Galois and Pataki Connections on Generalized Ordered Sets

Abstract

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(\leq )=(X,\leq ) 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)\leq y implies x \leq g(y). While, the function f is called increasingly upper φ-semiregular, for some function φ of X to itself, if f(u)\leq f(v) implies u\leq \varphi (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.