Galois and Pataki Connections on Generalized Ordered Sets

  • Árpád Száz Department of Mathematics, University of Debrecen, H-4002 Debrecen, P.O. Box 400, Hungary
Keywords: ordered sets, Galois connections, uniform spaces, continuous relations


In this paper, having in mind Galois and Pataki connections, we establish several basic theorems on increasingly seminormal and semiregular functions between gosets.

An ordered pair X(≤) = (X, ≤) consisting of a set X and a relation ≤ on X is called a goset (generalized ordered set).

A function f of one goset X to another Y is called increasingly upper g-seminormal, for some function g of Y to X, if  f (x) ≤ y implies xg(y).

While, the function f is called increasingly upper φ-semiregular, for some function  of X to itself, if  f (u) ≤  f (v) implies u ≤ φ(v).

The increasingly lower seminormal (semiregular) functions are defined by the reverse implications. Moreover, a function is called increasingly normal (regular) if it is both increasingly upper and lower seminormal (semiregular).

The results obtained extend and supplement several former results of O. Ore and the present author on Galois and Pataki connections. Namely, the pairs ( f, g) and ( f, φ) may be called increasing Galois and Pataki connections if the function f is increasingly g‑normal and φ-regular, respectively.

How to Cite
Száz, Árpád. (2019). Galois and Pataki Connections on Generalized Ordered Sets. Earthline Journal of Mathematical Sciences, 2(2), 283-323.