Ranks and Approximations for Families of Ordered Theories

Rank values for various families of ordered theories are described as depending on the languages under consideration; a description of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{e}$$\end{document}-total transcendence in terms of these languages is also given. Approximations of ordered theories are studied, including approximations by finite and countably categorical orders. Closures are studied and ranks are described for families of ordered theories, including the families of o-minimal and weakly o-minimal theories of various signatures, as well as theories of pure linear orders with various constraints on the discrete parts.