Search Result for "algebra":
Wordnet 3.0

NOUN (1)

1. the mathematics of generalized arithmetical operations;

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The Collaborative International Dictionary of English v.0.48: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]

The Collaborative International Dictionary of English v.0.48: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

WordNet (r) 3.0 (2006):algebra
n 1: the mathematics of generalized arithmetical operations

The Free On-line Dictionary of Computing (30 December 2018):algebra

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)
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