Scott’s formula

There is a well-known formula in domain theory, which, given a monotonic map f from a basis B of a continuous poset X to a dcpo Y, produces the largest continuous map f’ defined on the whole of X and below f on B. This is Scott’s formula: f’(x) ≝ sup_{b ∈ B, b ≪ x} f(b). Generalizing this to the setting where X is a c-space, I will explain when exactly f’ coincides with f on B, hence when f’ is a continuous extension of f. Also, using a motivation from the theory of continuous valuations, I will describe conditions under which any algebraic laws satisfied by f transfer to f’. Read the full post.