Representing expansions of bounded distributive lattices with Galois connections in terms of rough sets

• We study expansions of bounded distributive lattices equipped with a Galois connection.
• Our results generalize well-known representation theorems by Jónsson and Tarski presented for Boolean algebras with operators.
• The representations are given in terms of rough sets approximation operators and Alexandrov topologies.
• Heyting algebras with a Galois connection can be extended to spatial Heyting algebras with rough set approximation operators.
• Heyting–Brouwer algebras with a Galois connection can be extended to weakly atomic Heyting–Brouwer algebras with rough set approximation operators.

This paper studies expansions of bounded distributive lattices equipped with a Galois connection. We introduce GC-frames and canonical frames for these algebras. The complex algebras of GC-frames are defined in terms of rough set approximation operators. We prove that each bounded distributive lattice with a Galois connection can be embedded into the complex algebra of its canonical frame. We show that for every spatial Heyting algebra L equipped with a Galois connection, there exists a GC-frame such that L is isomorphic to the complex algebra of this frame, and an analogous result holds for weakly atomic Heyting–Brouwer algebras with a Galois connection. In each case of representation, given Galois connections are represented by rough set upper and lower approximations.