Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Representations of domains via closure spaces in the quantale-valued setting

Guojun Wu, Wei Yao, Qingguo Li

Abstract

With a commutative unital quantale $L$ as the truth value table, this study focuses on the representations of $L$-domains by means of $L$-closure spaces. First, the notions of interpolative generalized $L$-closure spaces and directed closed sets are introduced. It is proved that in an interpolative generalized $L$-closure space (resp., $L$-closure space), the collection of directed closed sets with respect to the inclusion $L$-order forms a continuous $L$-dcpo (resp., an algebraic $L$-dcpo). Conversely, it is shown that every continuous $L$-dcpo (resp., algebraic $L$-dcpo) can be reconstructed by an interpolative generalized $L$-closure space (resp., $L$-closure space). Second, when $L$ is integral, the notion of dense subspaces of generalized $L$-closure spaces is introduced. By means of dense subspaces, an alternative representation for algebraic $L$-dcpos is given. Moreover, the concept of $L$-approximable relations between interpolative generalized $L$-closure spaces is introduced. Consequently, a categorical equivalence between the category of interpolative generalized $L$-closure spaces (resp., $L$-closure spaces) with $L$-approximable relations and that of continuous $L$-dcpos (resp., algebraic $L$-dcpos) with Scott continuous mappings is established.

Representations of domains via closure spaces in the quantale-valued setting

Abstract

With a commutative unital quantale as the truth value table, this study focuses on the representations of -domains by means of -closure spaces. First, the notions of interpolative generalized -closure spaces and directed closed sets are introduced. It is proved that in an interpolative generalized -closure space (resp., -closure space), the collection of directed closed sets with respect to the inclusion -order forms a continuous -dcpo (resp., an algebraic -dcpo). Conversely, it is shown that every continuous -dcpo (resp., algebraic -dcpo) can be reconstructed by an interpolative generalized -closure space (resp., -closure space). Second, when is integral, the notion of dense subspaces of generalized -closure spaces is introduced. By means of dense subspaces, an alternative representation for algebraic -dcpos is given. Moreover, the concept of -approximable relations between interpolative generalized -closure spaces is introduced. Consequently, a categorical equivalence between the category of interpolative generalized -closure spaces (resp., -closure spaces) with -approximable relations and that of continuous -dcpos (resp., algebraic -dcpos) with Scott continuous mappings is established.

Paper Structure

This paper contains 10 sections, 30 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Commutative unital quantale, the truth value table
  4. fuzzy sets and fuzzy posets
  5. fuzzy domains
  6. Categorical equivalent
  7. The representation of continuous $L$-dcpos
  8. The representation of algebraic $L$-dcpos
  9. Approximable $L$-relations
  10. Conclusion

Key Result

Lemma 2.1

(residuatedHajek) Suppose that $(L,\otimes, u)$ is a commutative unital quantale. Then for all $a,b,c\in L$ and $\{a_i|\ i\in I\},\ \{b_j|\ j\in J\}\subseteq L$, (Q1) $u\leq a\rightarrow b\Longleftrightarrow a\leq b$; (Q2) $0\rightarrow a=1$; (Q3) $u\rightarrow a=a$; (Q4) $a\otimes(a\rightarrow b)\l

Theorems & Definitions (50)

  • Lemma 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 40 more