Topogenous orders and their applications on lattices

Abstract

In his influential book, [Cs´a63] A. Cs´asz´ar developed the well-known theory of syntopogenous ´ structures on a set. His intention was to create a comprehensive framework that simultaneously encompasses the study of topological, proximal, and uniform structures. In the same monograph, he demonstrated independently, along with Pervin [Per62], that every topological space possesses a compatible quasi-uniformity. A similar observation was noted for a uniform space, provided the topological space is completely regular. On the other hand, Herrlich in [Her74a] introduced the concept of “nearness” with the aim of unifying various topological structures. This Ph.D. thesis aims to investigate topogenous orders and their generalizations, such as quasi-uniformities, syntopogenous structures, on complete lattices which extend and generalize existing literature in this field. We explore the study of quasi-uniformities through the lens of syntopogenous structures, and establish a Galois connection between these two constructs. Furthermore, we provide conditions under which certain Cs´asz´ar structures are order isomorphic to quasi-uniformities on a complete lattice. As Cs´asz´ar structures are deeply rooted in pointfree topology, our research naturally extends into the realm of frames. We establish a correspondence between pre-nearness and Cs´asz´ar structures. In line with these ideas, we also delve into the relationship between pre-uniformities and entourage quasi-uniformities in the context of frames