On Strong Inclusions and Asymmetric Proximities in Frames

Abstract

Abstract:The strong inclusion, a specific type of subrelation of the order of a lattice with pseudocomplements, has been used in the concrete case of the lattice of open sets in topology for an expedient definition of proximity, and allowed for a natural point-free extension of this concept. A modification of a strong inclusion for biframes then provided a point-free model also for the non-symmetric variant. In this paper we show that a strong inclusion can be non-symmetrically modified to work directly on frames, without prior assumption of a biframe structure. The category of quasi-proximal frames thus obtained is shown to be concretely isomorphic with the biframe based one, and shown to be related to that of quasi-uniform frames in a full analogy with the symmetric case.