On a Description of Terminal Coalgebras and Iterative Theories

Abstract

A concrete description of terminal coalgebras, T, of all finitary endofunctors of Set is presented: each such a functor is a quotient of the polynomial endofunctor of some finitary signature Σ modulo some “basic” equations. Then T can be described as the algebra of all infinite ∑-labeled trees modulo the congruence obtained by applying the basic equations finitely or infinitely many times (a concept defined below). As a consequence, free iterative theories in the sense of Calvin Elgot are described over all finitary endofunctors of Set: they are the theories of all rational ∑-labeled trees (i.e., trees having only finitely many subtrees) modulo the above congruence.