A General Stone Representation Theorem

This note contains a Stone-style representation theorem for compact Haus-dorff spaces. The note is very much inspired by some existing representation theorems and is expository in nature. The first representation theorem is by Jung and Sünderhauf in [JS] and there is also a version of it for compact Hausdorff spaces noted by D. Moshier [Mo], the ideas of which were described to us by A. Jung. This covers §1 of the note. The other representation theorem uses normal lattices and was discovered recently by G. Plebanek [Pl]. We show in §2 that the two representation theorems are reducible to each other. Some notation and definitions might be specific to this note. The original, highly recommended paper [JS], deals with a much more general situation where neither the Hausdorff property nor compactness is assumed in the representation theorem, while the notes [Pl] develop the lattice representation theorem in a self-sufficient manner. Further results on representation theorems likely including the theorem described in §1 here, will be published as part of a future paper by Jung et al, and the work of Plebanek is done as part of a separate project on Banach spaces. 1 Spils Definition 1.1 A strong proximity involution lattice (spil) is given by a 1 is a bounded distributive lattice and the following additional axioms hold: (i) ≺ is a binary relation which is interpolating, meaning it satisfies ≺ 2 =≺ so for all a, b, c ∈ B