Terms and Infinite Trees as Monads Over a Signature

In this paper, we prove that the usual construction of terms and infinite trees over a signature is a particular case of a more general construction of monads over the category of sets. In this way, we obtain a family of semantical domains having a tree-like structure and appearing as the completions of the corresponding finite structures. Though it is quite different in its technical developments our construction should be compared with the one of De Bakker and Zucker which is very similar in spirit and motivation. We feel that one outcome of the present approach is that, due to its connection with Lawvere's algebraic theories, it should provide an interesting framework to deal with equational varieties of process algebras.