets X equipped with a map α :for all x, y, z ∈ X and r, s ∈ [0, ∞]. Partial metric spaces are metric spaces in which the distance from a point to itself may not be zero. Explicitly, (generalized) partial metric spaces are sets X equipped a with a map α :for all x, y, z ∈ X. As discovered by Höhle-Kubiak [17] and Pu-Zhang [29] and formalized later by Stubbe [35] in a more general setting, although partial metric spaces are not categories enriched in the quantale [0, ∞] + , one may construct a quantaloid of diagonals of the quantale [0, ∞] + , usually denoted by D[0, ∞] + , such that partial metric spaces are precisely categories enriched in the quantaloid D[0, ∞] + .It is now natural to ask whether it is possible to consider the probabilistic version of partial metric spaces or, equivalently, the partial version of probabilistic metric spaces. As far as we know, this topic has been considered recently by several authors under the name probabilistic partial metric spaces or fuzzy partial metric spaces; see, e.g. [1,31,37,38,39]. Following the categorical interpretation of partial metric spaces, this paper aims at a full-scale investigation of the quantaloid D∆ * of diagonals of the quantale ∆ * of distance distributions w.r.t. a continuous t-norm * on [0, 1], so that a categorical foundation of probabilistic partial metric spaces can be established, which was more or less neglected in the existing references.In general, let Q = (Q, &, 1) be a commutative and integral quantale (i.e., complete residuated lattice), in which p & q ≤ r ⇐⇒ p ≤ q → r for all p, q, r ∈ Q. With DQ denoting the quantaloid of diagonals of Q (see Proposition 2.2), a DQ-category (also called partial Q-category, see [14]) consists of a set X and a map α : X × X

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Section: A Appendix: Q-categories Vs Dq-categoriesmentioning

confidence: 99%

Towards probabilistic partial metric spaces: Diagonals between distance distributions

He¹,

Lai²,

Shen³

2019

Fuzzy Sets and Systems

Self Cite

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“…An L * -s-quantale is a generalization of the s-quantale fuzzifying the ordering relation underneath, and uses the notion of L-ordered set in the sense of [23,41,42]. By definition, an L-ordered set (X, R) consists of a set X and an L-order R on X, i.e.…”

Section: The Category Of L * -S-quantalesmentioning

confidence: 99%

“…μ). An L-ordered set (X, R) is called a complete L-lattice iff each μ ∈ L X has both μ and μ, or equivalently, each μ ∈ L X has μ [23,41,42]. As is observed in [23,42], an L-ordered set (resp.…”

Section: The Category Of L * -S-quantalesmentioning

confidence: 99%

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Stratified categorical fixed-basis fuzzy topological spaces and their duality

Demirci

2015

Fuzzy Sets and Systems

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“…Q-preordered sets have attracted wide attention in the fuzzy community; see [3,4,10,20,21,26,28,41,51] for instance. While Q-preordered sets defined by (1.i) are actually Q-preorders on crisp sets, recently Lai and Zhang and their co-authors have established the theory of Q-preorders on fuzzy sets especially when Q is a divisible quantale [30,34,48]; similar approaches have been adopted by Höhle and Kubiak for the construction of their quantalevalued preorders [19,22]. The key machinery involved in these works is that of categories enriched in a quantaloid [36,44,45,46], which is a special case of categories enriched in a bicategory [5,6,50].…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Galois connections on fuzzy sets

García

Lai

Shen

2018

Fuzzy Sets and Systems

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In fairly elementary terms this paper presents how the theory of preordered fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy sets, is established under the guidance of enriched category theory. Motivated by several key results from the theory of quantaloid-enriched categories, this paper develops all needed ingredients purely in order-theoretic languages for the readership of fuzzy set theorists, with particular attention paid to fuzzy Galois connections between preordered fuzzy sets.

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Quantale-valued preorders: Globalization and cocompleteness

Cited by 27 publications

References 16 publications

Towards probabilistic partial metric spaces: Diagonals between distance distributions

Towards probabilistic partial metric spaces: Diagonals between distance distributions

Stratified categorical fixed-basis fuzzy topological spaces and their duality

Fuzzy Galois connections on fuzzy sets

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