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AUTHOR(S): Liviu Vladutu
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ABSTRACT This work develops a categorical pipeline that connects the algebraic semantics of fuzzy logic with the ordered K-theory of AF C^*-algebras (Approximately Finite-Dimensional C^*-algebras). Starting from the classical equivalence between MV-algebras (Many-Valued algebras, i.e., algebras of Łukasiewicz logic) and unital ℓ-groups (lattice-ordered abelian groups with a strong unit) established by Mundici, we show how these ordered structures naturally embed into dimension groups (ordered abelian groups with interpolation and unperforation), the ordered K0 invariants (Grothendieck ordered K-theory group of projections) that classify AF algebras. By composing the functors Ξ, F, and K_0^(-1), we construct a functorial correspondence: MV→u^l G→DimGrp→AF that assigns to each MV-algebra a unique AF C^*-algebra whose ordered K_0-group recovers its underlying fuzzy-logical structure. This provides an operator-algebraic semantics for many-valued reasoning, where MV-operations correspond to projection structure and truncated addition in the associated AF-algebra. As an illustrative example, we compute explicitly the AF algebra associated with the three-valued Łukasiewicz algebra Ł3 and show that it corresponds to the matrix algebra M_2 (C). The developed framework clarifies the conceptual and categorical role of ordered K-theory in fuzzy logic and suggests new connections between many-valued reasoning, dimension groups, and the structure theory of C^*-algebras. |
KEYWORDS K-theory; Fuzzy logic; MV-algebras; AF C*-algebras; Dimension groups; Lattice-ordered groups; Ordered K_0 invariants; Many-valued logic; Functorial equivalence; Operator algebras |
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Cite this paper Liviu Vladutu. (2026) Towards a K-Theoretic Justification of Fuzzy Logic. International Journal of Mathematical and Computational Methods, 11, 5-14 |
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