Algebraic structure

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, 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

Algebraic structures
Group-like Semigroup / Monoid Quasigroup and Loop Abelian group Magma
Ring-like Ring Semiring Near-ring Commutative ring Domain Integral domain Field
Lattice-like Lattice Semilattice Map of lattices
Module-like Module Group with operators Vector space
Algebra-like Algebra Associative Non-associative Graded Bialgebra

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.

• Magma or groupoid: S and a single binary operation over S.
  • Steiner magma: A commutative magma satisfying x(xy) = y.
    • Squag: an idempotent Steiner magma.
    • Sloop: a Steiner magma with distinguished element 1, such that xx = 1.
• Semigroup: an associative magma.
  • Monoid: a unital semigroup.
    • Group: a monoid with a unary operation, inverse, giving rise to an inverse element.
      • Abelian group: a commutative group.
  • Band: a semigroup of idempotents.
    • Semilattice: a commutative band. The binary operation can be called either meet or join.
      • Boundary algebra: a unital semilattice (equivalently, an idempotent commutative monoid) with a unary operation, complementation, denoted by enclosing its argument in parentheses, giving rise to an inverse element that is the complement of the identity element. The identity and inverse elements bound S. Also, x(xy) = x(y) holds.

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.

• Quasigroup: a cancellative magma. Equivalently, ∀x,y∈S, ∃!a,b∈S, such that xa = y and bx = y.
  • Loop: a unital quasigroup with a unary operation, inverse.
    • Moufang loop: a loop in which a weakened form of associativity, (zx)(yz) = z(xy)z, holds.
      • Group: an associative loop.

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.
  • Complemented lattice: a lattice with a unary operation, complementation, denoted by postfix " ' ", giving rise to an inverse element. That element and its complement bound the lattice.
• 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.
      • Interior algebra: a Boolean algebra with an added unary operation, the interior operator, denoted by postfix " ' " and obeying the identities x'x=x, x"=x, (xy)'=x'y', and 1'=1.
    • 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.

• Semiring: a ringoid such that S is a monoid under each operation. Each operation has a distinct identity element. Addition also commutes, and has an identity element that annihilates multiplication.
  • Commutative semiring: a semiring with commutative multiplication.
  • Ring: a semiring with a unary operation, additive inverse, giving rise to an inverse element -x, which when added to x, yields the additive identity element. Hence S is an abelian group under addition.
    • Rng: a ring lacking a multiplicative identity.
    • Commutative ring: a ring with commutative multiplication.
      • Boolean ring: a commutative ring with idempotent multiplication, equivalent to a Boolean algebra.
  • Kleene algebra: a semiring with idempotent addition and a unary operation, the Kleene star, denoted by postfix * and obeying the identities (1+x*x)x*=x* and (1+xx*)x*=x*. See "Lattice-like structures" for another structure having the same name.

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:

1. R are scalars, denoted by Greek letters. R is a ring under the binary operations of scalar addition and multiplication;
2. 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, {e₁, ... e_n}⊂M, where the positive integer n is the dimension of the free module. For every v∈M, there exist κ₁, ..., κ_n∈R such that v = κ₁e₁ + ... + κ_ne_n. Let 0 and 0 be the respective identity elements for module and scalar addition. If r₁e₁ + ... + r_ne_n = 0, then r₁ = ... = 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:

1. 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.
2. 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.
  • Integral domain: a domain whose multiplication commutes. Also a commutative cancellative ring.
    • Euclidean domain: an integral domain with a function f: S→N satisfying the division with remainder property.
• 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.
    • Ordered field: a field whose elements are totally ordered.
      • Real field: a Dedekind complete ordered field.

The following structures are not varieties for reasons in addition to S≠{0}:

• Simple ring: a ring having no ideals other than 0 and S.
  • Weyl algebra:
• Artinian ring: a ring whose ideals satisfy the descending chain condition.

Composite Systems: Vector Spaces, and Algebras over Fields. Two Sets, M and R, and at least three binary operations.

The members of:

1. M are vectors, denoted by lower case letters. M is at minimum an abelian group under vector addition, with distinguished member 0.
2. 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 and with a norm, namely a function M → R that is positive homogeneous, subadditive, 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.
      • Symmetric algebra: a commutative algebra with unital vector multiplication.

Composite Systems: Multilinear algebras. Two sets, V and K. Four binary operations:

1. 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.
2. 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.
3. 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. v₁ ∧ v₂ ∧ ... ∧ v_k = 0 if and only if v₁, ..., 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 outer (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.
      • Grassmann-Cayley algebra: a geometric algebra without an inner product.

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

• Nonzero N under addition (+) is a magma.
• N under addition is a magma with an identity.
• Z under subtraction (−) is a quasigroup.
• Nonzero Q under division (÷) is a quasigroup.
• 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.
• 2x2 matrices(of non-zero determinant) with matrix multiplication form a group.
• Z under addition (+) is an abelian group.
• Nonzero Q under multiplication (×) is an abelian group.
• 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, Z is an abelian group under addition, as is the integers modulo n Z/nZ.
• A monoid is a category with a single object, in which case the composition of morphisms and the identity morphism interpret monoid multiplication and identity element, respectively.
• The Boolean algebra 2 is a boundary algebra.
• More examples of groups and list of small groups.

Lattices

• The normal subgroups of a group, and the submodules of a module, are modular lattices.
• Any field of sets, and the connectives of first-order logic, are models of Boolean algebra.
• The connectives of intuitionistic logic form a model of Heyting algebra.
• The modal logic S4 is a model of interior algebra.
• Peano arithmetic and most axiomatic set theories, including ZFC, NBG, and New foundations, can be recast as models of relation algebra.

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).
• Sets of hypercomplex numbers were early prototypes of algebraic structures now called rings.

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.
  • R contains several interesting subfields, the algebraic, the computable, and the definable numbers.
• 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.

• Ordered group: a group with a compatible partial order. I.e., S is partially ordered.
• Linearly ordered group: a group whose S is a linear order.
• Archimedean group: a linearly ordered group for which the Archimedean property holds.
• Lie group: a group whose S has a compatible smooth manifold structure.
• Topological group: a group whose S has a compatible topology.
• Topological vector space: a vector space whose M has a compatible topology; a superset of normed vector spaces.
• Banach spaces, Hilbert spaces, Inner product spaces
• Vertex operator algebras

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. A forgetful functor between categories of algebraic structures "forgets" a part of a 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
• universal property.