Algebraic structure

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In higher mathematics, "algebraic structure" is a loosely-defined phrase referring to the mathematical objects traditionally studied in the field of abstract algebra: sets with operations.

The word "structure" can refer to a specific mathematical object or an even more abstract concept. For example, the monster group simultaneously is an algebraic structure, and it has an algebraic structure: the structure shared by all groups. This article uses both senses of the phrase.

Contents 1 In the sense of universal algebra 2 Allowing axioms other than identities 3 Examples 4 Allowing additional structure 5 Categories 6 See also

In the sense of universal algebra

In universal algebra, one studies algebraic structures consisting of a set and a collection of operations defined on the set which are required to satisfy certain identities.

Simple structures

• Set: a set is a degenerate algebraic structure, one that has zero operations defined on it
• Pointed set: a set S with a distinguished element s of S
• Unary system: a set S with a unary operation, i.e. a function S → S
• Pointed unary system: a unary system with a distinguished element (such objects occur in discussions of the Peano axioms)

Group-like structures (One binary operation)

• Magma or groupoid: a set with a single binary operation
• Quasigroup: a magma in which the binary operation always has an inverse
• Loop: a quasigroup with an identity element
• Semigroup: an associative magma
• Monoid: a semigroup with an identity element
• Group: a monoid in which every element has an inverse, or equivalently, an associative loop
• Abelian group: a commutative group

Ring-like structures (two binary operations)

• Semiring: a set which forms a semi-group under two different binary operations, where one of them ("addition") commutes, satisfying distributivity. This the same as a ring, but without additive inverses.
• Ring: a set with an abelian group operation as addition, together with a monoid operation as multiplication, satisfying distributivity
• Rng: a ring without a multiplicative identity.
• Commutative ring: a ring whose multiplication is commutative
• Kleene algebra: an idempotent semiring with additional unary operator (the Kleene star); these are modeled on regular expressions

Modules

• Module over a given ring R: a set with an abelian group operation as addition, together with an additive unary operation of scalar multiplication for every element of R, with an associativity condition linking scalar multiplication to multiplication in R
• Vector space: a module over a field (see below for fields)

Algebras

• Algebra: a module or vector space together with a bilinear operation as multiplication
• Associative algebra: an algebra whose multiplication is associative
• Commutative algebra: an associative algebra whose multiplication is commutative
• Lie algebra: a non-associative algebra important in geometry

Lattices

• Lattice: a set with two commutative, associative, idempotent operations satisfying the absorption law
• Boolean algebra: a distributive, complemented lattice

Allowing axioms other than identities

One broadening of the concept of algebraic structure is to study sets with operations that must satisfy axioms other than identities. The above structures are all formal systems, they consist purely of definitions and do not place any restrictive conditions on the structures. In the definition of a field below there is the restrictive condition 0 (the additive identity) ≠ 1 (the multiplicative identity). For this to be a purely formal structure no such condition should be placed. However, if 0=1 then the structure collapses. Hence, a necessary restriction that 0 ≠ 1 needs to be placed to insure that we have a useful mathematical entity. Although these structures undoubtedly have an algebraic flavor, they suffer from defects not found in universal algebra. For example, there does not exist a product of two integral domains, nor a free field over any set.

• Integral domain: a ring with 0 ≠ 1 that has no zero divisors other than 0
• Division ring, also called a skew field: an integral domain with an inverse operation (the inverse operation is not defined for the additive identity)
• Field: a commutative division ring
• Artinian ring: a ring that satisfies the descending chain condition on ideals.

Examples

• The (non-zero) natural numbers with addition (+) is a magma.
• The non-negative integers under addition is a magma with an identity.
• The integers Z with subtraction (−) form a quasigroup.
• The nonzero rationals Q with division (÷) form a quasigroup.

Groups

• 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 integers Z with addition (+) form an abelian group.
• The non-zero rationals Q 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^maⁿ = a^{m + n} = a^{n + m} = aⁿa^m = yx. In particular, the integers Z form an abelian group under addition, as do the integers modulo n Z/nZ.
• Further examples can be found in examples of groups.

Rings

• The natural numbers (including zero), with the ordinary addition and multiplication is a commutative semiring.
• The set R[X] of all polynomials 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 Z_n = Z/nZ of integers modulo n (as an additive group the cyclic group of order n ) forms a ring with n elements (see modular arithmetic).

Integral domain

• The integers with the two operations of addition and multiplication form an integral domain.
• The p-adic integers.

Fields

• The rational numbers with addition and multiplication form a field.
• The real numbers R, under the usual operations of addition and multiplication.
  • The real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers, and the definable numbers.
• 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 C, under the usual operations of addition and multiplication.
• An algebraic number field is a finite field extension of the rational numbers Q, that is, a field containing Q which has finite dimension as a vector space over Q. 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 F_q, Z/qZ, 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: 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 additional non-algebraic structures, such as topological spaces. The algebraic structure is required to be somehow compatible with the additional structure.

• Ordered group: a group with a compatible partial order
• Linearly ordered group: a group with a compatible linear order
• Archimedean group: a linearly ordered group for which the Archimedean property holds
• Topological group: a group with a compatible topology
• Lie group: a group with a compatible manifold structure
• Graded algebra: an algebra with a "grading"
• Clifford algebra: an associative algebra determined by quadratic form on a vector space
• Topological vector space: a vector space with a compatible topology

Categories

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

• algebraic
• essentially algebraic
• presentable
• locally presentable
• monadic

functors and categories.

See also

• signature (universal algebra)cs:Algebraická struktura

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