"Structure" can refer to a specific mathematical object or to a more abstract concept. For example, the Monster Group both "is" an algebraic structure, and "has" a structure in common with all other Group s. This entry will employ both meanings of "structure."
The structures below are grouped first by the number of sets and binary operations each requires, then ranked informally by the number of required identities.
(a set ''S'', '''no''' binary operation):
- Set : a set is a degenerate algebraic structure, having zero operations.
- Pointed Set : ''S'' has a distinguished element ''s''.
- Unary system: ''S'' has a Unary Operation , i.e. a function ''S'' → ''S''.
- Pointed unary system: a unary system that is also a pointed set, e.g., any model of the Peano Axioms .
(a set ''S'', '''one''' Binary Operation ):
(a set ''S'', '''two''' Binary Operations ):
- Ringoid: one binary operation (multiplication) Distributes over the other (addition).
- s. Addition also commutes.
- Boolean Algebra : an idempotent commutative semiring.
- --- ).
- --- addition, and two unary operations subject to Involution . ''S'' is the Cartesian Square of some set.
Kleene and relation algebras require additional axioms not discussed in this entry.
- Ring : a semiring with an additive inverse. Hence ''S'' is an abelian group under addition.
- --- Rng : a ring lacking a multiplicative identity.
- --- Commutative Ring : a ring whose multiplication commutes.
(a set ''S'', '''two''' Binary Operations ):
- and Join semilattice. The two operations satisfy the Absorption Law and form a dual pair of operations.
- over the other.
- , giving rise to an Inverse Element , a lattice bound. One of meet or join can be defined in terms of the other and complementation.
- Heyting Algebra : a Boolean algebra with an added binary operation, relative pseudo-complement.
- , and a unary operation, Converse . ''S'', the Cartesian Square of some set, is a Monoid under relative product, whose identity element is distinct from the Boolean bounds. Relative product is also residuated, and distributes over meet or join. Converse obeys involution.
(two sets, ''M'' and ''R'', '''three''' Binary Operation s):
- Module : ''M'' is an abelian group under addition, ''R'' is a ring. Scalar multiplication, a mapping from ''M''X''R'' to ''M'', associates in that for any two elements ''r'', ''s'' of ''R'', and any element ''x'' of ''M'', ''r''(''sx'') = (''rs'')''x'', has an identity element, and distributes over addition.
- . "Field" is defined below.
The above algebraic structures are all Formal Systems , i.e., Universal Algebra structures that can all be defined solely by Definition s and axiomatic identities. All operations are defined for all values of ''S'', hence no operation is partial.
The concept of algebraic structure can be extended to study sets with operations satisfying axioms other than identities. For example, the definition of an integral domain given below includes just such a restrictive condition, namely 0 ≠ 1 (the additive and multiplicative identities must be distinct). This is necessary, because if we admit 0=1, the notion of integral domain collapses. Although structures with restrictive conditions still retain an undoubted algebraic flavor, they suffer from defects absent from Universal Algebra structures. For example, no product of two integral domains exists, nor does a free field over any set.
(a set ''S'', '''two''' Binary Operation s. 0 is the additive identity.):
- on Ideals .
- is 0.
- Division Ring (or ''skew field''): a ring with a multiplicative inverse for all elements of ''S'' other than 0.
- Field : a division ring whose multiplication commutes.
- Ordered Field . The starting point for much of mathematics, including real and functional analysis.
(two sets, ''A'' and ''K'', '''three''' Binary Operation s):
- Every group is a loop, because ''a'' --- ''x'' = ''b'' If And Only If ''x'' = ''a''−1 --- ''b'', and ''y'' --- ''a'' = ''b'' if and only if ''y'' = ''b'' --- ''a''−1.
- The Integer s with addition (+) form an abelian group.
- The non-zero Rationals with Multiplication (×) form an abelian group.
- Two by two Matrices with multiplication form a group (non commutative).
- Every Cyclic Group ''G'' is abelian, because if ''x'', ''y'' are in ''G'', then ''xy'' = ''a''''m''''a''''n'' = ''a''''m'' + ''n'' = ''a''''n'' + ''m'' = ''a''''n''''a''''m'' = ''yx''. In particular, the Integer s form an abelian group under addition, as do the Integers Modulo ''n'' /''n''.
- Further examples can be found in Examples Of Groups .
- The Natural Number s (including Zero ), with the ordinary addition and multiplication is a commutative semiring.
- The set ''R'' {Link without Title} of all Polynomial s over some coefficient ring ''R'' forms a ring.
- Two by two Matrices with addition and multiplication form a ring (non commutative).
- ''Finite ring'': If ''n'' is a positive integer, then the set ''n'' = /''n'' of integers modulo ''n'' (as an additive group the Cyclic Group of order ''n'' ) forms a ring with ''n'' elements (see Modular Arithmetic ).
- The integers with the two operations of addition and multiplication form an integral domain.
- The P-adic Integers .
- The Rational Numbers with addition and multiplication form a field.
- The Real Numbers , under the usual operations of addition and multiplication.
- ---The real numbers contain several interesting subfields: the real Algebraic Number s, the Computable Number s, and the Definable Number s.
- When the real numbers are given the usual ordering they form a ''complete ordered field'' which is categorical — it is this structure that provides the foundation for most formal treatments of Calculus .
- The Complex Numbers , under the usual operations of addition and multiplication.
- An Algebraic Number Field is a finite field extension of the Rational Number s , that is, a field containing which has finite dimension as a Vector Space over . Such 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 ''q'', '''Z'''/''q'''''Z''', or GF(''q''). Every other finite field is isomorphic to one of these fields. Such fields are often called a Galois Field , whence the notation GF(''q'').
- ---In particular, for a given prime number ''p'', the set of integers modulo ''p'' is a finite field with ''p'' elements: ''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 .
Algebraic structures can also be defined on sets with added structure of a non-algebraic nature, such as Topological Space s. The added structure must be compatible, in some sense, with the algebraic structure.
- .
- .
- holds.
- Topological Group : a group with a compatible topology.
- structure.
- Graded Algebra : a field or commutative ring with extra structure known as "grading".
- --- associative algebra over a Vector Space ''V'' and a Field ''K'' with a bilinear "exterior product".(denoted by infix ∧) "alternating" on ''V''. Letting all letters denote elements of ''V'', ''v''∧''v''=0, ''u''∧''v'' = -''v''∧''u'', and whenever are Linearly Dependent .
Every algebraic structure has its own notion of Homomorphism , a Function that is compatible with the given operation(s).
In this way, every algebraic structure defines a Category .
For example, the category of groups has all groups as objects and all group homomorphisms as morphisms.
This category, being a Concrete Category , may be regarded as a category of sets with extra Structure in the category-theoretic sense.
Similarly, the category of topological groups (with continuous group homomorphisms as morphisms) 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
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