Proximity biframes and compactifications of completely regular ordered spaces

Guram Bezhanishvili and Patrick J. Morandi

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.

Published 2015-11-04.

http://www.tac.mta.ca/tac/volumes/30/44/30-44.pdf

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