Abstract. In our PhD thesis ([4]), we showed that for the study of denotational semantics of linear logic ([6]), it is crucial to generalize the standard notion of monad on a category. An earlier generalization was already given by Hoofman ([9]): semimonads. But it was not suitable for our problem. This is why we introduced another generalization: weak monads. In this note, we present this new notion and give some examples.

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A prime algebraic lattice can be characterised as isomorphic to the downwards-closed subsets, ordered by inclusion, of its complete primes. It is easily seen that the downwards-closed subsets of a partial order form a completely distributive algebraic lattice when ordered by inclusion. The converse also holds; any completely distributive algebraic lattice is isomorphic to such a set of downwards-closed subsets of a partial order. The partial order can be recovered from the lattice as the order of the lattice restricted to its complete primes. Consequently prime algebraic lattices are precisely the completely distributive algebraic lattices. The result extends to Scott domains. Several consequences are explored briefly: the representation of Berry’s dIdomains by event structures; a simplified form of information systems for completely distributive Scott domains; and a simple domain theory for concurrency.

université de Savoie