On Transition Constructions for Automata -- A Categorical Perspective

We investigate the transition monoid construction for deterministic automata in a categorical setting and establish it as an adjunction. We pair this adjunction with two other adjunctions to obtain two endofunctors on deterministic automata, a comonad and a monad, which are closely related, respectively, to the largest set of equations and the smallest set of coequations satisfied by an automaton. Furthermore, we give similar transition algebra constructions for lasso and {\Omega}-automata, and show that they form adjunctions. We present some initial results on sets of equations and coequations for lasso automata.