PROXIMITY BIFRAMES AND COMPACTIFICATIONS OF COMPLETELY REGULAR ORDERED SPACES.

We generalize the concept of a strong inclusion on a biframe [Sch93] to that of a proximity on a biframe, which is related to the concept of a strong bi-inclusion on a frame introduced in [PP12b]. We also generalize the concept of a bi-compactification of a biframe [Sch93] to that of a compactification of a biframe, and prove that the poset of compactifications of a biframe L is isomorphic to the poset of proximities on L. As a corollary, we obtain Schauerte's characterization of bi-compactifications of a biframe [Sch93]. In the spatial case this yields Blatter and Seever's characterization of compactifications of completely regular ordered spaces [BS76] and a characterization of bi-compactifications of completely regular bispaces. [ABSTRACT FROM AUTHOR]