We generalize the concept of a strong inclusion on a biframe to that of a proximity on a biframe, which is related to the concept of a strong bi-inclusion on a frame introduced by Picado and Pultr. We also generalize the concept of a bi-compactification of a biframe 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. In the spatial case this yields Blatter and Seever's characterization of compactifications of completely regular ordered spaces and a characterization of bi-compactifications of completely regular bispaces.
Keywords: Frame; biframe; bispace; ordered space; compactification; proximity
2010 MSC: 06D22; 54E55; 54D35; 54E05
Theory and Applications of Categories, Vol. 30, 2015, No. 44, pp 1469-1500.