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

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.