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The characterization for the sobriety of $L$-convex spaces

Guojun Wu, Wei Yao

Abstract

With a commutative integral quantale $L$ as the truth value table, this study focuses on the characterizations of the sobriety of stratified $L$-convex spaces, as introduced by Liu and Yue in 2024. It is shown that a stratified sober $L$-convex space $Y$ is a sobrification of a stratified $L$-convex space $X$ if and only if there exists a quasihomeomorphism from $X$ to $Y$; a stratified $L$-convex space is sober if and only if it is a strictly injective object in the category of stratified $S_0$ $L$-convex spaces.

The characterization for the sobriety of $L$-convex spaces

Abstract

With a commutative integral quantale as the truth value table, this study focuses on the characterizations of the sobriety of stratified -convex spaces, as introduced by Liu and Yue in 2024. It is shown that a stratified sober -convex space is a sobrification of a stratified -convex space if and only if there exists a quasihomeomorphism from to ; a stratified -convex space is sober if and only if it is a strictly injective object in the category of stratified -convex spaces.

Paper Structure

This paper contains 5 sections, 12 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sober $L$-convex spaces
  4. The characterization of sobriety
  5. Conclusion remarks

Key Result

Lemma 2.3

(Yao-Lu) For each mapping $f: X\longrightarrow Y$, the Zadeh extensions$f^{\rightarrow}$ and $f^{\leftarrow}$are $L$-order-preserving with respect the inclusion $L$-order, and $f^{\rightarrow}$ is left adjoint to $f^{\leftarrow}$, denoted $f^{\rightarrow}\dashv f^{\leftarrow}$; this means that

Theorems & Definitions (31)

  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Example 2.6
  • Definition 2.7
  • Example 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 21 more