General algebra

The part of algebra concerned with the study of various algebraic systems, including the theory of groups, rings, modules, semi-groups, lattices, etc. Such areas as the study of matrices and linear equations, algebraic geometry and algebraic number theory, multi-linear algebra, etc., remain outside the scope of general algebra. The distinction of general algebra as a part of algebra is rather conventional, and its boundaries are not fixed. For example, it is hard to say whether field theory, the theory of finite groups and the theory of finite-dimensional Lie algebras should be regarded as general algebra.

If a universal algebra is provided with an order or a topology compatible with the operations, then one has a partially ordered or topological algebra, respectively. The investigation of these objects is also regarded as belonging to general algebra. The most developed theories are those of partially ordered and topological groups and rings.

The origin of the development of general algebra goes back to the 19th century, when certain finite groups and finite-dimensional algebras were investigated. However, modern general algebra is associated with the penetration of set-theoretic modes of thought into algebra and is a product of the 20th century. Thus, the first monograph (O.Yu. Shmidt, see [1]) in which a group is not assumed to be finite by definition appeared only in 1916. Initially this reorientation concerned group theory, and then ring theory. The results of this reorientation are reflected in the monograph of B.L. van der Waerden, published in 1930–1931 in German [2].

The revelation of general features in the theory of groups and rings led to the consideration of lattices, universal algebras and categories. The appearance of the theory of models and algebraic systems is associated with the discovery of links between algebra and mathematical logic. The desire to find out to what extent various results in group theory depend on the presence of inverse elements or on associativity gave rise to semi-groups and quasi-groups. All these newly-appearing sections of general algebra found after some time their own problems, their own methods of development and their own fields of application (for example, semi-groups have turned out to be particularly important in the algebraic theory of automata). In addition, these trends in turn began to influence the classical areas of general algebra from which they had arisen. Thus, the concept of a variety of algebras, which crystallized in the theory of universal algebras, plays an important role in modern group and ring theory. As another example one can mention the study of classes of groups and rings defined by properties of their lattices of subgroups and normal subgroups, ideals and subrings, as well as problems concerned with lattice isomorphisms. See also Universal algebra.

#### References[edit]

[1] |  O.Yu. Shmidt, "Abstract theory of groups" , Freeman (1966) (Translated from Russian)

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[2] |  B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) MR1541390 Zbl 1032.00002 Zbl 1032.00001 Zbl 0903.01009 Zbl 0781.12003 Zbl 0781.12002 Zbl 0724.12002 Zbl 0724.12001 Zbl 0569.01001 Zbl 0534.01001 Zbl 0997.00502 Zbl 0997.00501 Zbl 0316.22001 Zbl 0297.01014 Zbl 0221.12001 Zbl 0192.33002 Zbl 0137.25403 Zbl 0136.24505 Zbl 0087.25903 Zbl 0192.33001 Zbl 0067.00502
[3] |  A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) MR0158000 Zbl 0121.25901
[4] |  A.G. Kurosh, "General algebra. Lectures for the academic year 1969/70" , Moscow (1974) (In Russian) MR0392756
[5] |  A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian) Zbl 0266.08001
[6] |  M. Suzuki, "Structure of a group and the structure of its lattice of subgroups" , Springer (1956) MR0083487 Zbl 0070.25406

#### Comments[edit]

In the West, the phrase "general algebra" is not used as a technical term, except as a synonym for universal algebra.

In fact, infinite groups did made their appearance before in monographs (see, e.g., [a1], Section 14, where it is clearly stated that "the number of distinct operations contained in a group may be either finite or infinite" and [a2]).

#### References[edit]

[a1] |  W. Burnside, "Theory of groups of finite order" , Cambridge Univ. Press (1897) MR0069818 Zbl 28.0118.03

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[a2] |  A. Schoenflies, "Kristallsysteme und Kristallstruktur" , Teubner (1891)