Nearly realcompact frames

A Tychonoff space X   has been defined by Blair and van Douwen to be nearly realcompact if βX−υXβX−υX is dense in βX−XβX−X. We extend this concept to frames. Notwithstanding the appearance of remainders in the definition – notions which do not exist, in general, in the category of frames – we obtain a conservative definition of nearly realcompact frames. We give several frame-theoretic characterisations of these types of frames. We also have a ring-theoretic characterisation which states that L   is nearly realcompact precisely if the intersection of the free maximal ideals of the ring RLRL coincides with the intersection of its hyper-real maximal ideals. We show that P-frames and paracompact frames are nearly realcompact. Hence, Boolean frames are nearly realcompact.