Algebras of Complemented Subsets

Complemented subsets were introduced by Bishop, in order to avoid complementation in terms of negation. In his two approaches to measure theory Bishop used two sets of operations on complemented subsets. Here we study these two algebras and we introduce the notion of Bishop algebra as an abstraction of their common structure. We translate constructively the classical bijection between subsets and boolean-valued functions by establishing a bijection between the proper classes of complemented subsets and of strongly extensional, boolean-valued, partial functions. Avoiding negatively defined concepts, most of our results are within minimal logic.