- Algebraic structure
In

algebra , a branch ofpure mathematics , an**algebraic structure**consists of one or more sets 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 inuniversal algebra .Abstractly, 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

semigroup s and magmas.**tructures whose axioms are all identities**If the axioms defining a structure are all identities, the structure is a variety (not to be confused with

algebraic variety in the sense ofalgebraic 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 noconnective s, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part ofuniversal 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 ofbinary operation s (0 to 4) they require. Incidentally, no structure mentioned in this entry requires an operation whosearity 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 "A"x"A"→"B" or "A"x"B"→"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

theorem s of "A" are theorems of "B"; theconverse 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 theabsorption 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 singleunary operation over "S".

* Pointed unary system: a unary system with "S" a pointed set.**Group-like structures**:**One**binary operation , denoted byconcatenation . Formonoid s, boundary algebras, and sloops, "S" is apointed set .

* Magma or groupoid: "S" and a singlebinary operation over "S".

**Steiner magma: Acommutative magma satisfying "x"("xy") = "y".

*** Squag: anidempotent Steiner magma.

*** Sloop: a Steiner magma with distinguished element 1, such that "xx" = 1.

*Semigroup : anassociative magma.

**Monoid : aunital semigroup.

*** Group: a monoid with aunary operation , inverse, giving rise to aninverse element .

****Abelian group : acommutative group.

** Band: a semigroup ofidempotent s.

***Semilattice : acommutative band. The binary operation can be called either meet or join.

**** Boundary algebra: aunital semilattice (equivalently, anidempotent commutativemonoid ) with aunary operation , complementation, denoted by enclosing its argument in parentheses, giving rise to aninverse element that is the complement of theidentity element . The identity and inverse elements bound "S". Also, "x"("xy") = "x"("y") holds.**Three**binary operation s.Quasigroups are listed here, despite their having 3 binary operations, because they are (nonassociative) magmas. Quasigroups feature 3 binary operations only because establishing the quasigroupcancellation 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: aunital 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 morebinary operations , including meet and join, connected by theabsorption law . "S" is both a meet and join semilattice, and is apointed 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, thegreatest lower bound and theleast 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 aninverse 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, theinterior operator , denoted by postfix ' and obeying the identities x'x=x, x"=x, (xy)'=x'y', and 1'=1.

*****Relation algebra : an interior algebra whose interior operator is called converse. "S" is always theCartesian square of some set, and is amonoid under an added residuated binary operation, relative product, whose identity element is distinct from the Boolean bounds. Relative product distributes over meet or join.

***Heyting algebra : a bounded distributive lattice with an added binary operation,relative pseudo-complement , denoted byinfix " ' ", 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 andmultiplication , with multiplication distributing over addition. Semirings arepointed set s.

*Semiring : a ringoid such that "S" is amonoid under each operation. Each operation has a distinctidentity element . Addition also commutes, and has anidentity element that annihilates multiplication.

**Commutative semiring : a semiring with commutative multiplication.

** Ring: a semiring with a unary operation, additive inverse, giving rise to aninverse element equal to the additiveidentity 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 withidempotent multiplication, equivalent to a Boolean algebra.

**Kleene algebra : a semiring withidempotent addition and a unary operation, theKleene 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:

#"R" are scalars, denoted by Greek letters. "R" is a ring under the binary operations of scalar addition and multiplication;

#"M" are "module elements" (often but not necessarily vectors), denoted by Latin letters. "M" is anabelian group under addition. There may be otherbinary operation s.The "scalar multiplication" of scalars and module elements is a function "R"x"M"→"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 freebasis , {"e"_{1}, ... "e"_{"n"}}⊂"M", where the positive integer "n" is thedimension of the free module. For every "v"∈"M", there exist κ_{1}, ..., κ_{n}∈"R" such that "v" = κ_{1}"e"_{1}+ ... + κ_{n}"e"_{n}. Let**0**and 0 be the respective identity elements for module and scalar addition. If "r"_{1}"e"_{1}+ ... + "r"_{n}"e"_{n}=**0**, then "r"_{1}= ... = "r"_{n}= 0.

*Algebra over a ring (also "R-algebra"): a (free) module where "R" is acommutative ring . There is a second binary operation over "M", called multiplication and denoted by concatenation, which distributes over module addition and isbilinear : α("xy") = (α"x")"y" = "x"(α"y").

*Jordan ring: analgebra over a ring whose module multiplication commutes, does not associate, and respects theJordan identity .Vector space s, closely related to modules, are defined in the next section.**tructures 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:

#The starting point for all structures in this section is a "nontrivial" ring, namely one such that "S"≠{0}, 0 being the additiveidentity 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.

#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 space s. Moreover, much of theoretical physics can be recast as models ofmultilinear algebra s. Although structures with nonidentities retain an undoubted algebraic flavor, they suffer from defects varieties do not have. For example, neither the product ofintegral domain s nor a free field over any set exist.**Arithmetics**:**Two**binary operation s, addition and multiplication. "S" is aninfinite set . Arithmetics are pointed unary systems, whoseunary operation isinjective successor, and with distinguished element 0.

*Robinson arithmetic . Addition and multiplication arerecursive ly defined by means of successor. 0 is theidentity 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 anaxiom schema ofinduction . 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 operation s, addition and multiplication. "S" is nontrivial, i.e., "S"≠{0}.

* Domain: a ring whose solezero divisor is 0.

**Integral domain : a domain whose multiplication commutes. Also a commutativecancellative 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 anabelian group under multiplication.

***Ordered field : a field whose elements are totally ordered.

****Real field : aDedekind 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 thedescending chain condition .**Composite Systems: Vector Spaces, and Algebras over Fields**. Two Sets, "M" and "R", and at least**three**binary operations.The members of:

#"M" are vectors, denoted by lower case letters. "M" is at minimum anabelian group under vector addition, with distinguished member**0**.

#"R" are scalars, denoted by Greek letters. "R" is a field, nearly always the real orcomplex field , with 0 and 1 as distinguished members.**Three**binary operations.

*Vector space : afree module ofdimension "n" except that "R" is a field.

**Normed vector space : a vector space with a norm, namely a function "M" → "R" that issymmetric ,linear , and positive definite.

***Inner product space (also "Euclidean" vector space): a normed vector space such that "R" is thereal field , whose norm is the square root of theinner product , "M"×"M"→"R". Let "i","j", and "n" be positive integers such that 1≤"i","j"≤"n". Then "M" has anorthonormal basis such that "e"_{i}•"e"_{j}= 1 if "i"="j" and 0 otherwise; seefree module above.

***Unitary space: Differs from inner product spaces in that "R" is thecomplex 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 adirect sum decomposition. Seegraded algebra below.**Four**binary operations.

*Algebra over a field : Analgebra 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 : analgebra over a field respecting theJacobi identity , whose vector multiplication, theLie bracket denoted ["u,v"] ,anticommute s, does not associate, and isnilpotent .

**Associative algebra : analgebra over a field , or a module, whose vector multiplication associates.

***Linear algebra : an associativeunital algebra with the members of "M" being matrices. Every matrix has adimension "n"x"m", "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 "n"x"m" and matrix "B" be "i"x"j". Then "AB" is defined if and only if "m=i"; "BA", if and only if "j=n". There also exists an "m"x"m" matrix "I" and an "n"x"n" matrix "J" such that "AI"="JA"="A". If "u" and "v" are vectors having the same dimensions, they have aninner product , denoted 〈"u","v"〉. Hence there is anorthonormal basis ; seeinner product space above. There is a unary function, thedeterminant , from square ("n"x"n" for any "n") matrices to "R".

***Commutative algebra : an associative algebra whose vector multiplication commutes.

****Symmetric algebra : a commutative algebra withunital vector multiplication.**Composite Systems:**. Two sets, "V" and "K".Multilinear algebra s**Four**binary operation s:

# The members of "V" aremultivector s (including vectors), denoted by lower case Latin letters. "V" is anabelian group undermultivector addition, and amonoid underouter product . The outer product goes under various names, and is multilinear in principle but usuallybilinear . The outer product defines the multivectors recursively starting from the vectors. Thus the members of "V" have a "degree" (seegraded algebra below). Multivectors may have aninner product as well, denoted "u"•"v": "V"×"V"→"K", that issymmetric ,linear , and positive definite; seeinner product space above.

# 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 thereal field , as multilinear algebras are designed to describe physical phenomena withoutcomplex number s.

# 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 withunital outer product. The members of "V" have adirect 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 anAbelian group under addition.

**Exterior algebra (also "Grassmann algebra"): a graded algebra whoseanticommutative outer product, denoted by infix ∧, is called theexterior product . "V" has anorthonormal basis . "v"_{1}∧ "v"_{2}∧ ... ∧ "v"_{k}= 0 if and only if "v"_{1}, ..., "v"_{k}arelinearly dependent . Multivectors also have aninner product .

***Clifford algebra : an exterior algebra with a symmetricbilinear 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.**Examples**Some recurring universes:

**N**=natural numbers ;**Z**=integers ;**Q**=rational numbers ;**R**=real number s;**C**=complex number s.**N**is a pointed unary system, and under addition and multiplication, is both the standard interpretation ofPeano arithmetic and a commutativesemiring .Boolean algebras are at once

semigroup s, lattices, and rings. They would even beAbelian group s if the identity and inverse elements were identical instead of complements.**Group-like structures**

*Nonzero**N**underaddition (+) is a magma.

***N**under addition is a magma with an identity.

***Z**undersubtraction (−) 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**undermultiplication (×) is an Abelian group.

*Everycyclic group "G" is Abelian, because if "x", "y" are in "G", then "xy" = "a"^{m}"a"^{n}= "a"^{m+n}= "a"^{n+m}= "a"^{n}"a"^{m}= "yx". In particular,**Z**is an Abelian group under addition, as is the integers modulo "n"**Z**/"n**"Z**.

*Amonoid is a category with a single object, in which case the composition of morphisms and theidentity morphism interpret monoid multiplication and identity element, respectively.

* The Boolean algebra**2**is a boundary algebra.

*Moreexamples of groups andlist of small groups .Lattice s

* Thenormal subgroup s of a group, and thesubmodules of a module, are modular lattices.

* Anyfield of sets , and theconnective s offirst-order logic , are models of Boolean algebra.

* The connectives ofintuitionistic logic form a model ofHeyting algebra .

* Themodal logic S4 is a model ofinterior algebra .

*Peano arithmetic and most axiomatic set theories, includingZFC , NBG, andNew foundations , can be recast as models ofrelation algebra .**Ring-like structures**

* The set "R" [X] of allpolynomial s 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**/n**Z**of integers modulo "n" (the additivecyclic group of order "n" ) forms a ring having "n" elements (seemodular arithmetic ).Integral domain s

***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 anordered field and iscategorical . The resultingreal field grounds real andfunctional analysis .

****R**contains several interesting subfields, the algebraic, the computable, and thedefinable number s.

*Analgebraic number field is a finite field extension of**Q**, that is, a field containing**Q**which has finite dimension as avector space over**Q**. Algebraic number fields are very important innumber theory .

*If "q" > 1 is a power of aprime number , then there exists (up to isomorphism ) exactly onefinite field with "q" elements, usually denoted**F**_{"q"}, or in the case that "q" is itself prime, by**Z**/"q**"Z**. Such fields are calledGalois field s, 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**/"p**"Z**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; seemodular 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 compatiblepartial order . I.e., "S" is partially ordered.

*Linearly ordered group : a group whose "S" is alinear order .

*Archimedean group : a linearly ordered group for which theArchimedean property holds.

*Lie group : a group whose "S" has a compatible smoothmanifold structure.

*Topological group : a group whose "S" has a compatibletopology .

*Topological vector space : a vector space whose "M" has a compatibletopology ; a superset ofnormed vector space s.**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 ofhomomorphism , namely any function compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a category. For example, thecategory of groups has all groups as objects and allgroup homomorphism s as morphisms. Thisconcrete category may be seen as acategory of sets with added category-theoretic structure. Likewise, the category oftopological group s (whose morphisms are the continuousgroup homomorphism s) is acategory of topological spaces with extra structure.There are various concepts in category theory that try to capture the algebraic character of a context, for instance

*algebraic

*essentially algebraic

*presentable

*locally presentable

*monadic functors and categories

*universal property .**ee also***

arity

*category theory

*free object

*list of algebraic structures

*list of first order theories

*signature

*variety**References***

*A monograph available free online:

*Category theory:

*

***External links*** [

*http://math.chapman.edu/cgi-bin/structures Jipsen's algebra structures.*] Includes many structures not mentioned here.

* [*http://mathworld.wolfram.com/topics/Algebra.html Mathworld*] page on abstract algebra.

*Stanford Encyclopedia of Philosophy : [*http://plato.stanford.edu/entries/algebra/ Algebra*] byVaughan Pratt .

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