Abstract
The study of pure linear orderings—sets equipped solely with a linear (total) order—has deep historical roots in mathematical logic and order theory. Initial investigations trace back to Cantor’s work on ordinal numbers, which laid the foundation for understanding different sizes of ordered sets. Early 20th-century research by Hausdorff and others explored order types and their classification. Later, developments in model theory and set theory refined the structural properties of pure linear orderings, including their rigidity, embeddability, and definability in various logical frameworks. So, pure linear ordering and its classification are one of the classical mathematical questions. Descriptions of o-minimal and weakly o-minimal pure linear orderings are known, as well as we know that any pure linear ordering has an o-superstable elementary theory. The aim of this paper is to start investigation of pure linear ordering that have o-ω-stable elementary theory. So, we give the complete description of pure linear ordering of Morley o-rank 1. Key words: Pure linear ordering, o-minimal structure, o-stable theory, Dedekind’s cut, ordered structure, Morley o-rank.