algebra

algebra (ˈældʒɪbrə)
—n
1.	a branch of mathematics in which arithmetical operations and relationships are generalized by using alphabetic symbols to represent unknown numbers or members of specified sets of numbers
2.	the branch of mathematics dealing with more abstract formal structures, such as sets, groups, etc
[C14: from Medieval Latin, from Arabic al-jabr the bone-setting, reunification, mathematical reduction]
algebraist
—n

algebra

Arabic numerals to the West. The accent shifted 17c. from second syllable to first. The word was used in Eng. 15c.-16c. to mean "bone-setting," probably from the Arabs in Spain.

algebra   (āl'jə-brə)  Pronunciation Key  A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or quantities and express general relationships that hold for all members of a specified set.

algebra definition

A branch of mathematics marked chiefly by the use of symbols to represent numbers, as in the use of a2 + b2 = c2 to express the Pythagorean theorem.

algebra definition

mathematics, logic
1. A loose term for an algebraic structure.
2. A vector space that is also a ring, where the vector space and the ring share the same addition operation and are related in certain other ways.
An example algebra is the set of 2x2 matrices with real numbers as entries, with the usual operations of addition and matrix multiplication, and the usual scalar multiplication. Another example is the set of all polynomials with real coefficients, with the usual operations.
In more detail, we have:
(1) an underlying set,
(2) a field of scalars,
(3) an operation of scalar multiplication, whose input is a scalar and a member of the underlying set and whose output is a member of the underlying set, just as in a vector space,
(4) an operation of addition of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a vector space or a ring,
(5) an operation of multiplication of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a ring.
This whole thing constitutes an `algebra' iff:
(1) it is a vector space if you discard item (5) and
(2) it is a ring if you discard (2) and (3) and
(3) for any scalar r and any two members A, B of the underlying set we have r(AB) = (rA)B = A(rB). In other words it doesn't matter whether you multiply members of the algebra first and then multiply by the scalar, or multiply one of them by the scalar first and then multiply the two members of the algebra. Note that the A comes before the B because the multiplication is in some cases not commutative, e.g. the matrix example.
Another example (an example of a Banach algebra) is the set of all bounded linear operators on a Hilbert space, with the usual norm. The multiplication is the operation of composition of operators, and the addition and scalar multiplication are just what you would expect.
Two other examples are tensor algebras and Clifford algebras.
[I. N. Herstein, "Topics in Algebra"].
(1999-07-14)