Pre A ∗ ‐Algebras and Their Applications in Fuzzy Set Theory
Gebregziabiher Girum Gebreset, Jonnalagadda Venkateswara Rao, Habtu Alemayehu Atsbaha
Source abstract
This paper studies Pre A ∗ ‐algebras and their role as an algebraic foundation for fuzzy set theory. By relaxing the key Boolean axioms of distributivity and complementation, Pre A ∗ ‐algebras provide a robust algebraic structure for reasoning with uncertainty. We construct a fundamental three‐element algebra, A = {0, 1, 2}, representing truth, falsity, and an absorbing indeterminacy. We demonstrate how this structure models fuzzy membership and establishes clear links with standard logical connectives, including t‐norms, t‐conorms, fuzzy negation, and a multivalued XOR. Applications in decision‐making, data processing, and fuzzy control illustrate the relevance of Pre A ∗ ‐algebras as a distinct and advantageous basis for multivalued systems.
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