Outline of algebraic structures

In universal algebra, a branch of pure mathematics, an algebraic structure consists of a set closed under one or more operations, satisfying a number of axioms, including none. Abstract algebra is primarily the study of algebraic structures and their properties. 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 satisfy. For example, all groups are also semigroups and magmas.

Other web lists of algebraic structures, organized alphabetically, include Jipsen and PlanetMath. These resources mention many structures not included below, and include more information about each structure than is presented here. This entry is more revealing of the hierarchical connections among algebraic structures than are Jipsen and PlanetMath.

Generalities

Structures are listed below in approximate order of increasing complexity as follows:

• Structures that are varieties precede those that are not;
• Simple structures built on one set, the universe S, precede composite ones built on two;
• If A and B are two sets making up a composite structure, that structure may include functions of the form AxA→B or AxB→A.
• Structures are then ordered by the number and arities of the operations they contain. No structure mentioned in this entry requires an operation whose arity exceeds 2.

If structure B is under structure A and more indented, then all theorems of A are theorems of B; the converse usually does not hold.

Varieties

Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over a set that is part of the definition of the structure. Hence identities contain no sentential connectives, existentially quantified variables, or relations of any kind other than equality and the operations the structure allows. Nonidentities can often be recast as identities. For example, the lattice inequality α≤β can always be recast as the lattice identity α∧β=α.

If the axioms defining an algebraic structure are all identities--or can be recast as identities--the structure is a variety (not to be confused with algebraic variety in the sense of algebraic geometry).

Simple structures

No binary operation.

• Set: a degenerate algebraic structure having no operations.
• Pointed set: S has one or more distinguished elements. While pointed sets are near-trivial, they lead to discrete spaces, which are not.
  • Bipointed set: S has exactly two distinguished elements.
• Unary system: S and a single unary operation over S.
• Pointed unary system: a unary system with S a pointed set.

Group-like structures

• All group-like structures feature a primary (and usually unique) binary operation, denoted here by concatenation. This binary operation usually (i.e., for semigroups and hoops) associates. Hence brackets are not required to resolve the order of operation. When associativity is not the case (quasigroups, hoops, combinatory algebra), an embedded period indicates the grouping. Example: xy.z or x.yz.
• Steiner magmas, abelian groups, logic algebras, semilattices, and hoops feature a primary binary operation that commutes by assumption. Commutativity may be added to any group-like structure for which it is not already the case.
• Group, logic algebras, lattices, and loops feature a unary operation, denoted here by enclosure in parentheses.
• For monoids, loops, sloops, and combinatory algebras, S is a pointed set.

See magma for a:

• Diagram summarizing the connections among several of the better-known group-like structures;
• Description of the many properties that a magma may possess.

One binary operation.

• Magma or groupoid: S is closed under a single binary operation.
  • 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.
      • Boolean group: a monoid with xx = identity element.
      • Group: a monoid with a unary operation, inverse, giving rise to an inverse element equal to the identity element. thus (b)ba=a holds in all groups.
        • Abelian group: a commutative group. The single axiom yxz(yz)=x suffices.
        • Group with operators: a group with a set of unary operations over S, with each unary operation distributing over the group operation.
      • Logic algebra: a commutative monoid (equivalently, a unital semilattice) with an inverse called complementation. The inverse element is the complement of the identity element 1. 1 and (1) are lattice bounds for S. Hence (x)x=(1), x(1)=(1), and ((x))=x are all axioms. Logic algebras are also order algebras.
        • MV-algebra: a logic algebra satisfying the axiom ((x)y)y = ((y)x)x.
        • Boundary algebra: a logic algebra satisfying the axiom (xy)y = (x)y, from which it can be proved that boundary algebra is a distributive lattice. x(1)=(1) and ((x))=x are now provable.
    • Band: an idempotent semigroup.
      • Rectangular band: a band satisfying the axiom xyz = xz.
      • Normal band: a band satisfying the axiom xyzx = xzyx.

The following two structures form a bridge connecting magmas and lattices:

• Semilattice: a commutative band. The group operation is called meet or join.

• Lattice: a semilattice with a unary operation, dualization, denoted (x) and satisfying the absorption law, x(xy) = (x(xy)) = x. xx = x is now provable.

• Order algebra: an idempotent magma satisfying yx=xyx=xy. Hence idempotence holds in the following wide sense. For any subformula x of formula z: (i) all but one instance of x may be erased; (ii) x may be duplicated at will anywhere in z.
• Combinatory algebra: The elements of S are higher order functions, and the binary operation is function composition. S and K are distinguished elements (known as combinators), implicitly defined by the axioms Sxyz = xz.yz and Kxy = x. These are the only axioms. The distinguished elements I and 1 are defined as I=SK.K and 1=S.KI. The properties Ix=x and 1xy=xy are provable. The only group-like structure with the expressive power of set theory.

Two binary operations.

• Hoop: a commutative monoid with a second binary operation, denoted by infix →, satisfying the axioms x→.y→z = xy.→z, x→x = 1, and x→y.x = y→x.y.

Three binary operations.

Quasigroups feature 3 binary operations because axiomatizing the cancellation property by means of identities alone requires two binary operations in addition to the group operation. Quasigroups being nonassociative, periods indicate the grouping.

• Quasigroup: a cancellative magma. Equivalently, ∀x,y∈S, ∃a,b∈S, such that xa = y and bx = y.
  • Loop: a unital quasigroup. Every element of S has, provably, a unique left and right inverse.
    • Bol loop: A loop satisfying either a.b.ac = a.ba.c (left) or ca.b.a = c.ab.a (right).
      • Moufang loop: a left and right bol loop. More simply, a loop satisfying zx.yz = z.xy.z.
      • Bruck loop: a bol loop whose inverse satisfies (ab) = (a)(b).
    • Group: an associative loop.

Lattices-like structures

The binary operations meet (infix ∧) and join (infix ∨), which characterize all structures in this section, are always idempotent, by assumption or proof. The only lattice-like structure that does not associate is latticoids.

One binary operation, one of meet or join. These structures are also magmas; see the preceding section.

• Semilattice: the binary operation commutes and associates.
  • Lattice: a semilattice with a unary operation, dualization, denoted by enclosure . The binary and unary operations interact via a form of the absorption law, x(xy) = x = (x(xy)).

Two binary operations, meet and join. By duality, interchanging all meets and joins preserves truth.

N.B. Lattice has another mathematical meaning unrelated to this section, namely a discrete subgroup of the real vector space Rⁿ that spans Rⁿ.

• Latticoid: meet and join commute but do not associate.
• Skew lattice: meet and join associate but do not commute.
• Lattice: a commutative skew lattice, an associative latticoid, and both a meet and join semilattice. Meet and join interact via the absorption law.
  • Bounded lattice: a lattice with two distinguished elements, the greatest (1) and the least element (0), such that x∨1=1 and x∨0=x. Dualizing requires interchanging 0 and 1. A bounded lattice is a pointed set.
  • Involutive lattice: a lattice with a unary operation, denoted by postfix ', and satisfying x" = x and (x∨y)' = x' ∧y' .
  • Complemented lattice: a lattice with a unary operation, complementation, denoted by postfix ', such that x∧x' = 0 and 1=0'. 0 and 1 bound S.
    • Orthocomplemented lattice: a complemented lattice satisfying x" = x and x∨y=y ↔ y' ∨x' = x' (complementation is order reversing).
    • DeMorgan algebra: a complemented lattice satisfying x" = x and (x∨y)' = x' ∧y' . Also a bounded involutive lattice.
  • Modular lattice: a lattice satisfying the modular identity, x∨(y∧(x∨z)) = (x∨y)∧(x∨z).
    • Arguesian lattice: a modular lattice satisfying the identity .
    • Distributive lattice: a lattice in which each of meet and join distributes over the other. Distributive lattices are modular, but the converse need not hold.
      • Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.
        • Boolean algebra with operators: a Boolean algebra with one or more added operations. All added operations (a) evaluate to 0 if any argument is 0, and (b) are join preserving, i.e., distribute over join if binary. If postfix * denotes an additional unary operation, then 0* = 0 and (x∨y)* = x*∨y*.
          • Modal algebra: Boolean algebra with a single added unary operation, the modal operator.
            • Derivative algebra: a modal algebra whose added unary operation, the derivative operator, satisfies x**∨x*∨x = x*∨x.
            • Interior algebra: a modal algebra whose added unary operation, the interior operator, satisfies x*∨x = x and x** = x*. The dual is a closure algebra.
              • Monadic Boolean algebra: a closure algebra whose added unary operation, the existential quantifier, denoted by prefix ∃, satisfies the axiom ∃(∃x)' = (∃x)'. The dual operator, ∀x := (∃x' )' is the universal quantifier.

Two structures whose intended interpretation is first order logic:

• Polyadic algebra: a monadic Boolean algebra with a second unary operation, denoted by prefixed S. I is an index set, J,K⊂I. ∃ maps each J into the quantifier ∃(J). S maps I→I transformations into Boolean endomorphisms on S. σ, τ range over possible transformations; δ is the identity transformation. The axioms are: ∃(∅)a=a, ∃(J∪K) = ∃(J)∃(K), S(δ)a = a, S(σ)S(τ) = S(στ), S(σ)∃(J) = S(τ)∃(J) (∀i∈I-J, such that σi=τi), and ∃(J)S(τ) = S(τ)∃(τ^-1J) (τ injective).
• Cylindric algebra: Boolean algebra augmented by cylindrification operations.

Three binary operations:

• Boolean semigroup: a Boolean algebra with an added binary operation that associates, distributes over join, and is annihilated by 0.

• Implicative lattice: a distributive lattice with a third binary operation, implication, that distributes left and right over each of meet and join.
• Brouwerian algebra: a distributive lattice with a greatest element and a third binary operation, denoted by infix " ' ", satisfying ((x∧y)≤z)∧(y≤x)' z.

• Heyting algebra: a Brouwerian algebra with a least element, whose third binary operation, now called relative pseudo-complement, satisfies the identities x'x=1, x(x'y) = xy, x' (yz) = (x'y)(x'z), and (xy)'z = (x'z)(y'z).

Four binary operations:

• Residuated lattice: a Brouwerian algebra with a least element and a fourth binary operation, denoted by infix ⊗, such that (⊗,1) is a commutative monoid obeying the adjointness property ((x≤y)' z) ↔ (x⊗y≤z).

• Residuated Boolean algebra: a residuated lattice whose lattice part is a Boolean algebra.

• Relation algebra: an interior algebra whose interior operator is called converse. S, the Cartesian square of some set, is a monoid under an added residuated binary operation, relative product, whose identity element is distinct from the Boolean bounds. Relative product distributes over join. The converse of a function is its inverse, and the relative product of two functions is their composition.

• Action algebra: a Kleene lattice with a left and right residual. Hence combines a residuated semilattice ⟨∨, •, 1, ←, →⟩ with a Kleene lattice ⟨∨, 0, •, 1, *⟩.

Lattice-ordered structure: S includes distinguished elements and is closed under additional operations, such that the axioms for a semigroup, monoid, group, or a ring are satisfied.

Ring-like structures

Two binary operations, addition and multiplication. That multiplication has a 0 is either an axiom or a theorem.

• Shell: addition and multiplication have respective identity elements 0 and 1.
• Ringoid: multiplication distributes over addition..
  • Nonassociative ring: a ringoid that is an abelian group under addition.
    • Lie ring: a nonassociative ring whose multiplication anticommutes and satisfies the Jacobi identity.
    • Jordan ring: a nonassociative ring whose multiplication commutes and satisfies the Jordan identity.
  • Newman algebra: a ringoid that is also a shell. There is a unary operation, inverse, denoted by a postfix "'", such that x+x'=1 and xx'=0. The following are provable: inverse is unique, x"=x, addition commutes and associates, and multiplication commutes and is idempotent.
  • Semiring: a ringoid that is also a shell. Addition and multiplication associate, addition commutes.
    • Commutative semiring: a semiring whose multiplication commutes.
  • Rng: a ringoid that is an Abelian group under addition and 0, and a semigroup under multiplication.
    • Ring: a rng that is a monoid under multiplication and 1.
      • Commutative ring: a ring with commutative multiplication.
        • Boolean ring: a commutative ring with idempotent multiplication, modeling Boolean algebra

N.B. These definitions do not command universal assent:

• Some employ "ring" to denote what is here called a rng, and refer to a ring in the above sense as a "ring with identity";
• Some define a semiring as having no identity elements.

Modules and Algebras

Two sets, R and S. Elements of R are scalars, denoted by Greek letters. Elements of S are denoted by Latin letters. For every ring R, there is a corresponding variety of R-modules.

• Module: S is an abelian group with operators, each unary operator indexed by R. The operators are scalar multiplication RxS→S, which commutes, associates, is unital, 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.
  • Vector space: A module such that R is a field.
    • Group ring:
• Algebra over a ring (also R-algebra): a module where R is a commutative ring. There is a second binary operation over S, called multiplication and denoted by concatenation, which distributes over module addition and is bilinear: α(xy) = (αx)y = x(αy).
  • Algebra over a field: An algebra over a ring whose R is a field.
    • Associative algebra: an algebra over a field or ring, whose vector multiplication associates.
      • Commutative algebra: an associative algebra whose vector multiplication commutes.
      • Incidence algebra: an associative algebra such that the elements of S are the functions f [a,b]: [a,b]→R, where [a,b] is an arbitrary closed interval of a locally finite poset. Vector multiplication is defined as a convolution of functions.
      • Group algebra:
    • Jordan algebra: an algebra over a field whose vector multiplication commutes, may or may not associate, and satisfies the Jordan identity.
    • Lie algebra: an algebra over a field satisfying the Jacobi identity. The vector multiplication, the Lie bracket denoted [u,v], anticommutes, usually does not associate, and is nilpotent.

Partial order for nonlattices

If the partial order relation ≤ is added to a structure other than a lattice, the result is a partially ordered structure. These are discussed in Birkhoff (1967: chpts. 13-15, 17) using a differing terminology. Examples include:

• Ordered magma, semigroup, monoid, group, and vector space: In each case, S is partially ordered;
• Linearly ordered group and ordered ring: S is linearly ordered;
  • Archimedean group: a linearly ordered group for which the Archimedean property holds.

Structures that are not varieties

The structures in this section are not varieties because they cannot be axiomatized with identities alone. Most of the nonidentities are of three very simple kinds:

1. The requirement that S (or R or K) be 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. Axioms involving multiplication, holding for all members of S (or R or K) except 0. In order for an algebraic structure to be a variety, the domain of each operation must be an entire underlying set; there can be no partial operations.
3. "0 is not the successor of anything," included in nearly all arithmetics.

Many mathematical structures that are not varieties are nevertheless of fundamental importance, either by virtue of their foundational nature (Peano arithmetic, ubiquity (the real field), or richness (e.g., fields, normed vector spaces). Moreover, a great deal of theoretical physics can be recast using the nonvarieties called multilinear algebras. Although structures with nonidentities retain an undoubted algebraic flavor, they suffer from defects varieties do not share. For example, neither the product of integral domains nor a free field over any set exists.

No operations. Functions or relations may be present:

• Multiset: S is a multiset and N is the set of natural numbers. There is a multifunction m: S→N such that m(x) is the multiplicity of x∈S.
• Graph: S is the field of a symmetric binary relation.

Arithmetics

If the name of a structure in this section includes the word "arithmetic," the structure features one or both of the binary operations addition and multiplication. If both operations are included, the recursive identity defining multiplication links them. Arithmetics necessarily have infinite models.

• Cegielski arithmetic (Smorynski 1991): A commutative cancellative monoid under multiplication. 0 annihilates multiplication, and xy=1 if and only if x and y are both 1. Other axioms and one axiom schema govern order, exponentiation, divisibility, and primality; consult Smorynski. Adding the successor function and its axioms as per Dedekind algebra render addition recursively definable, resulting in a system with the expressive power of Robinson arithmetic.

In the structures below, addition and multiplication, if present, are recursively defined by means of successor, denoted by prefix σ. 0 is the axiomatic identity element for addition, and annihilates multiplication. Both axioms hold for semirings.

• Dedekind algebra (Potter 2004: 90): A pointed unary system whose operation, called successor, is injective. S is pointed and not a variety because it has a unique member, named 0, not included in the range of successor. Dedekind algebras are fragments of Skolem arithmetic.
  • Skolem arithmetic (Boolos and Jeffrey 2002: 73-76): Also called primitive recursive arithmetic. Not an algebraic structure because there is no fixed set of operations of fixed adicity. Skolem arithmetic is a Dedekind algebra with projection functions, whose arguments are functions and that return any desired argument of a function. The identity function is the projection function whose arguments are all unary operations. Composite operations of any adicity, including addition and multiplication, may be constructed using function composition and primitive recursion. Induction becomes a theorem.
    • Kalmar arithmetic: Skolem arithmetic with different primitive functions.
  • Dedekind-Peano structure: A Dedekind algebra with an axiom schema of induction.
    • Presburger arithmetic: A Dedekind-Peano structure with recursive addition.

Arithmetics above this line are decidable. Those below are incompletable.

• Robinson arithmetic: Presburger arithmetic with recursive multiplication.

• Peano arithmetic: Robinson arithmetic with an axiom schema of induction. The semiring axioms for N (other than x+0=x and x0=0, included in the recursive definitions of addition and multiplication) are now theorems.

• Heyting arithmetic: Peano arithmetic with intuitionist logic as background logic.

Lattices that are not varieties

• Semimodular lattice:
• Kleene lattice: a bounded distributive lattice with a unary involution, denoted by postfix ', satisfying the axioms (x∨y)' = x'∨y', x" = x, and (x∧x')∨(y∨y') = y∨y'.
• Part algebra: a Boolean algebra with no least element 0, so that the complement of 1 is not defined.

Two sets, Φ and D.

• Information algebra: D is a lattice, and Φ is a commutative monoid under combination, an idempotent operation. The operation of focussing, f: ΦxD→Φ satisfies the axiom f(f(φ,x),y)=f(φ,x∧y) and distributes over combination. Every element of Φ has an identity element in D under focussing.

Field-like structures

Two binary operations, addition and multiplication. S is nontrivial, i.e., S≠{0}.

• Kleene algebra: a semiring with idempotent addition and a unary operation, the Kleene star, denoted by postfix * and satisfying (1+x*x)x*=x*=(1+xx*)x*.
• 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 (also 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. Recapitulating: addition and multiplication commute, associate, and are unital. S is closed under a two-sided additive inverse, nonzero S under a two-sided multiplicative inverse. Multiplication distributes over addition.
    • Algebraically closed field: a field F such that all polynomial equations with coefficients in F also have roots in F.
    • Ordered field: a field whose elements are totally ordered. Sums and products of positive elements are positive.
      • 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.

Vector spaces that are not varieties

The following composite structures are extensions of vector spaces that are not varieties. Two sets: M is a set of vectors and R is a set of scalars.

Three binary operations.

• 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 Euclidian 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.
  • 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.
• Graded vector space: a vector space such that the members of M have a direct sum decomposition. See graded algebra below.

Structures that build on the notion of vector space:

• Matroid:
• Antimatroid:

Multilinear algebras

Four binary operations. Two sets, V and K:

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 scalar multiplication of scalars and multivectors, V×K→V, has the same properties as module scalar multiplication.

• Symmetric algebra: a unital commutative algebra with vector multiplication.
• Universal enveloping algebra: Given a Lie algebra L over K, the "most general" unital associative K-algebra A, such that the Lie algebra A_L contains L.
  • Hopf algebra:
• 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.
  • Tensor algebra: A graded algebra such that V includes all finite iterations of a binary operation over V, called the tensor product. All multilinear algebras can be seen as special cases of tensor algebra.
    • 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 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.
        • Grassmann-Cayley algebra: a geometric algebra without an inner product.

Structures with topologies or manifolds

These algebraic structures are not varieties, because the underlying set either has a topology or is a manifold, characteristics that are not algebraic in nature. This added structure must be compatible in some sense, however, with the algebraic structure. The case of when the added structure is partial order is discussed above, under varieties.

Topology:

• Topological group: a group whose S has a topology;
• Topological vector space: a normed vector space whose R has a topology.

Manifold:

• Lie group: a group whose S has a smooth manifold structure.

Categories

Two classes, O and C. The elements of set O are objects. The elements of C, which may be a proper class, are morphisms defined over O.

• There are two functions, c, d: C→O. d(x) is the domain of morphism x, and c(x) is its codomain.
• There is a single binary partial operation over C, called composition and denoted by concatenation. xy is defined iff c(x)=d(y). If xy is defined, d(xy) = d(x) and c(xy) = c(y).

Category: Composition associates, and has left and right identity elements, the domain and codomain of x, respectively, so that d(x)x = x = xc(x). Letting φ and γ stand for either unary operation, φ(γ(x)) = γ(x). The variety most closely related to categories are monoids.

Examples

Some recurring universes: N=natural numbers; Z=integers; Q=rational numbers; R=real numbers; C=complex numbers.

Arithmetics

• N is a pointed unary system, and the standard interpretation of Peano arithmetic.
• The universe of singletons forms a Dedekind-Peano structure if {x} interprets the successor of x, and the null set interprets 0 (Lewis 1991).

Group-like structures

• Nonzero N under addition is a magma.
• Z under addition (+) is an abelian group.
• Z under subtraction (−) is a quasigroup.
• Nonzero Q under multiplication (×) is an abelian group.
• 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.
• Invertible 2x2 matrices form a group under matrix multiplication.
• The permutations preserving the partition of a set induced by an equivalence relation form a group under function composition and inverse.
• 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 are the integers modulo n, Z/nZ.
• The set of all functions X→X, X any nonempty set, is a monoid under function composition and the identity function.
• In category theory, the following are monoids under composition of morphisms and the identity morphism:
  • A category with a single object;
  • The set of all endomorphisms of an object X in category C .
• The Boolean algebra 2 is a boundary algebra.
• MV-algebras characterize multi-valued and fuzzy logics.
• More examples of groups and list of small groups.

Boolean algebras (BA) are primarily viewed as Boolean (complemented distributive) lattices. They are also ortholattices, rings, commutative monoids, and Newman algebras, and would be abelian groups if the BA identity and inverse elements were identical.

Lattices

• The subgroups of a group, normal or not, the subrings and ideals of a ring, the submodules of a module, and the subspaces of a vector space, ordered by set inclusion, all form modular lattices.
• The preceding holds for the sublattices of a lattice, if the lattice includes the empty set.
• The closed sets of a topological space form a lattice under finite unions and intersections. The open sets, ordered by inclusion, form a lattice under arbitrary unions.
• The equivalence relations on any set, ordered by set inclusion, form a lattice.
• Any field of sets, and the connectives of first-order logic, are models of Boolean algebra. See Lindenbaum-Tarski algebra.
• The connectives of intuitionistic logic form a model of Heyting algebra.
• The modal logics K, S4, S5, and wK4 are models of modal algebra, interior algebra, monadic Boolean algebra, and derivative algebra, respectively.
• Peano arithmetic and most axiomatic set theories, including ZFC, NBG, and New Foundations, can be recast as models of relation algebra.

Ring-like structures

• N is a commutative semiring under addition and multiplication.
• The set R[X] of all polynomials over some coefficient ring R is a ring.
• 2x2 matrices under 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).

Field-like structures

• Z is an integral domain under addition and multiplication.
• Each of Q, R, C, and the p-adic integers is a field under addition and multiplication.
• Q and R are ordered fields, totally ordered by "<" in the usual way
• R is the only Dedekind complete ordered field, as the axioms for such a field are categorical. The real field R grounds real and functional analysis.
  • R contains several interesting subfields, the algebraic, the computable, and the definable numbers.
• C is an algebraically closed field.
• An algebraic number field in number theory is a finite field extension of Q, that is, a field containing Q which has finite dimension as a vector space over Q.
• 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.

See also

• abstract algebra
• algebraic structure
• arity
• category theory
• free object
• universal algebra
• signature (universal algebra)
• variety (universal algebra)
• list of abstract algebra topics
• list of linear algebra topics
• list of first order theories
• list of mathematics lists

References

• Garrett Birkhoff, 1967. Lattice Theory, 3rd ed, AMS Colloquium Publications Vol. 25. American Mathematical Society.
• --------, and Saunders MacLane, 1999 (1967). Algebra, 2nd ed. New York: Chelsea.
• George Boolos and Richard Jeffrey, 1980. Computability and Logic, 2nd ed. Cambridge Univ. Press.
• Dummit, David S., and Foote, Richard M., 2004. Abstract Algebra, 3rd ed. John Wiley and Sons.
• Grätzer, George, 1978. Universal Algebra, 2nd ed. Springer.
• David K. Lewis, 1991. Part of Classes. Blackwell.
• Michel, Anthony N., and Herget, Charles J., 1993 (1981). Applied Algebra and Functional Analysis. Dover.
• Potter, Michael, 2004. Set Theory and its Philosophy, 2nd ed. Oxford Univ. Press.
• Smorynski, Craig, 1991. Logical Number Theory I. Springer-Verlag.

A monograph available free online:

• Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.

External links

• Jipsen's algebra structures. Includes many structures not mentioned here.
• Mathworld page on abstract algebra.
• Stanford Encyclopedia of Philosophy: Algebra by Vaughan Pratt.