In this paper we revisit some pioneering efforts to equip Petri nets with compact operational models for expressing causality. The models we propose have a bisimilarity relation and a minimal representative for each equivalence class, and they can be fully explained as coalgebras on a presheaf category on an index category of partial orders. First, we provide a set-theoretic model in the form o...

For each regular language L we describe a family of canonical nondeterministic acceptors (nfas). Their construction follows a uniform recipe: build the minimal dfa for L in a locally finite variety V , and apply an equivalence between the finite V-algebras and a category of finite structured sets and relations. By instantiating this to different varieties we recover three well-studied canonical...

We de ne the continuum up to order isomorphism, and hence up to homeomorphism via the order topology, in terms of the nal coalgebra of either the functor N×X , product with the set of natural numbers, or the functor 1+N×X . This makes an attractive analogy with the de nition of N itself as the initial algebra of the functor 1+X , disjoint union with a singleton. We similarly specify Baire space...

We present a general coalgebraic setting in which we define finite and infinite behaviour with B\"uchi acceptance condition for systems whose type is monad. The first part of the paper devoted to presenting construction monad suitable modelling (in)finite behaviour. second focuses on concepts (coalgebraic) automaton its ($\omega$-) end Kleene-type theorems regular input. framework instantiated ...

In seeking a unified study of computational effects, in particular in order to give a general operational semantics agreeing with the standard one for state, one must take account of the coalgebraic structure of state. Axiomatically, one needs a countable Lawvere theory L, a comodel C, typically the final one, and a model M , typically free; one then seeks a tensor C ⊗ M of the comodel with the...

For each regular language L we describe a family of canonical nondeterministic acceptors (nfas). Their construction follows a uniform recipe: build the minimal dfa for L in a locally finite variety V, and apply an equivalence between the category of finite V-algebras and a suitable category of finite structured sets and relations. By instantiating this to different varieties, we recover three w...

A labelled transition system can be understood as a coalgebra for a particular endofunctor on the category of sets. Generalizing, we are led to consider coalgebras for arbitrary endofunctors on arbitrary categories. Bisimulation is a crucial notion in the theory of labelled transition systems. We identify four definitions of bisimulation on general coalgebras. The definitions all specialize to ...

We present a framework for extending Stone’s representation theorem for distributive lattices to representation theorems for distributive lattices with operators. We proceed by introducing the definition of algebraic theory of operators over distributive lattices. Each such theory induces a functor on the category of distributive lattices such that its algebras are exactly the distributive latt...