nLab
overlap algebra
Contents
Idea
Overlap algebras are one possible constructive version of complete Boolean algebras . In the presence of excluded middle , the category of overlap algebras is equivalent to the category of complete Boolean algebras and continuous homomorphisms.

However, in the absence of excluded middle , the powerset of a set is no longer a complete Boolean algebra , but it is an overlap algebra.

Definition
(See Definition 2.1 in OA .)

An overlap algebra is an overt frame $L$ such that if for all $z\in L$ the positivity of $z\wedge x$ implies the positivity of $z\wedge y$ , then we have $x\le y$ .

Morphisms of overlap algebras are precisely open morphisms of frames .

Thus, in the presence of excluded middle the category of overlap algebras is equivalent to the category of complete Boolean algebras , see Proposition 5.2 in OA .

Properties
In the presence of excluded middle , overlap algebras coincide with complete Boolean algebras . (Proposition 2.2 and Proposition 5.2 in OA .)

References
Francesco Ciraulo? , Michele Contente? , Overlap Algebras: a Constructive Look at Complete Boolean Algebras , arXiv:1904.13320