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Generalizing Substitution

Published online by Cambridge University Press:  15 January 2004

Tarmo Uustalu
Tarmo Uustalu*
Affiliation:
Inst. of Cybernetics, Tallinn Technical Univ., Akadeemia tee 21, EE-12618 Tallinn, Estonia; tarmo@cs.ioc.ee. The work was largely done during the author's postdoctoral leave to Dep. de Informática, Univ. do Minho, Braga, Portugal.
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Abstract

It is well known that, given an endofunctor H on a category C , the initial (A+H-)-algebras (if existing), i.e. , the algebras of (wellfounded) H-terms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Moss [17] and Aczel, Adámek, Milius and Velebil [12] 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. the inverses of the final T'(A,-)-coalgebras for any endobifunctor T' on any category C such that the functors T'(-,X) uniformly carry a monad structure.

Keywords

Algebras of termsnon-wellfounded termssubstitutioniteration of guarded substitution rulesmonadshyperfunctionsfinitely or possibly infinitely branching trees.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 37 , Issue 4: Fixed Points in Computer Science (FICS'02) , October 2003 , pp. 315 - 336
Copyright
© EDP Sciences, 2003

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