Compactification of closed preordered spaces

A topological preordered space admits a Hausdorff closed preorder compactification if and only if it is Tychonoff and the preorder is represented by the family of continuous isotone functions. We construct the largest Hausdorff closed preorder compactification for these spaces and clarify its relation with Nachbin’s compactification. Under local compactness the problem of the existence and identification of the smallest Hausdorff closed preorder compactification is considered.