# zbMATH — the first resource for mathematics

Vorlesungen über allgemeine Algebra. (Лекции по общей алгебре.) (Russian) Zbl 0105.24901
Moskva: Staatsverlag für physikalisch-mathematische Literatur, 396 S. (1962).
Throughout this book, the style is both pleasant and lucid, the choice of topics both apt and refreshing. Chapter 2 introduces groupoids, semigroups and groups; Chapter 3, universal algebra and groups with multiple operators [P. J. Higgins, Proc. Lond. Math. Soc. (3) 6, 366–416 (1956; Zbl 0073.01704)]; Chapter 4, lattices; Chapter 5, operator groups, modules and linear algebra; Chapter 6, ordered and topological groups along with valuations.
To give an indication of the richness of content, we append a detailed list of topics for Chapter 1: sets, mappings, inverses, axiom of choice; binary relations on a set, their products, inverses, the identity relation, the sharing of reflexivity, transitivity, symmetry and/or anti-symmetry by a relation and its inverse, the induced binary relation in a subset, $$n$$-ary relations; equivalence relations, partitions, unions and intersections of equivalence relations, the quotient of a set by an equivalence relation; partial and linear ordering with examples, order isomorphisms, embedding by an order isomorphism, the realization of a partially ordered set as a set of subsets under inclusion, inverse order isomorphisms; the minimal condition, the descending chain condition, the induction principle (in a most elegant form for partially ordered sets), the equivalence of these three conditions, construction of functions by induction, well ordering, maximal elements; upper and lower bounds of subsets, maximal chains, the equivalence of the axiom of choice, Zermelo’s theorem, Hausdorff’s theorem and the theorem of Kuratowski and Zorn.
Reviewer: F. Haimo

##### MSC:
 00A05 Mathematics in general 06-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures 08-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general algebraic systems 13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra 15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra 20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory 22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups