Constructive Suprema

Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics, which can be viewed as the constructive core of mathematics and whose theorems can be translated into many formal systems of computable mathematics. The relationship between two classically equivalent notions of supremum is examined in detail. Whereas the classical least upper bound is based on the negative concept of partial order, the other supremum is based on the positive notion of excess relation. Equivalent conditions of existence are obtained for both suprema in the general case of a partially ordered set; other equivalent conditions are obtained for subsets of a lattice and, in particular, for subsets of R.