This chapter introduces the concept of \( n \)-free Boolean algebras, where \( n \) is a positive integer or \( \omega \). A Boolean algebra \( A \) is \( n \)-free over a subset \( X \) if every \( n \)-preserving function from \( X \) to any Boolean algebra \( B \) extends to a unique homomorphism. The chapter defines \( n \)-preserving functions and explores the properties of \( n \)-independence, which generalizes the notion of disjointness in infinite sets. It also discusses the relationship between \( n \)-free Boolean algebras and graph spaces, hypergraph spaces, and their duals. Key results include the equivalence of \( n \)-freeness and having an \( n \)-independent generating set, and the construction of \( n \)-free Boolean algebras from hypergraphs. The chapter concludes with discussions on algebraic constructions and the closure properties of these structures under various operations.This chapter introduces the concept of \( n \)-free Boolean algebras, where \( n \) is a positive integer or \( \omega \). A Boolean algebra \( A \) is \( n \)-free over a subset \( X \) if every \( n \)-preserving function from \( X \) to any Boolean algebra \( B \) extends to a unique homomorphism. The chapter defines \( n \)-preserving functions and explores the properties of \( n \)-independence, which generalizes the notion of disjointness in infinite sets. It also discusses the relationship between \( n \)-free Boolean algebras and graph spaces, hypergraph spaces, and their duals. Key results include the equivalence of \( n \)-freeness and having an \( n \)-independent generating set, and the construction of \( n \)-free Boolean algebras from hypergraphs. The chapter concludes with discussions on algebraic constructions and the closure properties of these structures under various operations.