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Presenting Morphisms of Distributive Laws

2015
*
Conference on Algebra and Coalgebra in Computer Science
*

A format for well-behaved translations between structural operational specifications is derived from a notion

doi:10.4230/lipics.calco.2015.190
dblp:conf/calco/KlinN15
fatcat:aabse6lr5bcyziargok7lytbai
*of**distributive**law**morphism*, previously studied by Power and Watanabe. ... A simple*distributive**law**of*an endofunctor Σ over an endofunctor B is a natural transformation λ : ΣB =⇒ BΣ. ... So far in this paper we only treated simple SOS, to explain the general idea*of**presenting**distributive**law**morphisms*in a relatively basic setting. ...##
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A unified treatment of structural definitions on syntax for capture-avoiding substitution, context application, named substitution, partial differentiation, and so on
[article]

2022
*
arXiv
*
pre-print

We introduce a category-theoreticabstraction

arXiv:2204.03870v1
fatcat:fkldci7cnjdmlicsnqmsscugai
*of*a syntax with auxiliary functions, called an admissiblemonad*morphism*. ... Relying on an abstract form*of*structural recursion,we then design generic tools to construct admissible monad morphismsfrom basic data. ... From*distributive**laws*Let us now show how to construct admissible monad*morphisms*from*distributive**laws*. ...##
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Presenting Distributive Laws

2015
*
Logical Methods in Computer Science
*

In this paper we describe how to obtain a

doi:10.2168/lmcs-11(3:2)2015
fatcat:s4rlgr4wkrhsjde2hzwxnzicva
*distributive**law*for a monad with an equational*presentation*from a*distributive**law*for the underlying free monad. ...*Distributive**laws**of*a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. ... Thus, from Proposition 4.10 and Corollary 4.6 we obtain a*distributive**law*κ*of*P ω ((−) * ) over Id × F such that i • q : λ ⇒ κ is a*morphism**of**distributive**laws*, i.e., κ is*presented*by λ and the equations ...##
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Presenting Distributive Laws
[chapter]

2013
*
Lecture Notes in Computer Science
*

In this paper we describe how to obtain a

doi:10.1007/978-3-642-40206-7_9
fatcat:i3bxe3y3n5cctkdzbv2fzquffu
*distributive**law*for a monad with an equational*presentation*from a*distributive**law*for the underlying free monad. ...*Distributive**laws**of*a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. ... Thus, from Proposition 3 and Corollary 1 we obtain a*distributive**law*κ*of*P ω (Id * ) over 2 × Id A such that i • q : λ ⇒ κ is a*morphism**of**distributive**laws*, i.e., κ is*presented*by λ and the equations ...##
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Structural Operational Semantics and Modal Logic, Revisited

2010
*
Electronical Notes in Theoretical Computer Science
*

Secondly, a more concrete understanding

doi:10.1016/j.entcs.2010.07.019
fatcat:5ndmofo7l5h6nfnod5ilzfxgdu
*of*the assumptions*of*the theorem is provided, where proving compositionality amounts to finding a syntactic*distributive**law*between two collections*of*predicate ... Firstly, a more abstract, conceptual proof*of*the main compositionality theorem is given, based on an understanding*of*modal logic as a study*of*coalgebras in slice categories*of*adjunctions. ... Connections ρ and ζ are simply*morphisms**of*endofunctors, just as λ and χ are endomorphisms on them; also sliced*distributive**laws*χ λ are*distributive**law**morphisms*in the sense*of*[24] . ...##
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Factorisations of distributive laws

2016
*
Journal of Pure and Applied Algebra
*

Here we define the category

doi:10.1016/j.jpaa.2015.09.008
fatcat:sf7n7yitwnccpg6dd4574epfjq
*of*factorisations*of*a*distributive**law*, show that it acts on this construction, and give some explicit examples. ... Kassel MSC: 16T05; 18C15; 18C20; 19D55 Recently, Böhm and Ştefan constructed duplicial (paracyclic) objects from*distributive**laws*between (co)monads. ... The aim*of*the*present*paper is to extend this procedure to any duplicial object defined by a*distributive**law*. ...##
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The Tensor Product of Triples as Multilinear Product

1993
*
Revista Matemática Complutense
*

In this paper we introduce a notion

doi:10.5209/rev_rema.1993.v6.n1.17839
fatcat:2tbdajli5nfkrp3r7qiknewxxe
*of*multilinear product ¡br triples in Set, which if it is given by a*distributive**law*Ihen coincides with the one given by Hunge. ... Wc also demonstrate that the tensor product*of*two triples, if there exist, is an initial object jo a suitable category*of*multilinear products. ... The aim*of*the*present*paper is to introduce a notion*of*multilinear product for triples in Set, which if it is given by a*distributive**law*then coincides with the one given by Bunge, ami to demonstrate ...##
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A Category Theoretic View of Nondeterministic Recursive Program Schemes

2011
*
Annual Conference for Computer Science Logic
*

The Kleisli category A M

doi:10.4230/lipics.csl.2011.496
dblp:conf/csl/Schwencke11
fatcat:eemeg2qyybahzj5p2cx2uebusu
*of*a monad (M, η, µ) on a category A is given as follows: the objects*of*A M are the same objects as the ones*of*A; the*morphisms**of*A M between X and Y are all*morphisms*X → M ... and by Jf = η Y · f : X → M Y on*morphisms*f : X → Y . ... Proposition 3. 5 . 5 For the canonical*distributive**law*λ*of*a polynomial Set-functor H over the monad Definition 4. 1 . 1 Given a*distributive**law*δ : N M → M N*of*monads, a δ-*distributive**law**of*an ...##
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Interacting Frobenius Algebras are Hopf

2016
*
Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16
*

However the more general theory does not arise from a

doi:10.1145/2933575.2934550
dblp:conf/lics/DuncanD16
fatcat:nfolwrzdqvcjtpjj2iz63t7d3u
*distributive**law*. ... Bonchi, Sobocinski, and Zanasi [9] have shown that, given a suitable*distribution**law*, a pair*of*Hopf algebras forms two Frobenius algebras. ... The*distributive**laws*λF , ρ and σ form a*distributive*series*of*monads [12] , and hence determine a PROP FG*presented*by the equations*of*M + PG + C and equations (P1), (P2) and (F). ...##
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String diagrams for free monads (functional pearl)

2016
*
SIGPLAN notices
*

Full terms

doi:10.1145/3022670.2951947
fatcat:oxckyyqapnc7vmhnuatu2nyv34
*of*use are available: Abstract We show how one can reason about free monads using their universal properties rather than any concrete implementation. ... Acknowledgments Diagrams similar to the ones shown in Section 3 first appeared in an unpublished appendix to a paper by the*present*authors and Jeremy Gibbons (2015) . ... The authors would like to thank Tom Schrijvers and Alexander Vandenbroucke for their notes on an early draft*of*this paper, and the anonymous reviewers for their constructive remarks. ...##
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String diagrams for free monads (functional pearl)

2016
*
Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming - ICFP 2016
*

Full terms

doi:10.1145/2951913.2951947
dblp:conf/icfp/PirogW16
fatcat:mces5ufbjfdadarzmqx7fmyyka
*of*use are available: Abstract We show how one can reason about free monads using their universal properties rather than any concrete implementation. ... Acknowledgments Diagrams similar to the ones shown in Section 3 first appeared in an unpublished appendix to a paper by the*present*authors and Jeremy Gibbons (2015) . ... The authors would like to thank Tom Schrijvers and Alexander Vandenbroucke for their notes on an early draft*of*this paper, and the anonymous reviewers for their constructive remarks. ...##
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Adding recursive constructs to bialgebraic semantics

2004
*
The Journal of Logic and Algebraic Programming
*

For a class

doi:10.1016/j.jlap.2004.03.005
fatcat:xi7bctaqjrfyjgyilipeuqvb4q
*of*recursive equations called regular unfolding rules, this construction yields*distributive**laws*as analyzed by Turi and Plotkin. ... Rather than interpreting recursive constructs by means*of*operational rules, separate recursive equations are added to semantic descriptions*of*languages. ... Gordon Plotkin's insightful remarks motivated the developments*presented*in Section 8. Thanks are ...##
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Interacting Frobenius Algebras are Hopf
[article]

2016
*
arXiv
*
pre-print

Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable

arXiv:1601.04964v1
fatcat:jbxzvcqsgjey7bhlhmzg2lkwoa
*distributive**law*, a pair*of*Hopf algebras forms two Frobenius algebras. ... We recover the system*of*Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a*distributive**law*. ... be composed via a*distributive**law*ρ : C; PG → PG; C, yielding a PROP*presented*by the equations*of*C + PG and equation (P2). ...##
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Weakening and Iterating Laws using String Diagrams
[article]

2022
*
arXiv
*
pre-print

We also provide examples

arXiv:2205.03640v1
fatcat:iohqvnvw4veq7ooo6ixvwubhqe
*of*new weak*distributive**laws*arising from iteration. ... The main contribution*of*this paper is to generalise the associativity*of*iterated composition to weak*distributive**laws*. ... The*presentation**of*the*present*paper -including the string diagrammatic approach -is greatly inspired by the one*of*Street-weak*distributive**laws*in [31] . ...##
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Factorisations of distributive laws
[article]

2014
*
arXiv
*
pre-print

Here we define the category

arXiv:1409.7521v1
fatcat:vj4es5bfivc5lemg6ewtqx45lm
*of*factorisations*of*a*distributive**law*, show that it acts on this construction, and give some explicit examples. ... Recently, Boehm and Stefan constructed duplicial (paracyclic) objects from*distributive**laws*between (co)monads. ... The aim*of*the*present*paper is to extend this procedure to any duplicial object defined by a*distributive**law*. ...
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