Title
A unified algebraic framework for vagueness and granularity in fuzzy and rough set theories
Authors
[ 1 ] Department of Informatics, Systems and Communication (DISCo), University of Milano–Bicocca | [ 2 ] Department of Economics and Business, University of Catania | [ 3 ] Instytut Informatyki, Wydział Informatyki i Telekomunikacji, Politechnika Poznańska | [ 4 ] Instytut Badań Systemowych PAN | [ P ] employee
Scientific discipline (Law 2.0)
Year of publication
2026
Published in
Article type
scientific article
Publication language
english
Keywords
EN
- Boolean algebra
- Brouwer-Zadeh distributive lattices
- Vagueness
- Ambiguity
- Uncertainty
- Graduality
- Granularity
- Pawlak operator
- Fuzzy sets
- Rough sets
Abstract
EN We investigate the identification of the Boolean algebra ℘(𝑋) of all subsets of a universe 𝑋 with the set E(𝑋) = {0, 1}^𝑋 of characteristic functions, focusing on the expression of the complement operation in three equivalent forms. Building on this, we extend the framework to fuzzy sets 𝑓 ∶ 𝑋 → [0, 1], which form a complete distributive lattice equipped with three distinct complement operations: • the Zadeh (Kleene) complement 𝑓′ (𝑥) = 1 − 𝑓(𝑥), • the Brouwer (intuitionistic) complement 𝑓∼ (𝑥) = 𝜒 {𝑥∈𝑋∶𝑓 (𝑥)=0} , and • the anti-Brouwer complement 𝑓♭ (𝑥) = 𝜒 {𝑥∈𝑋∶𝑓 (𝑥)≠1} . We analyze the algebraic properties and interrelations of these complements. This leads to the study of Brouwer–Zadeh (BZ) algebras, which generalize Boolean algebras to lattices with two non-standard complements linked via 𝑎∼′ = 𝑎∼∼ , implying 𝑎∼ ≤ 𝑎′ . We show that any BZ algebra naturally decomposes into three non-overlapping components: a Zadeh algebra, a Brouwer algebra, and an anti-Brouwer algebra. Furthermore, we derive two significant structures from BZ algebras: • the Pawlak Rough Approximation Structure (RAS), and • the Kuratowski Abstract Topological Structure (ATS). To model information granularity and graduality, we introduce the Pawlak operator into the BZ framework, yielding the Pawlak–Brouwer–Zadeh (PBZ) lattice. This enriched structure allows us to formally distinguish between vagueness and ambiguity in RAS and to model granularity in fuzzy sets, providing an algebraic foundation for shadowed sets and related granular models.
Date of online publication
25.09.2025
Pages (from - to)
122726-1 - 122726-56
Comments
Article Number: 122726
License type
CC BY (attribution alone)
Open Access Mode
czasopismo hybrydowe
Open Access Text Version
final published version
Date of Open Access to the publication
in press
Full text of article
Access level to full text
public
Ministry points / journal
200
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