Order embedding theorems and multi-utility representation of the preorder

Abstract

As an improvement of Fletcher-Lindgren order embedding theorem, we show that every completely regular ordered space X is order embedded in the Tychonoff ordered cube of infinite weight of X. For that purpose, we evaluate the minimal cardinality of continuous functions which represent multi-utility of the (pre)order on X. Moreover, for a topological preordered space X which admits a continuous multi-utility representation (or a completely regular ordered space X) and its compact subspace, similar descriptions by maps to Tychonoff ordered cube are also given.