Bialgebras and Realizations

In this paper we consider when a linear functional on a bialgebra is realized by the action of the bialgebra on a finite object. This depends on whether the action of the bialgebra on the functional is finite. We consider two specific cases: the Myhill–Nerode Theorem, which gives a condition for a language to be accepted by a finite automaton, and Fliess’ Theorem, which gives a condition for for the input/output maps of a control system to be realized by action on a finite dimensional state space. If H is a bialgebra, then the linear dual H∗ is an algebra with