On the category of L-fuzzy automata, coalgebras and dialgebras

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 Σ-semiautomata LFSA ( Σ ) to the category LFTCRL(Σ), a category of complete residuated lattices corresponding to L-fuzzy Σ-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 LFSA ( Σ ) is a category of T 1 -coalgebras/ ( T 2 , T 3 ) -dialgebras. Moreover, we establish a relationship between the category of T 1 -coalgebras and the category of ( T 2 , T 3 ) -dialgebras.