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In algebra, a branch of
pure
mathematics, an algebraic structure consists
of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the
study of algebraic structures and their properties. The notion of
algebraic structure has been
formalized in universal algebra.
As an abstraction, an "algebraic structure" is the collection of
all possible models of a given set of axioms. More
concretely, an algebraic structure is any particular model of some
set of axioms. For example, the monster group both "is" an algebraic
structure in the concrete sense, and abstractly, "has" the group
structure in common with all other groups. This article employs both
meanings of "structure."
This definition of an algebraic structure should not be taken as
restrictive. Anything that satisfies the axioms defining a
structure is an instance of that structure, regardless of how many
other axioms that instance happens to have. For example, all groups
are also semigroups and
magmas.
Structures whose
axioms are all identities
If the axioms defining a structure are all identities, the structure is a variety (not to be confused
with algebraic variety in the sense of algebraic
geometry). Identities are equations formulated using only the
operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. Identities contain no
connectives, existentially
quantified variables, or relations of any kind other than the
allowed operations. The study of varieties is an important part of
universal
algebra.
All structures in this section are varieties. Some of these
structures are most naturally axiomatized using one or more
nonidentities, but are nevertheless varieties because there exists
an equivalent axiomatization, one perhaps less perspicuous,
composed solely of identities. Algebraic structures that are not
varieties are described in the following section, and differ from
varieties in their metamathematical properties.
In this section and the following one, structures are listed in
approximate order of increasing complexity, operationalized as
follows:
- Simple structures requiring but one set, the universe S, are listed
before composite ones requiring two sets;
- Structures having the same number of required sets are then
ordered by the number of binary operations (0 to 4) they
require. Incidentally, no structure mentioned in this entry
requires an operation whose arity exceeds 2;
- Let A and B be the two sets that make up a
composite structure. Then a composite structure may include 1 or 2
functions of the form
AxA→B or
AxB→A;
- Structures having the same number and kinds of binary
operations and functions are more or less ordered by the number of
required unary and 0-ary (distinguished
elements) operations, 0 to 2 in both cases.
The indentation structure employed in this section and the one
following is intended to convey information. If structure
B is under structure A and more indented, then
all theorems of A
are theorems of B; the converse does not hold.
Ringoids and lattices can be clearly distinguished
despite both having two defining binary operations. In the case of
ringoids, the two operations are linked by the distributive law; in the case of lattices,
they are linked by the absorption law. Ringoids also tend to
have numerical models, while lattices tend to have set-theoretic models.
Simple structures: No binary
operation:
- Set:
a degenerate algebraic structure having no operations.
- Pointed set:
S has one or more distinguished elements, often 0, 1, or
both.
- Unary system: S and a single unary operation
over S.
- Pointed unary system: a unary system with S a pointed
set.
Group-like structures:
One binary operation, denoted by concatenation. For
monoids, boundary algebras, and sloops, S
is a pointed
set.
Three binary operations. Quasigroups are
listed here, despite their having 3 binary operations, because they
are (nonassociative) magmas. Quasigroups feature 3 binary
operations only because establishing the quasigroup cancellation property by means of
identities alone requires two binary operations in addition to the
group operation.
Lattice: Two
or more binary operations, including meet and join, connected
by the absorption
law. S is both a meet and join semilattice, and is a pointed set if and only
if S is bounded. Lattices often have no unary operations.
Every true statement has a dual, obtained by replacing every instance of
meet with join, and vice versa.
- Bounded lattice: S has two
distinguished elements, the greatest lower
bound and the least upper
bound. Dualizing
requires replacing every instance of one bound by the other, and
vice versa.
- Modular
lattice: a lattice in which the modular identity holds.
- Distributive lattice: a lattice in
which each of meet and join distributes over the other.
Distributive lattices are modular, but the converse does not hold.
- Kleene
algebra: a bounded distributive lattice with a unary operation
whose identities are x"=x, (x+y)'=x'y', and (x+x')yy'=yy'. See
"ring-like structures" for another structure having the same
name.
- Boolean algebra: a
complemented distributive lattice. Either of meet or join can be
defined in terms of the other and complementation.
- Heyting
algebra: a bounded distributive lattice with an added binary
operation, relative
pseudo-complement, denoted by infix " ' ", and governed by the axioms x'x=1,
x(x'y) = xy, x'(yz) = (x'y)(x'z), (xy)'z = (x'z)(y'z).
Ringoids: Two binary operations, addition and multiplication, with multiplication distributing over
addition. Semirings are pointed sets.
N.B. The above definition of ring does not command universal
assent. Some authorities employ "ring" to denote what is here
called a rng, and refer to a ring in the above sense as a "ring
with identity."
Modules: Composite Systems Defined
over Two Sets, M and R: The members
of:
- R are scalars, denoted by Greek letters.
R is a ring under the binary operations of scalar addition
and multiplication;
- M are module elements (often but not
necessarily vectors), denoted by Latin letters.
M is an abelian group under addition. There may
be other binary operations.
The scalar multiplication of scalars and module
elements is a function RxM→M which
commutes, associates (∀r,s∈R,
∀x∈M, r(sx) =
(rs)x ), has 1 as identity element, and
distributes over module and scalar addition. If only the
pre(post)multiplication of module elements by scalars is defined,
the result is a left (right) module.
- Free module: a
module having a free basis,
{e_{1}, ...
e_{n}}⊂M, where the positive
integer n is the dimension of the free module. For every
v∈M, there exist κ_{1}, ...,
κ_{n}∈R such that v =
κ_{1}e_{1} + ... +
κ_{n}e_{n}. Let 0 and 0
be the respective identity elements for module and scalar addition.
If r_{1}e_{1} + ... +
r_{n}e_{n} = 0,
then r_{1} = ... = r_{n} =
0.
- Algebra over a ring (also
R-algebra): a (free) module where R is a commutative
ring. There is a second binary operation over M,
called multiplication and denoted by concatenation, which
distributes over module addition and is bilinear: α(xy) =
(αx)y = x(αy).
- Jordan
ring: an algebra over a
ring whose module multiplication commutes, does not associate,
and respects the Jordan identity.
Vector spaces,
closely related to modules, are defined in the next section.
Structures
with some axioms that are not identities
The structures in this section are not varieties because they
cannot be axiomatized with identities alone. Nearly all of the
nonidentities below are one of two very elementary kinds:
- The starting point for all structures in this section is a
"nontrivial" ring, namely one such that S≠{0}, 0 being the
additive identity element. The nearest thing to
an identity implying S≠{0} is the nonidentity 0≠1, which
requires that the additive and multiplicative identities be
distinct.
- Nearly all structures described in this section include
identities that hold for all members of S except 0. In
order for an algebraic structure to be a variety, its operations
must be defined for all members of S; there can be no
partial operations.
Structures whose axioms unavoidably include nonidentities are
among the most important ones in mathematics, e.g., fields
and vector spaces.
Moreover, much of theoretical physics can be recast as models of multilinear
algebras. Although structures with nonidentities retain an
undoubted algebraic flavor, they suffer from defects varieties do
not have. For example, neither the product of integral
domains nor a free field over any set exist.
Arithmetics: Two binary
operations, addition and multiplication. S is an infinite set.
Arithmetics are pointed unary systems, whose unary operation
is injective successor, and
with distinguished element 0.
- Robinson arithmetic. Addition and
multiplication are recursively defined by means of successor. 0 is the identity
element for addition, and annihilates multiplication. Robinson
arithmetic is listed here even though it is a variety, because of
its closeness to Peano arithmetic.
- Peano arithmetic. Robinson arithmetic with
an axiom schema of
induction. Most ring and field
axioms bearing on the properties of addition and multiplication are
theorems of Peano arithmetic or of proper extensions thereof.
Field-like structures: Two binary
operations, addition and multiplication. S is
nontrivial, i.e., S≠{0}.
- Domain: a ring whose sole zero divisor is 0.
- Division
ring (or sfield, skew field): a ring in which
every member of S other than 0 has a two-sided
multiplicative inverse. The nonzero members of S form a group
under multiplication.
- Field: a division ring whose
multiplication commutes. The nonzero members of S form an
abelian group
under multiplication.
The following structures are not varieties for reasons in
addition to S≠{0}:
Composite Systems: Vector Spaces, and Algebras over
Fields. Two Sets, M and R, and at least
three binary operations.
The members of:
- M are vectors, denoted by lower case letters.
M is at minimum an abelian group under vector addition, with
distinguished member 0.
- R are scalars, denoted by Greek letters.
R is a field, nearly always the real or complex field, with 0 and 1 as
distinguished members.
Three binary operations.
- Vector space:
a free module of dimension n except
that R is a field.
- Normed vector space: a vector space
with a norm, namely a function
M → R that is symmetric, linear, and positive definite.
- Inner product space (also
Euclidean vector space): a normed vector space such that
R is the real
field, whose norm is the square root of the inner product,
M×M→R. Let i,j, and
n be positive integers such that
1≤i,j≤n. Then M has an orthonormal
basis such that e_{i}•e_{j} =
1 if i=j and 0 otherwise; see free module above.
- Unitary space: Differs from inner
product spaces in that R is the complex field, and
the inner product has a different name, the hermitian inner product, with different
properties: conjugate symmetric, bilinear, and positive definite. See
Birkhoff and MacLane (1979: 369).
- Graded vector space: a vector space
such that the members of M have a direct sum decomposition. See graded algebra
below.
Four binary operations.
- Algebra over a field: An algebra over a ring except that R
is a field instead of a commutative ring.
- Jordan
algebra: a Jordan ring except that R is a
field.
- Lie algebra: an
algebra over a field respecting
the Jacobi
identity, whose vector multiplication, the Lie bracket denoted
[u,v], anticommutes, does not associate, and is nilpotent.
- Associative algebra: an algebra
over a field, or a module, whose vector
multiplication associates.
- Linear
algebra: an associative unital algebra with the members of M
being matrices. Every matrix has a dimension
nxm, n and m positive integers.
If one of n or m is 1, the matrix is a vector; if
both are 1, it is a scalar. Addition of matrices is defined only if
they have the same dimensions. Matrix multiplication, denoted by
concatenation, is the vector multiplication. Let matrix A
be nxm and matrix B be
ixj. Then AB is defined if and only if
m=i; BA, if and only if j=n. There also
exists an mxm matrix I and an
nxn matrix J such that
AI=JA=A. If u and v
are vectors having the same dimensions, they have an inner product, denoted
〈u,v〉. Hence there is an orthonormal
basis; see inner product space above. There is
a unary function, the determinant, from square
(nxn for any n) matrices to
R.
- Commutative algebra: an associative
algebra whose vector multiplication commutes.
Composite Systems: Multilinear algebras. Two
sets, V and K. Four binary
operations:
- The members of V are multivectors (including vectors), denoted by
lower case Latin letters. V is an abelian group under multivector addition,
and a monoid under outer product. The
outer product goes under various names, and is multilinear in principle
but usually bilinear. The
outer product defines the multivectors recursively starting from
the vectors. Thus the members of V have a "degree" (see graded algebra
below). Multivectors may have an inner product as
well, denoted u•v:
V×V→K, that is symmetric, linear, and positive definite; see inner
product space above.
- The properties and notation of K are the same as those
of R above, except that K may have -1 as a
distinguished member. K is usually the real field, as
multilinear algebras are designed to describe physical phenomena
without complex
numbers.
- The multiplication of scalars and multivectors,
V×K→V, has the same properties as the
multiplication of scalars and module elements that is part of a module.
- Graded
algebra: an associative algebra with unital outer product. The members of V
have a direct sum
decomposition resulting in their having a "degree," with vectors
having degree 1. If u and v have degree
i and j, respectively, the outer product of
u and v is of degree i+j. V
also has a distinguished member 0 for each
possible degree. Hence all members of V having the same
degree form an abelian group under addition.
- Exterior
algebra (also Grassmann algebra): a graded algebra
whose anticommutative outer product, denoted by
infix ∧, is called the exterior product.
V has an orthonormal basis.
v_{1} ∧ v_{2} ∧ ... ∧
v_{k} = 0 if and only if v_{1},
..., v_{k} are linearly
dependent. Multivectors also have an inner product.
- Clifford
algebra: an exterior algebra with a symmetric bilinear form
Q: V×V→K. The special case
Q=0 yields an exterior algebra. The exterior product is
written 〈u,v〉. Usually,
〈e_{i},e_{i}〉 = -1 (usually) or 1
(otherwise).
- Geometric algebra: an exterior
algebra whose exterior (called geometric) product is
denoted by concatenation. The geometric product of parallel
multivectors commutes, that of orthogonal vectors anticommutes. The
product of a scalar with a multivector commutes. vv yields
a scalar.
Examples
Some recurring universes: N=natural numbers; Z=integers;
Q=rational numbers;
R=real
numbers; C=complex numbers.
N is a pointed unary system, and under addition
and multiplication, is both the standard interpretation of Peano arithmetic and a commutative semiring.
Boolean algebras are at
once semigroups, lattices, and
rings.
They would even be abelian groups if the identity and
inverse elements were identical instead of complements.
Group-like structures
Lattices
Ring-like structures
- The set R[X] of all polynomials over some coefficient ring
R is a ring.
- 2x2 matrices with matrix addition and
multiplication form a ring.
- If n is a positive integer, then the set
Z_{n} =
Z/nZ of integers modulo
n (the additive cyclic group of order n ) forms a
ring having n elements (see modular arithmetic).
Integral domains
- Z under addition and multiplication is an
integral domain.
- The p-adic
integers.
Fields
- Each of Q, R, and
C, under addition and multiplication, is a
field.
- R totally ordered by "<" in the usual way
is an ordered
field and is categorical. The resulting real field grounds real and functional
analysis.
- An algebraic number field is a
finite field extension of Q, that is, a field
containing Q which has finite dimension as a vector space over
Q. Algebraic number fields are very important in
number
theory.
- If q > 1 is a power of a prime number, then there exists (up to isomorphism) exactly one finite field with
q elements, usually denoted
F_{q}, or in the case that
q is itself prime, by
Z/qZ. Such fields are
called Galois fields, whence the alternative
notation GF(q). All finite fields are isomorphic to some
Galois field.
- Given some prime number p, the set
Z_{p} =
Z/pZ of integers modulo
p is the finite field with p elements:
F_{p} = {0, 1, ...,
p − 1} where the operations are defined by
performing the operation in Z, dividing by
p and taking the remainder; see modular
arithmetic.
Allowing additional
structure
Algebraic structures can also be defined on sets with added
structure of a non-algebraic nature, such as a topology. The added structure must be
compatible, in some sense, with the algebraic structure.
Category
theory
The discussion above has been cast in terms of elementary abstract and
universal
algebra. Category theory is another way of
reasoning about algebraic structures (see, for example, Mac Lane
1998). A category is a collection of objects with
associated morphisms. Every algebraic structure has its
own notion of homomorphism, namely any function compatible with the
operation(s) defining the structure. In this way, every algebraic
structure gives rise to a category. For example, the category of
groups has all groups as objects and all group
homomorphisms as morphisms. This concrete category may be seen as a category of
sets with added category-theoretic structure. Likewise, the
category of topological groups (whose morphisms
are the continuous group homomorphisms) is a category of topological
spaces with extra structure.
There are various concepts in category theory that try to
capture the algebraic character of a context, for instance
See also
References
- MacLane, Saunders; Birkhoff,
Garrett (1999), Algebra (2nd ed.), AMS Chelsea, ISBN
978-0-8218-1646-2
- Michel,
Anthony N.; Herget, Charles J. (1993), Applied Algebra and
Functional Analysis, New York: Dover Publications, ISBN
978-0-486-67598-5
A monograph available free online:
Category theory:
External
links