A set-theoretic approach to algebraic L-domains

In this paper, the notion of locally algebraic intersection structure is introduced for algebraic L-domains. Essentially, every locally algebraic intersection structure is a family of sets, which forms an algebraic L-domain ordered by inclusion. It is shown that there is a locally algebraic intersection structure which is order-isomorphic to a given algebraic L-domain. This result extends the classic Stone’s representation theorem for Boolean algebras to the case of algebraic L-domains. In addition, it can be seen that many well-known representations of algebraic L-domains, such as logical algebras, information systems, closure spaces, and formal concept analysis, can be analyzed in the framework of locally algebraic intersection structures. Then, a set-theoretic uniformity across different representations of algebraic L-domains is established.