Result on Partially Ordered Compact Spaces

Partially ordered compact spaces were considered in the paper. E denote a compact Harsdorff space, C(E) denote the set of all continuous real valued functions on the compact Harsdorff space E, C+(E) is the set {f ∈ C(E) : f (t) ≥ 0, ∀t ∈ E} . ≤ denotes a relation of partial order given in E. That is, a binary relation that is reflexive (x ≤ x∀x ∈ E) antisymmetric x ≤ y and y ≤ ximpliesx = y, ∀x, y ∈ E) and transitive (x ≤ y and y ≤ zimpliesx ≤ z, ∀x, y, z ∈ E) and the class of all monotonic increasing functions belonging to C + (E), is denoted by F(E, ≤). This expository paper established that ≤ is the relation of partial order determined by F(E, ≤) if and only if E is monotonically separated relatives to ≤ . Also it is shown that there is a one-to-one correspondence F → ≤F between the uniformly closed semialgebras with identity and type 1 in C(E) and the relations of partial order in E relative to which E is monotonically separated.