On Minimal Fuzzy Realization in Category Theoretic Setting

Shailendra Singh (), Amarjit Kaur Sahni and Jayanti Tripathi Pandey ()
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Shailendra Singh: Department of Mathematics & Computing, Indian Institute of Technology (ISM), Dhanbad 826004, India
Amarjit Kaur Sahni: Department of Mathematics, AIAS, Amity University, UP, India
Jayanti Tripathi Pandey: Department of Mathematics, AIAS, Amity University, UP, India

New Mathematics and Natural Computation (NMNC), 2022, vol. 18, issue 03, 889-917

Abstract: This paper aims to study the minimal fuzzy realization for a fuzzy language with membership values in a complete residuated lattice by using category theory. Specifically, we introduce the concept of a category 𠒞𠒯ℛℒ(Σ), whose object-class is complete transition residuated lattices corresponding to deterministic Σ-semiautomata. We give the categorical characterization of reachability and observability maps for a given deterministic fuzzy automaton. In another direction, we demonstrate that the category 𠒟𠒮𠒜(Σ) is a subcategory of the categories ℱ𠒞𠒜(Σ) of F1-coalgebras and ℱ𠒟𠒜(Σ) of (F2,F3)-dialgebras. Also, we discuss the concept of bisimulation between F1-coalgebras. Next, we introduce a general theory of minimal fuzzy realization for a given fuzzy language in a category theory setting. Strikingly, we demonstrate that all minimal fuzzy realization for a given fuzzy language is one of a kind up to isomorphism.

Keywords: Category; complete residuated lattice; deterministic fuzzy automaton; fuzzy language; fuzzy realization (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S1793005722500429

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