Function Spaces in the Category of Directed Suprema Preserving Maps1

provide the notation and terminology for this paper. Let F be a function. We say that F is uncurrying if and only if the conditions (Def. 1) are satisfied. (Def. 1)(i) For every set x such that x ∈ dom F holds x is a function yielding function, and (ii) for every function f such that f ∈ dom F holds F(f) = uncurry f. We say that F is currying if and only if the conditions (Def. 2) are satisfied. (Def. 2)(i) For every set x such that x ∈ dom F holds x is a function and π 1 (x) is a binary relation, and (ii) for every function f such that f ∈ dom F holds F(f) = curry f. We say that F is commuting if and only if the conditions (Def. 3) are satisfied. (Def. 3)(i) For every set x such that x ∈ dom F holds x is a function yielding function, and (ii) for every function f such that f ∈ dom F holds F(f) = commute(f). Let us observe that every function which is empty is also uncurrying, currying, and commuting. Let us note that there exists a function which is uncurrying, currying, and commuting. Let F be an uncurrying function and let X be a set. One can check that FX is uncurrying. Let F be a currying function and let X be a set. Note that FX is currying. Next we state two propositions: (1) Let X, Y , Z, D be sets. Suppose D ⊆ (Z Y) X. Then there exists a many sorted set F indexed by D such that F is uncurrying and rng F ⊆ Z [: X,Y :] .