# Ranks for Families of Theories and Their Spectra

Authors
• 1 Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090, Russia , Novosibirsk (Russia)
• 2 Novosibirsk State Technical University, Novosibirsk, 630073, Russia , Novosibirsk (Russia)
Type
Published Article
Journal
Lobachevskii Journal of Mathematics
Publisher
Publication Date
Dec 13, 2021
Volume
42
Issue
12
Pages
2959–2968
Identifiers
DOI: 10.1134/S1995080221120313
Source
Springer Nature
Keywords
Disciplines
• Article
AbstractWe define ranks and degrees for families of theories, similar to Morley rank and degree, as well as Cantor–Bendixson rank and degree, and the notion of totally transcendental family of theories. Bounds for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e$$\end{document}-spectra with respect to ranks and degrees are found. It is shown that the ranks and the degrees are preserved under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document}-closures and values for the ranks and the degrees are characterized. Criteria for totally transcendental families in terms of cardinality of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document}-closure and of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e$$\end{document}-spectrum value, for a countable language, are proved.