On the category of \(L\)-fuzzy automata, coalgebras and dialgebras. (English) Zbl 1522.18006
Summary: This paper is towards the study of \(L\)-fuzzy automata from the categorical point of view, where \(L\) is a complete residuated lattice. We introduce a functor having both the left/right adjoint from the category of \(L\)-fuzzy \(\Sigma\)-semiautomata \(\mathbf{LFSA}(\Sigma)\) to the category \(\mathbf{LFTCRL}(\Sigma)\), a category of complete residuated lattices corresponding to \(L\)-fuzzy \(\Sigma\)-semiautomata. The \(L\)-fuzzy response map of an \(L\)-fuzzy automaton leads us to provide a characterization of an \(L\)-fuzzy regular language. Interestingly, we show that the category \(\mathbf{LFSA}(\Sigma)\) is a category of \(\mathcal{T}_1\)-coalgebras/\((\mathcal{T}_2,\mathcal{T}_3)\)-dialgebras. Moreover, we establish a relationship between the category of \(\mathcal{T}_1\)-coalgebras and the category of \((\mathcal{T}_2, \mathcal{T}_3)\)-dialgebras.
Keywords:
categories; complete residuated lattices; \(L\)-fuzzy automata; \(L\)-fuzzy languages; \(L\)-fuzzy response mapsReferences:
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