Algebra

algebra
    n 1: the mathematics of generalized arithmetical operations
    

Mathematics \Math`e*mat"ics\, n. [F. math['e]matiques, pl., L.
   mathematica, sing., Gr. ? (sc. ?) science. See {Mathematic},
   and {-ics}.]
   That science, or class of sciences, which treats of the exact
   relations existing between quantities or magnitudes, and of
   the methods by which, in accordance with these relations,
   quantities sought are deducible from other quantities known
   or supposed; the science of spatial and quantitative
   relations.
   [1913 Webster]

   Note: Mathematics embraces three departments, namely: 1.
         {Arithmetic}. 2. {Geometry}, including {Trigonometry}
         and {Conic Sections}. 3. {Analysis}, in which letters
         are used, including {Algebra}, {Analytical Geometry},
         and {Calculus}. Each of these divisions is divided into
         pure or abstract, which considers magnitude or quantity
         abstractly, without relation to matter; and mixed or
         applied, which treats of magnitude as subsisting in
         material bodies, and is consequently interwoven with
         physical considerations.
         [1913 Webster]
    

Algebra \Al"ge*bra\, n. [LL. algebra, fr. Ar. al-jebr reduction
   of parts to a whole, or fractions to whole numbers, fr.
   jabara to bind together, consolidate; al-jebr
   w'almuq[=a]balah reduction and comparison (by equations): cf.
   F. alg[`e]bre, It. & Sp. algebra.]
   1. (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.
      [1913 Webster]

   2. A treatise on this science.
      [1913 Webster] Algebraic
    

algebra

   <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)