Infinite trees and completely iterative theories: a coalgebraic view

Generalizing Substitution

“…A monad's property of complete iterativeness in the sense of Aczel et al [2] is, for a given, but unspecified notion of guardedness of morphisms, definable as follows. Definition 1.5.…”

“…The example of non-wellfounded H-terms, investigated in Moss [17] and Aczel et al [2], is considerably more interesting. Substitution is definable also for nonwellfounded H-terms, but the definition is not as simple any more as for wellfounded H-terms.…”

“…In Aczel et al's [2] terminology (which builds on that of Elgot et al [8,9]), their monad is completely iterative. Definition 2.7.…”

“…Moss [17] and Aczel, Adámek, Milius and Velebil [2] have shown that a similar monad, which even enjoys the additional special property of having iterations for all guarded substitution rules (complete iterativeness), arises from the inverses of the final (A + H−)-coalgebras (if existing), i.e., the algebras of non-wellfounded H-terms. We show that, upon an appropriate generalization of the notion of substitution, the same can more generally be said about the initial T (A, −)-algebras resp.…”

“…Manes [14]. In contrast, the comparably fundamental fact that a similar monad, which even enjoys the additional property of having iterations for all guarded substitution rules (complete iterativeness), arises from the inverses of the final (A+H−)-coalgebras (if existing), i.e., the algebras of non-wellfounded H-terms over different A, was only recently pointed out by Moss [17] and Aczel, Adámek, Milius and Velebil [1,2] (their substitution and solution theorems), although the "down-to-earth" universal-algebra case, where C = Set and H is a polynomial endofunctor, was settled in the 1970s by Elgot and colleagues [9].…”