# Generalizing substitution

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2003)

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

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topUustalu, Tarmo. "Generalizing substitution." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 37.4 (2003): 315-336. <http://eudml.org/doc/244874>.

@article{Uustalu2003,

abstract = {It is well known that, given an endofunctor $H$ on a category $\mathbb \{C\}$, 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 [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 $T^\{\prime \}(A,\{-\})$-algebras resp. the inverses of the final $T^\{\prime \}(A,\{-\})$-coalgebras for any endobifunctor $T^\{\prime \}$ on any category $\mathbb \{C\}$ such that the functors $T^\{\prime \}(\{-\}, X)$ uniformly carry a monad structure.},

author = {Uustalu, Tarmo},

journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et 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},

number = {4},

pages = {315-336},

publisher = {EDP-Sciences},

title = {Generalizing substitution},

url = {http://eudml.org/doc/244874},

volume = {37},

year = {2003},

}

TY - JOUR

AU - Uustalu, Tarmo

TI - Generalizing substitution

JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

PY - 2003

PB - EDP-Sciences

VL - 37

IS - 4

SP - 315

EP - 336

AB - It is well known that, given an endofunctor $H$ on a category $\mathbb {C}$, 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 [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 $T^{\prime }(A,{-})$-algebras resp. the inverses of the final $T^{\prime }(A,{-})$-coalgebras for any endobifunctor $T^{\prime }$ on any category $\mathbb {C}$ such that the functors $T^{\prime }({-}, X)$ 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/244874

ER -

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