# Product Rules and Distributive Laws

Abstract : We give a categorical perspective on various product rules, including Brzozowski’s product rule $(st)_a = s_a t + o(s) t_a$ and the familiar rule of calculus $(st)_a = s_a t + s t_a$ It is already known that these product rules can be represented using distributive laws, e.g. via a suitable quotient of a GSOS law. In this paper, we cast these product rules into a general setting where we have two monads S andT, a (possibly copointed) behavioural functor F, a distributive law of T over S, a distributive law of S over F, and a suitably defined distributive law $TF \Rightarrow FST$ We introduce a coherence axiom giving a sufficient and necessary condition for such triples of distributive laws to yield a new distributive law of the composite monad ST over F, allowing us to determinize FST-coalgebras into lifted F coalgebras via a two step process whenever this axiom holds.
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Conference papers
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Cited literature [28 references]

https://hal.inria.fr/hal-01446036
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### Citation

Joost Winter. Product Rules and Distributive Laws. 13th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Apr 2016, Eindhoven, Netherlands. pp.114-135, ⟨10.1007/978-3-319-40370-0_8⟩. ⟨hal-01446036⟩

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