This paper is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics. The paper has a survey character, too. The most frequently used algebras like cylindric-, relation-, polyadic-, and quasi-polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. In some (...) directions we tried to give an overview of the most recent results and research trends, too. (shrink)

For every k ∈ ω, there is an infinite set $A_k \subseteq \omega$ and a d(k) ∈ ω such that for all $Q_0, Q_1 \subseteq A_k$ where |Q 0 | = |Q 1 or $d(k) , the structures $\langle \omega, +, Q_0\rangle$ and $\langle \omega, +, Q_1\rangle$ are indistinguishable by first-order sentences of quantifier depth k whose atomic formulas are of the form u = v, u + v = w, and Q(u), where u, v, and w are variables.

Translator's summary The translated paper is an extract, published in 1945, of an unpublished thesis, of both historical and technical import, dealing with notions of definability and their relation to invariance under automorphisms. The author develops a metamathematical Galois theory, and discusses and anticipates some aspects of higher-order model theory in an informal but conceptually rich manner.

We use a fundamental theorem of Vaught, called the covering theorem in [V] (cf. theorem 0.1 below) as well as a generalization of it (cf. Theorem $0.1^\ast$ below) to derive several known and a few new results related to the logic $L_{\omega_1\omega}$. Among others, we prove that if every countable model in a $PC_{\omega_1\omega}$ class has only countably many automorphisms, then the class has either $\leq\aleph_0$ or exactly $2^{\aleph_0}$ nonisomorphic countable members (cf. Theorem $4.3^\ast$) and that the class of countable (...) saturated structures of a sufficiently large countable similarity type is not $PC_{\omega_1\omega}$ among countable structures (cf. Theorem 5.2). We also give a simple proof of the Lachlan-Sacks theorem on bounds of Morley ranks ($\s 7$). (shrink)