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Generalized (m,n)-Fuzzy BL-Subalgebras: Algebraic Foundations, Power-Implication Structures

Articles in Press

Document Type : Research Article

Authors

Department of Mathematics‎, ‎Payame Noor University (PNU)‎, ‎Iran.‎

10.30473/coam.2025.75088.1320

Abstract

This paper offers the idea of (anti) (m‎, ‎n)-fuzzy BL-subalgebras as a novel extension of classical BL-algebras within the fuzzy mathematical framework. ‎The proposed structures generalize various types of fuzzy subalgebras, including (anti) intuitionistic, (anti) Pythagorean, (anti) Fermatean, and (anti) q-rung orthopair fuzzy BL-subalgebras for q >= 1. Fundamental algebraic properties and equivalent characterizations of (m,n)-fuzzy BL-subalgebras are established through the notion of value-cuts. Furthermore, the concept of power-implication preserving (PIP) BL-algebras is introduced, and it is shown that a PIP BL-algebra exists for every prime number. Several closure properties of (m,n)-fuzzy BL-subalgebras under combination operations are also derived within this framework. From an applied perspective, the developed theoretical results can serve as a mathematical foundation for modeling and reasoning in fuzzy control systems and optimization processes, particularly in decision-making environments characterized by uncertainty and graded information.

Highlights

  • Introduces the new notion of (anti) (m,n)-fuzzy BL-subalgebras, extending classical BL-algebras and unifying several existing fuzzy algebraic structures, including intuitionistic, Pythagorean, Fermatean, and q-rung orthopair fuzzy BL-subalgebras.
  • Establishes fundamental algebraic properties and equivalent characterizations of (m,n)-fuzzy BL-subalgebras through value-cut techniques, linking fuzzy semantics with crisp algebraic representations.
  • Develops the concept of power-implication preserving (PIP) BL-algebras, proving the existence of a PIP BL-algebra for every prime number and revealing a notable interaction between fuzzy algebra and number theory.
  • Demonstrates several closure properties of (m,n)-fuzzy BL-subalgebras under combination operations such as intersection and union, ensuring algebraic consistency within the proposed framework.
  • Introduces and analyzes (m,n)-fuzzy nil radical BL-subalgebras, with results showing stability of these structures under homomorphic images.
  • Provides a mathematical foundation suitable for applications in fuzzy control, optimization, and decision-making systems, particularly in environments involving uncertainty and graded information.

Keywords

Main Subjects

References
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Control and Optimization in Applied Mathematics

Articles in Press, Accepted Manuscript
Available Online from 01 December 2025
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