Abstract

ing from these examples we propose the following definition: Definition 17 The relation R is said to be F-well-founded if and only if, for all relations S , the equation 19 X:: X = S ffl F:X ffl R has a unique solution. 2 By design, R is well-founded (in the conventional sense) if and only if it is id-well-founded, where id is the identity relator. Moreover, the converse of any initial F -algebra is F-well-founded. A stronger statement can be made: Theorem 18 Suppose R is an F -coalgebra that is a bijection between F:R? and R? . Then the following are all equivalent: (a) R is F-well-founded, (b) R is F -reductive, (c) R[ is an initial F -algebra. 2 (The equivalence between (b) and (c) has already been observed.) One of the fundamental properties of reductivity is that it implies well-foundedness. This is theorem 7 of [3]. The converse is not true. Let R be a nonempty, well-founded relation (for example the relation succ[ on natural numbers) . Then it is easy to show th...

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