Generalizing Substitution

Tarmo Uustalu

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 37, Issue: 4, page 315-336
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Uustalu, Tarmo. "Generalizing Substitution." RAIRO - Theoretical Informatics and Applications 37.4 (2010): 315-336. <http://eudml.org/doc/92726>.

@article{Uustalu2010,
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. },
author = {Uustalu, Tarmo},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Algebras of terms; non-wellfounded terms; substitution; iteration of guarded substitution rules; monads; hyperfunctions; finitely or possibly infinitely branching trees.; terms; guarded substitution},
language = {eng},
month = {3},
number = {4},
pages = {315-336},
publisher = {EDP Sciences},
title = {Generalizing Substitution},
url = {http://eudml.org/doc/92726},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Uustalu, Tarmo
TI - Generalizing Substitution
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 4
SP - 315
EP - 336
AB - 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.
LA - eng
KW - Algebras of terms; non-wellfounded terms; substitution; iteration of guarded substitution rules; monads; hyperfunctions; finitely or possibly infinitely branching trees.; terms; guarded substitution
UR - http://eudml.org/doc/92726
ER -

References

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