Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On unification of categories associated with F -transforms and fuzzy pretopological spaces as Qua category

Abha Tripathi, S. P. Tiwari

TL;DR

This work addresses unifying categories linked to F-transforms and fuzzy pretopological spaces by formulating Qua, a category of success measures of answers and transformations. It develops and relates four core categories—LSpaceFP, LFtrans_downarrow, LFPrTop, and LFCInt—through isomorphisms and adjoint functors, and embeds them into Qua to illuminate their connections. The paper also constructs and relates fuzzy lower transformation systems and lower/upper pretopologies within Qua, establishing a network of functors that connect these perspectives. The resulting framework offers a principled, category-theoretic lens for unifying transform-based and topology-based fuzzy structures, with adjoint pairs suggesting robust dualities and potential generalizations to broader fuzzy-relational settings.

Abstract

In this contribution, our motive is to unify the categories associated with F-transforms and fuzzy pretopological spaces as a new category Qua, whose object classes are success measurements of answers and morphisms are pairs of success measurements of transformations. Specifically, the categories of spaces with L-valued fuzzy partitions, L-valued fuzzy lower transformation systems, L-valued fuzzy pretopological spaces, and Cech L-valued fuzzy interior spaces with the morphisms as pairs of L-valued fuzzy relations between the underlying sets of corresponding objects have intriguing relationships with the category Qua.

On unification of categories associated with F -transforms and fuzzy pretopological spaces as Qua category

TL;DR

This work addresses unifying categories linked to F-transforms and fuzzy pretopological spaces by formulating Qua, a category of success measures of answers and transformations. It develops and relates four core categories—LSpaceFP, LFtrans_downarrow, LFPrTop, and LFCInt—through isomorphisms and adjoint functors, and embeds them into Qua to illuminate their connections. The paper also constructs and relates fuzzy lower transformation systems and lower/upper pretopologies within Qua, establishing a network of functors that connect these perspectives. The resulting framework offers a principled, category-theoretic lens for unifying transform-based and topology-based fuzzy structures, with adjoint pairs suggesting robust dualities and potential generalizations to broader fuzzy-relational settings.

Abstract

In this contribution, our motive is to unify the categories associated with F-transforms and fuzzy pretopological spaces as a new category Qua, whose object classes are success measurements of answers and morphisms are pairs of success measurements of transformations. Specifically, the categories of spaces with L-valued fuzzy partitions, L-valued fuzzy lower transformation systems, L-valued fuzzy pretopological spaces, and Cech L-valued fuzzy interior spaces with the morphisms as pairs of L-valued fuzzy relations between the underlying sets of corresponding objects have intriguing relationships with the category Qua.
Paper Structure (7 sections, 45 theorems, 51 equations, 6 figures)

This paper contains 7 sections, 45 theorems, 51 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Category theory
  4. Complete residuated and co-residuated lattice
  5. Categories of spaces with $L$-valued fuzzy partitions and $L$-valued fuzzy lower transformation systems
  6. Categorical view of $L$-valued fuzzy partition with $L$-valued fuzzy pretopology and $L$-valued fuzzy interior operator
  7. Conclusion

Key Result

Proposition 3.1

Spaces with $L$-valued fuzzy partitions with their $L$-valued fuzzy FP-maps form a category.

Figures (6)

  • Figure 1: Diagram for Definition \ref{['adj0']}.
  • Figure 2: Diagram for Proposition \ref{['com']}.
  • Figure 3: Diagram for Proposition \ref{['adj']}.
  • Figure 4: Diagram is equivalent to the diagram in Figure \ref{['fig:6']}.
  • Figure 5: Diagram for Proposition \ref{['adj1']}.
  • ...and 1 more figures

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.1
  • Remark 2.2
  • Definition 2.8
  • ...and 56 more