Basic Tools, Increasing Functions, and Closure Operations in Generalized Ordered Sets

Abstract

Having in mind Galois connections, we establish several consequences of the following definitions.An ordered pair X( ≤ ) = (X, ≤ ) consisting of a set X and a relation ≤ on X is called a goset (generalized ordered set).For any x ∈ X and $A \subseteq X$, we write x ∈ ub _X (A) if a ≤ x for all a ∈ A, and $x \in \mathop{\mathrm{int}}\nolimits _{X}(A)$ if $\mathop{\mathrm{ub}}\nolimits _{X}(x) \subseteq A$, where $\mathop{\mathrm{ub}}\nolimits _{X}(x) =\mathop{ \mathrm{ub}}\nolimits _{X}{\bigl (\{x\}\bigr )}$.Moreover, for any $A \subseteq X$, we also write $A \in \mathcal{U}_{X}$ if $A \subseteq \mathop{\mathrm{ub}}\nolimits _{X}(A)$, and $A \in \mathcal{T}_{X}$ if $A \subseteq \mathop{\mathrm{int}}\nolimits _{X}(A)$. And in particular, $A \in \mathcal{E}_{X}$ if $\mathop{\mathrm{int}}\nolimits _{X}(A)\neq \emptyset$ .A function f of one goset X to another Y is called increasing if u ≤ v implies f(u) ≤ f(v) for all u, v ∈ X.In particular, an increasing function $\varphi$ of X to itself is called a closure operation if $x \leq \varphi (x)$ and $\varphi {\bigl (\varphi (x)\bigr )} \leq \varphi (x)$ for all x ∈ X.The results obtained extend and supplement some former results on increasing functions and can be generalized to relator spaces.

In Honor of Constantin Carathéodory