Outline of algebraic structures

In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties below, are included among the algebraic structures by tradition.

Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath. These lists mention many structures not included below, and may present more information about some structures than is presented here.

Generalities

An algebraic structure consists of one or two sets closed under some operations, functions, and relations, satisfying a number of axioms, including none. 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.

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 underlying set S precede composite structures built on two;
• If A and B are the two underlying 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. The heap, a group-like structure, is the only structure mentioned in this entry requiring an operation whose arity exceeds 2.

If structure B is under structure A and more indented, then A is interpretable in B, meaning that all theorems of A are theorems of B. The converse is usually not the case.

A structure is trivial if the cardinality of S is less than 2, and is otherwise nontrivial.

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.

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). Nonidentities can often be recast as identities. For example, any lattice inequality of the form α≤β can always be recast as the identity α∧β=α.

An important result is that given any variety C and any underlying set X, the free object F(X)∈C exists.

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

Group-like structures

	Closure	Associative	Identity	Cancellation	Commutative
Partial magma	Unneeded	Unneeded	Unneeded	Unneeded	Unneeded
Semigroupoid	Unneeded	Required	Unneeded	Unneeded	Unneeded
Small category	Unneeded	Required	Required	Unneeded	Unneeded
Groupoid	Unneeded	Required	Required	Required	Unneeded
Commutative Groupoid	Unneeded	Required	Required	Required	Required
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Commutative magma	Required	Unneeded	Unneeded	Unneeded	Required
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded
Commutative quasigroup	Required	Unneeded	Unneeded	Required	Required
Associative quasigroup	Required	Required	Unneeded	Required	Unneeded
Commutative-and-associative quasigroup	Required	Required	Unneeded	Required	Required
Unital magma	Required	Unneeded	Required	Unneeded	Unneeded
Commutative unital magma	Required	Unneeded	Required	Unneeded	Required
Loop	Required	Unneeded	Required	Required	Unneeded
Commutative loop	Required	Unneeded	Required	Required	Required
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Commutative semigroup	Required	Required	Unneeded	Unneeded	Required
Monoid	Required	Required	Required	Unneeded	Unneeded
Commutative monoid	Required	Required	Required	Unneeded	Required
Group	Required	Required	Required	Required	Unneeded
Abelian group	Required	Required	Required	Required	Required

See magma for a list of the many properties that a group-like structures may possess. The diagram to the right summarizes the defining properties of:

• The better-known group-like structures, from the least (magmas) to the most restrictive (groups);
• The related notions of category and groupoid.

All group-like structures feature a primary (and often unique) binary or ternary operation, which usually (e.g., for semigroups and hoops) associates. This operation will nearly always be denoted here by concatenation, and when it is binary, will be called "group product." If group product associates, brackets are not required to resolve the order of operation. When group product does not associate (e.g., quasigroups, semiheaps, loops, implication algebras), an embedded period indicates the grouping. Examples: xy.z, x.yz.

For Steiner magmas, abelian groups, logic algebras, bands, equivalence algebras, and hoops, group product also commutes. Commutativity may be added to any group-like structure for which it is not already the case.

Groups, logic algebras, lattices, and loops feature a unary operation, denoted here by enclosure in parentheses.

For monoids, loops, and sloops, S is a pointed set.

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.
  • Equivalential algebra: a magma satisfying xx.y=y, xy.z.z=xy, and xy.xzz.xzz=xy.
  • Implicational calculus: a magma satisfying xy.x=x, x.yz=y.xz, and xy.y=yx.x.
  • Equivalence algebra: an idempotent magma satisfying xy.x=x, x.yz=xy.xz, and xy.z.y.x = xz.y.x.
  • Semigroup: an associative magma.
    • Semigroup with involution: a semigroup with a unary operation, involution, denoted by enclosure in parentheses, such that ((x))=x and (xy) = (y)(x).
    • Equivalential calculus: a commutative semigroup satisfying yyx=x.
    • Monoid: a unital semigroup.
      • Boolean group: a monoid with xx = identity element.
      • Group: a monoid with a unary operation, inverse, denoted by enclosure in parentheses, and satisfying (a)a = identity element.
        • Abelian group: a commutative group. The single axiom yxz(yz)=x suffices.
        • Group with operators: a group closed under one or more unary operations, with each such operation distributing over group product.
        • Algebraic group:
          • Reductive group: an algebraic group such that the unipotent radical of the identity component of S is trivial.
      • Logic algebra: a commutative monoid with a unary operation, complementation, denoted by enclosure in parentheses, and satisfying x(1)=(1) and ((x))=x. 1 and (1) are lattice bounds for S.
        • MV-algebra: a logic algebra satisfying the axiom ((x)y)y = ((y)x)x.
        • Boundary algebra: a logic algebra satisfying (x)x=1 and (xy)y = (x)y, from which it can be proved that boundary algebra is a distributive lattice. (0)=1, (1)=0, ((x))=x and xx=x are now provable.
  • Order (algebra): an idempotent magma satisfying yx=xy.x, xy=xy.y, x:xy.z=x.yz, and xy.z.y=xz.y. 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.
    • Band: an associative order algebra, and 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 binary 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.

Two binary operations.

• Rack: Infix ◅ and ▻ denote the two operations. The axioms are x▻(y▻z) = (x▻y)▻(x▻z), (z◅y)◅x = (z◅x)◅(y◅x), (x▻y)◅x = y, and x▻(y◅x) = y. A simpler but nonequational way of stating an important fact about racks is that ∀x,y∈S, there exists a unique z such that x▻z = y. Asserting this makes redundant the operation denoted by ◅.
  • Quandle: Either of the rack operations is idempotent.
• 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. In addition to group product, quasigroups feature 2 binary operations denoted by infix "/" and "\". These added operations permit axiomatizing the defining property of quasigroups, cancellation, by means of identities alone.

• Quasigroup: a cancellative magma. A quasigroup satisfies the axioms y = x(x\y) = x\(xy) = (y/x)x = (yx)/x. The following equivalent but nonvariety definition may be more intuitive. S is a quasigroup iff ∀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.

The following diagram summarizes two possible paths from magma to group.

File:Magma to Group.svg

NOTE:

• a = associativity, d = divisibility, e = identity, i = invertibility. d and i jointly imply the cancellation property.
• G = group, L = loop, M = magma, N = monoid, Q = quasigroup, S = semigroup.

One ternary operation, heap product, denoted xyz:

• Semiheap: S is closed under heap product, which para-associates: vwx.yz = v.wxy.z = vw.xyz.
  • Idempotent semiheap: A semiheap satisfying xxx = x.
    • Generalized heap: An idempotent semiheap satisfying yy.zzx = zz.yyx and xyy.zz = xzz.yy.
  • Heap: A semiheap satisfying yyx = xyy = x.
    • Group: A heap with distinguished element 1. The group product of x and y is defined as x1y, and the group inverse of x is defined as 1x1.

Lattice-like structures

The binary operations meet and join, which characterize nearly all structures in this section, are idempotent, by assumption or proof. Latticoids are the only lattice-like structure that do not associate. N.B. "Lattice" is also employed in a number of group-theoretic contexts, including to refer to a discrete subgroup of the real vector space Rⁿ that spans Rⁿ.

Some concepts from order theory that recur in lattice theory:

• Strict partial order: A set S with a relation that is transitive and irreflexive.
  • Preordered set: A strict partial order that is reflexive.
    • Directed set: A preordered set such that every pair of elements has an upper bound.
    • Partially ordered set: A preordered set with the antisymmetry property.
      • Totally ordered set: A partially ordered set with totality replacing reflexivity.

One binary operation, one of meet or join, and denoted by concatenation. The two structures below are also magmas; see the preceding section.

• Semilattice: the binary operation commutes and associates. Also a partially ordered set closed under one of pairwise greatest lower bound or least upper bound.
  • Lattice: a semilattice with a unary operation, dualization, denoted by enclosure within a pair of brackets. If xy denotes meet, (xy) denotes join, and vice versa. The binary and unary operations interact via a form of the absorption law, x(xy) = x = (x(xy)).

Two binary operations, meet (infix ∧) and join (infix ∨). Duality means that interchanging all meets and joins preserves truth.

• 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: x∧(x∨y) = x. Also a partially ordered set closed under pairwise greatest lower bound and least upper bound.
  • 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.
    • Complete lattice:
      • Algebraic lattice:
  • Involutive lattice: a lattice with a unary operation, denoted by postfix ', and satisfying x"=x and (x∨y)' = x' ∧y' .
  • Relatively complemented lattice:
  • 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).
      • Orthomodular lattice: an ortholattice such that (x ≤ y) → (x ∨ (x^⊥ ∧ y) = y) holds.
    • De Morgan 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).
    • Metric lattice:
    • Projective lattice:
    • 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, usually unary. Let a postfix * denote any added unary operation. Then 0* = 0 and (x∨y)* = x*∨y*. More generally, all added operations (a) evaluate to 0 if any argument is 0, and (b) are join preserving, i.e., distribute over join.
          • Modal algebra: a Boolean algebra with a single added operator, 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.

Three structures whose intended interpretations are 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).
• Relation algebra: S, the Cartesian square of some set, is a:

• Boolean algebra under join and complementation;
• Monoid under binary composition (infix •) and the identity element I such that 1=I '∨I;
• Residuated Boolean algebra by virtue of a second unary operation, converse (postfix ^{${\breve {\ }}$}) and the axiom (A^{${\breve {\ }}$}•(A•B)')∨B ' = B '.

Converse is an involution and distributes over composition so that (A•B)^{${\breve {\ }}$} = B^{${\breve {\ }}$}•A^{${\breve {\ }}$}. Converse and composition each distribute over join.

• Cylindric algebra: Boolean algebra augmented by unary 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). In pointless topology, a Heyting algebra is called a frame.

Four or more binary operations:

• Residuated semilattice: a semilattice under meet or join, a monoid under product, and two further binary operations, residuation, satisfying the axioms .
  • Action algebra: a residuated semilattice that is also Kleene lattice. Hence combines a 〈∨, •, 1, ←, →〉 algebra with a 〈∨, 0, •, 1, *〉 algebra.
  • 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: a residuated Boolean algebra with an added unary operation, converse. S, the Cartesian square of some set, is a monoid under an added residuated binary operation, composition with identity element I, such that 1=I '∨I. Composition and converse distribute over join. A$\oplus$B = (A '⊗B ')' defines a 4th binary operation, relational addition.

The following Hasse diagram illustrates some pathways from a very general concept, partially ordered sets, to the oldest and most researched lattice, Boolean algebra. This diagram reveals some of the hierarchical structure linking a number of important types of lattices, not all discussed above because some are not varieties. This diagram consists of a number of links of the form A→B, each meaning that structure A is included in structure B.

Lattice ordered structures

S includes distinguished elements and is closed under additional operations, so 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: Multiplication has left/right identity element of 1, and a zero element, 0, which is also the left/right identity element for addition.
• 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, isomorphic to Boolean algebra.
      • Differential ring: A ring with an added unary operation, derivation, denoted by prefix ∂ and satisfying the product rule, ∂(xy) = ∂xy+x∂y.

N.B. The above definitions of rng, ring, and semiring do not command universal assent:

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

Modules and algebras

These structures feature a set R whose elements are scalars, denoted by Greek letters, and a set S whose members are denoted by Latin letters. For every ring R, there is a corresponding variety of R-modules.

• Monoid ring: R is a ring and S is a monoid.
  • Group ring: a monoid ring such that S is a group.
    • Group algebra: a group ring whose group product commutes.
• 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.
  • Comodule: the dual of a module.
  • Vector space: A module such that R is a field.
  • 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.
        • Coalgebra: the dual of a unital associative algebra.
        • 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.
      • 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.
        • Kac-Moody algebra: a Lie algebra, usually infinite-dimensional, definable by generators and relations through a generalized Cartan matrix.
          • Generalized Kac-Moody algebra: a Kac-Moody algebra whose simple roots may be imaginary.
          • Affine Lie algebra: a Kac-Moody algebra whose generalized Cartan matrix is positive semi-definite and has corank 1.

Quasivarieties

A quasivariety is a variety with one or more axioms that are quasiidentities. Let Greek letters be metavariables denoting identities. A quasiidentity then takes the form (α₁∧,...,∧α_n) → β.

Magmas

A magma (sometimes called "groupoid", or "algebra of type (2)") is a set equipped with a binary operation.

Cancellative

• Semigroup: a semigroup satisfying the added axioms xa=ya→x=y, and bx=by→x=y.
• Monoid: a unital cancellative semigroup.

Combinatory logic

The elements of S are higher order functions, and concatenation denotes the binary operation of function composition.

• BCI algebra: a magma with distinguished element 0, satisfying the identities (xy.xz)zy = 0, (x.xy)y = 0, xx=0, xy=yx=0 → x=y, and x0 = 0 → x=0.
  • BCK algebra: a BCI algebra satisfying the identity x0 = x. x≤y, defined as xy=0, induces a partial order with 0 as least element.
• Combinatory logic: A combinator concatenates upper case letters. Terms concatenate combinators and lower case letters. Concatenation is left and right cancellative. '=' is an equivalence relation over terms. The axioms are Sxyz = xz.yz and Kxy = x; these implicitly define the primitive combinators S and K. The distinguished elements I and 1, defined as I=SK.K and 1=S.KI, have the provable properties Ix=x and 1xy=xy. Combinatory logic has the expressive power of set theory.
  • Extensional combinatory logic: Combinatory logic with the added quasiidentity (Wx=Vx)→(W=V), with W, V containing no instance of x.

Graphs

One set, V a finite set of vertices, and a binary relation E⊆V², adjacency, consisting of edges. No operations.

• Directed graph: E is irreflexive.
  • Graph algebra:
  • Directed acyclic graph: A directed graph with no path whose endpoints are the same element of V.
  • Graph: A directed graph such that E is symmetric. Dropping the requirement that E be irreflexive makes loops possible.
    • Connected graph: A graph such that a path connects any two vertices.
      • Tree: a connected graph with no cycles.
      • Cycle graph: a connected graph consisting of a single cycle.
      • Complete graph: a connected graph such that the path between any two vertices includes no other vertex. Hence for any two vertices x and y, (x,y) and (y,x) are both elements of E.
        • Tournament: A complete graph such that only one of (x,y) and (y,x) is an element of E.

Lattices

• 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'.
• Semidistributive lattice: a lattice satisfying the axiom (x∧y = x∧z)→(x∧y=x∧(y∨z)), and dually.

Ring-like

• 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*.

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Universal classes

• Quasitrivial groupoid: a magma such that xy = x or y.

• Integral domain: A commutative ring such that (xy=0)→((x=0)∨(y=0)). Also a domain whose multiplication commutes.
• Integral relation algebra: a relation algebra such that (xy=0)→((x=0)∨(y=0)).

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.
• Ordered field: a field whose S is totally ordered by '≤,' so that (a≤b)→(a+c≤b+c) and (0≤a,b)→ (0≤ab).

Nonvarieties

Nonvarieties cannot be axiomatized solely with identities and quasiidentities. Many 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.

Most of the classic results of universal algebra do not hold for nonvarieties. For example, neither the free field over any set nor the direct product of integral domains exists. Nevertheless, nonvarieties often retain an undoubted algebraic flavor.

There are whole classes of axiomatic formal systems not included in this section, e.g., logics, topological spaces, and this exclusion is in some sense arbitrary. Many of the nonvarieties below were included because of their intrinsic interest and importance, either by virtue of their foundational nature (Peano arithmetic), ubiquity (the real field), or richness (e.g., fields, normed vector spaces). Also, a great deal of theoretical physics can be recast using the nonvarieties called multilinear algebras.

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.

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 usually links them. Arithmetics necessarily have infinite models.

• Cegielski arithmetic: 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 an injective operation called successor, denoted by prefix σ. 0 is the axiomatic identity element for addition, and annihilates multiplication. Both axioms hold for semirings.

• Dedekind algebra, also called a Peano algebra: A pointed unary system by virtue of 0, the unique element of S not included in the range of successor. Dedekind algebras are fragments of Skolem arithmetic.
  • 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 the background logic.

• • Primitive recursive arithmetic: A Dedekind algebra with recursively defined addition, multiplication, exponentiation, and other primitive recursive operations as desired. A rule of induction replaces the axiom of induction. The background logic lacks quantification and thus is not first-order logic.
  • Skolem arithmetic (Boolos and Jeffrey 2002: 73-76): Not an algebraic structure because there is no fixed set of operations of fixed adicity. Skolem arithmetic is a Dedekind algebra with projection functions, indexed by n, whose arguments are functions and that return the nth 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. Mathematical induction becomes a theorem.
    • Kalmar arithmetic: Skolem arithmetic with different primitive functions.

The following arithmetics lack a connection between addition and multiplication. They are the simplest arithmetics capable of expressing all primitive recursive functions.

• Baby Arithmetic: Because there is no universal quantification, there are axiom schemes but no axioms. [n] denotes n consecutive applications of successor to 0. Addition and multiplication are defined by the schemes [n]+[p] = [n+p] and [n][p] = [np].
  • R: Baby arithmetic plus the binary relations "=" and "≤". These relations are governed by the schemes [n]=[p] ↔ n=p, (x≤[n])→(x=0)∨,...,∨(x=[n]), and (x≤[n])∨([n]≤x).

Lattices that are not varieties

• 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}. S-0 is S with 0 removed.

• Domain: a ring whose sole zero divisor is 0.
• Integral domain: a commutative ring, 0 ≠ 1, and having the zero-product property: (x≠0∧y≠0) → xy≠0. Hence there are no zero divisors.
  • Integrally closed domain
  • Unique factorization domain: an integral domain in which every nonzero element can be written as a product of a unit and prime elements of S.
  • Principal ideal domain: an integral domain in which every ideal is principal, i.e., can be generated by a single element.
  • Euclidean domain: an integral domain with a function f: S→N satisfying the division with remainder property.

• Division ring (also skew field, sfield): a ring such that S-0 is a group under multiplication.
  • Field: a division ring whose multiplication commutes. Recapitulating: S is an abelian group under addition and 0, S-0 is an abelian group under multiplication and 1≠0, and multiplication distributes over addition. x0 = 0 is a theorem.
    • Algebraically closed field: a field such that all polynomial equations whose coefficients are elements of S have all roots in S. This field is the complex numbers.
    • Ordered field: a field whose S is totally ordered by '≤', so that (a≤b)→(a+c≤b+c) and (0≤a,b)→ (0≤ab).
      • Real closed field: an ordered real field such that for every element x of S, there exists a y such that x = y² or -y². All polynomial equations of odd degree and whose coefficients are elements of S, have at least one root that in S.
      • Real field: a Dedekind complete ordered field.
        • Differential field: A real field with an added unary operation, derivation, denoted by prefix ∂, distributing over addition, ∂(x+y) = ∂x+ ∂y, and satisfying the product rule, ∂(xy) = ∂xy + x∂y.

The following field-like 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:
    • Group Hopf algebra:
• Graded algebra: an associative algebra with unital outer product. The members of V have a directram 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;
  • Discrete group: a topological group whose topology is discrete. Also a 0-dimensional Lie group.
• 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

Let there be two classes:

• O whose elements are objects, and
• M whose elements are morphisms defined over O.

Let x and y be any two elements of M. Then there exist:

• Two functions, c, d : M→O. d(x) is the domain of x, and c(x) is its codomain.
• A binary partial operation over M, 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 (if defined), and x has left and right identity elements, the domain and codomain of x, respectively, so that d(x)x = x = xc(x). Letting φ stand for one of c or d, and γ stand for the other, then φ(γ(x)) = γ(x). If O has but one element, the associated category is a monoid.

• Groupoid: Two equivalent definitions.
  • Category theory: A small category in which every morphism is an isomorphism. Equivalently, a category such that every element x of M, x(a,b), has an inverse x(b,a); see diagram in section 2.2.
  • Algebraic definition: A group whose product is a partial function. Group product associates in that if ab and bc are both defined, then ab.c=a.bc. (a)a and a(a) are always defined. Also, ab.(b) = a, and (a).ab = b.

Unclassified

• Incidence algebra: An associative algebra, defined for any locally finite poset and commutative ring with unity. Part of order theory.
• Group ring:
  • Group algebra:
• Path algebra: related to a quiver and a directed graph.
• Categorical algebra: an associative algebra defined for any locally finite category and commutative ring with unity. Generalizes group algebra and incidence algebra, as the concept of category generalizes group and poset.

Examples

Recurring underlying sets: 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.
• A Dedekind algebra is a free S-algebra on zero generators of type 〈1,0〉. Freeness implies that no two terms are equal. Very general results from the theory of free algebras, e.g., definition by recursion, and uniqueness up to isomorphism, are now applicable.
• A Dedekind-Peano structure is a free object with one generator.
• The universe of singletons forms a Dedekind-Peano structure if {x} interprets the successor of x, and the null set interprets 0.

Group-like structures

• Nonzero N under addition is a magma.
• Z under subtraction (−) is a quasigroup.
• Nonzero Q under division (÷) is a quasigroup.
• 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 are the integers modulo n, Z/nZ.
• 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.
• MV-algebras characterize multi-valued and fuzzy logics.

• 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 set of all endomorphisms of object X in category C is a monoid under composition of morphisms and the identity morphism.

Also see examples of groups, list of small groups, and list of finite simple groups.

Lattices

• The following structures, if ordered by set inclusion, all form modular lattices. The:
  • Subgroups of a group, normal or not;
  • Subrings and ideals of a ring;
  • Submodules of a module and the subspaces of a vector space;
  • Equivalence relations on any set;
  • Sublattices of any lattice including 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.
• A Boolean algebra (BA) is also an ortholattice, a Boolean ring, a commutative monoid, and a Newman algebra. The BA 2 is a boundary algebra. A BA would be an abelian group if (b)ba = a were a BA identity.
• 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.
• Any first-order theory whose sentences can be written in such a way that the quantifiers do not nest more than three deep, can be recast as a model of relation algebra. Such models include Peano arithmetic and most axiomatic set theories, including ZFC, NBG, and New Foundations.

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 '≤'.
• R is the:
  • Only Dedekind complete ordered field, as the axioms for such a field are categorical;
  • Real field grounding real and functional analysis;
  • Filed whose subfields include the algebraic, the computable, and the definable numbers.
• C is an algebraically closed field.
• Some facts about finite fields:
  • There exists a complete classification thereof.
  • 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.

Lie groups: See table of Lie groups and list of simple Lie groups.

See also

• abstract algebra
• algebraic structure
• arity
• category theory
• free object
• operation (mathematics)
• signature
• universal algebra
• variety (universal algebra)
• list of abstract algebra topics
• list of first-order theories
• list of linear algebra topics
• 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:
  • Alphabetical list of algebra structures; includes many not mentioned here.
  • Online books and lecture notes.
  • Map containing about 50 structures, some of which do not appear above. Likewise, most of the structures above are absent from this map.
• PlanetMath topic index.
• Hazewinkel, Michiel (2001) Encyclopaedia of Mathematics. Springer-Verlag.
• Mathworld page on abstract algebra.
• Stanford Encyclopedia of Philosophy: Algebra by Vaughan Pratt.