n. 1a. the branch of elementary mathematics that generalizes arithmetic by using variables to range over numbers, for example in arithmetical identities such as x + y = y + x.
b. in particular, the use of symbols standing for unknown quantities in order to determine their value by the elementary operations of arithmetic. 2. also called abstract algebra. the study of systems, such as rings, groups, and fields, endowed with finitary operations with specific properties. 3. any formal calculus used to model and study the properties of the entities that are the intended interpretation of their symbols, such as the algebra of logic, and the algebra of classes; thus one might construct an algebra of color properties. 4. (more specifically) a Boolean algebra, or sigma-algebra (σ-algebra), and in particular, the algebra of subsets or algebra of propositions. 5. any formal system with only functions and constants, but no relations except possibly identity. 6. a ring that is a module over a field. See also algebra over a field. See also linear algebra.