Perfect compactifications of frames.

Abstract

We study the compacti cations of frames. In particular, we study the compacti cations of frames which are perfect. That is, those compacti cations for which the right adjoint of the compacti cation mapping preserves disjoint binary joins. The Stone-C ech compacti cation of a completely regular frame and the Freudenthal compacti cation of a rim-compact frame are known to be examples of such compacti cations. We study the Freudenthal compacti cation of a rim-compact frame with an aim of providing more properties and characterizations of this compacti cation in the context of frames since this is less studied in the literature compared to the Stone-C ech compacti cation of frames. One of the main results that we obtain about the Freudenthal compacti cation of a rim-compact frame is that it is the minimal perfect compacti cation for this class of frames and the maximal -compacti cation. The notion of a full -compact basis is known in the context of spaces. We de ne an analogous concept in the context of frames and show that the Freudenthal compacti cation of a rim-compact frame arises from such a basis. We also establish the one-to-one correspondence between such bases and the -compacti cations of a rim-compact frame. The fact that the compacti cations arriving from such basis are zero-dimensional is also established.It is well known that a frame has the least compacti cation if and only if it is regular continuous. Some conditions under which the least compacti - cation of a regular continuous frame is perfect have been studied by Baboolal in [1] and the study is furthered herein. An N-point compacti cation of a space is any compacti cation whose remainder consists of N points. The N-star compacti cations of frames are known to be the frame analogue of the N-point compacti cations for spaces. It has been shown that the least compacti cation of a regular continuous frame is an example of an N-star compacti cation. We study the conditions under which a 2-star compacti- cation of a regular continuous frame is perfect and we conjecture that the results can be generalized to any N > 1: We prove that, under perfectness, the 2-star compacti cation of a regular continuous frame is the only N-star compacti cation. We also show some results related to the connectedness of the remainder of the 1-star compacti cation. Some contribution to the theory of compacti cations of frames not relating to perfectness has also been made. The concept and the construction of a freely generated frame is well known. We have shown that any compacti cation of a frame L can be realized as a frame freely generated by L subject to certain relations.