Order dense embeddings into Riesz spaces

The formation of the Dedekind completion can be traced back to Richard Dedekind’s construction of real numbers in 1872. In his related work, he defined real numbers as Dedekind cuts of rational numbers equipped with appropriate arithmetic operations. In 1937, MacNeille showed that Dedekind cuts can be evaluated in any partially ordered set, and a Dedekind complete set can be constructed in which they can be embedded, and he called this set the Dedekind complement of the initial set. The biggest advantage of pre-Riesz spaces is that they can be studied with the help of order dense embeddings in Riesz spaces. In this thesis, three types of order dense embedding are discussed. 1) Order dense embedding of Archimedean directed ordered vector space into Dedekind complete

Riesz space, 2) Order dense embedding of Pre-Riesz space into Riesz space, 3) Order dense embedding of an Archimedean ordered vector space with order unit into the space of continuous functions defined on a compact Hausdorff space. Consider the partially ordered X^{δ} set of Dedekind cuts of the directed partial ordered vector space X and the operations ⊕, ⊖ and ∗ on X^{δ} . The main goal is to produce a result similar to embedding the Archimedean space X into its Dedekind completion. The space X is densely embedded in a Riesz space that is not necessarily Dedekind complete, and the condition for X to be Archimedean is relaxed to as weak a condition as possible. For X to be densely embedded in a Riesz space, a necessary and sufficient condition is that X is a pre-Riesz space.