Algebras for the Partial Map Classifier Monad

a ≤ b iff a is the supremum of the subset {a} ∩ {b}. On any elementary topos E , one has the functor which to an object A associates the object TA = Ã which classifies partial maps into A (cf e.g. [J] 1.2). This functor T carries a monad structure T= (T, η, μ) ; it is a submonad of the power ”set” monad P= (P, η, μ), as described in, say, [AL],[Mi], or [J] 5.3. We shall analyze the category of algebras for T, for the category of sets, but our arguments and constructions will be intuitionistically pure, so that everything carries over to an arbitrary elementary topos. In fact, when applied to the ”usual” boolean category of sets, our notions are rather trivial, or even laughable; for instance, the category of T-algebras is, in this case, just the category of pointed sets, and the proof of this fact is trivial. From [Ma] , we know that the category of algebras for the power set monad P is the category of cocomplete posets, with the structure map being the formation of suprema. Since TA ⊆ PA consists of those subsets of A which have at most one element, it has been conjectured that the category of algebras for T should be some category of posets with the (weak!) cocompleteness property that any subset with at most one element has a supremum, but the question was how to construct an order on A out of an algebra structure ξ : TA → A; for the full power set monad such order is given by: