SAMS Subject Classification Number: 13

In early 1960, Császár [2] introduced the notion of syntopogenous structures for spaces with the intention of studying, in the same setting, topological, uniform and proximity spaces. Recently, in [1], these orders were extended to a general category. In particular, topogenous orders were shown to play vital roles among which the unification of neighborhood, closure and interior operators.

In [4], quasi-uniformities and syntopogenous structures were successfully studied. Motivated by the fact that many researchers have focused their attention on the observation that the important aspect of a topological space is not its set of points but its lattice of open subsets, in this talk, we present the point-free counterpart of the results in [1] and [4].

We define topogenous orders on the category of frames. We show that they embrace nuclei and interior operators and many orders on frames including the Császár order introduced in [3].

Following [3] and [5], for syntopogenous and quasi-uniformities on frames respectively, we study entourage quasi-uniformities using syntopogenous structures on frames. We establish a Galois connection between syntopogenous structures and quasi-uniformities on a frame. This Galois connection permits us to establish which syntopogenous structures are isomorphic to the en tourage quasi-uniformities. Furthermore, this isomorphism lies at the center of the observation that a quasi-uniformity on a frame is a family of nuclei. In concluding we show that our results can be extended to the weil quasi-uniformities.

[1] D. Holgate, M. Iragi, and A. Razafindrakoto, Topogenous and nearness structures on categories. Appl. Categor. Struct 24 (2016), 447-455.

[2] A. Császár, Foundations of general topology, Pergamon, 1963.

[3] S.H. Chung Cauchy completion of Csaszar frames. Journal of the Korean Mathematical Society, 42 (2005), 291–304.

[4] D. Holgate and M. Iragi, Quasi-uniformities and syntopogenous structures categories. Topology and its applications, 263 (2019), 16–25.

[5] P. Fletcher; W. N. Hunsaker; W.F. Lindgren, Totally boundend quasi-uniformities. Commentationes Mathematicae Universitatis Carolinae, 34 (1993), No. 3, 529-537.