algebra

American Heritage® Dictionary of the English Language, Fourth Edition

1. n. A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set.
2. n. A set together with a pair of binary operations defined on the set. Usually, the set and the operations include an identity element, and the operations are commutative or associative.

Century Dictionary and Cyclopedia

1. n. Formal mathematics; the analysis of equations; the art of reasoning about relations, more especially quantitative relations, by the aid of a compact and highly systematized notation. In ordinary algebra the relations between quantities are expressed by signs of equality, addition, subtraction, multiplication, etc. ( =, + , —, ×), or by the position of the quantities (as xy for x × y, and x for x to the y power), and the quantities themselves are denoted by letters. Quantities whose values are unknown or are assumed to be variable are denoted by the last letters of the alphabet, as x, y, z; known or constant quantities by a, b, c, etc.; and problems are solved by expressing all the data in the form of equations, and then transforming these according to certain rules. The conceptions of negative and imaginary quantities (see negative and imaginary) are employed. The term higher algebra usually means the theory of invariants. See invariant. Multiple algebra, or n-way algebra, introduces the conception of units of different denominations, which can, however, be multiplied together. Each such system has a multiplication table characterizing it.
2. n. Any special system of notation adapted to the study of a special system of relationship: as, “it is an algebra upon an algebra,” Sylvester.
3. n. A treatise on algebra.
4. n. Its abbreviation is algebra
5. n. Algebra of multiple units.

Wiktionary

1. n. uncountable, mathematics A system for computation using letters or other symbols to represent numbers, with rules for manipulating these symbols.
2. n. uncountable, mathematics The study of algebraic structures.
3. n. countable, mathematics A universal algebra.
4. n. countable, algebra An algebraic structure consisting of a module of a commutative ring along with an additional binary operation that is bilinear.
5. n. countable, set theory, analysis A collection of subsets of a given set, such that this collection contains the empty set, and the collection is closed under unions and complements (and thereby also under intersections and differences).
6. n. countable, mathematics One of several other types of mathematical structure.

GNU Webster's 1913

1. n. (Math.) That branch of mathematics which treats of the relations and properties of quantity by means of letters and other symbols. It is applicable to those relations that are true of every kind of magnitude.
2. n. A treatise on this science.

WordNet 3.0

1. n. the mathematics of generalized arithmetical operations