Algebraic structure 106364 222883059 2008-07-01T16:01:47Z Ht686rg90 7340433 /* Structures whose axioms are all identities */ corrected In [[algebra]], a branch of [[pure mathematics]], an '''algebraic structure''' consists of one or more [[set]]s [[Closure (mathematics)|closed]] under one or more [[Operation (mathematics)|operations]], satisfying some [[axiom|axioms]]. [[Abstract algebra]] is primarily the study of algebraic structures and their properties. The notion of algebraic [[structure (mathematical logic)|structure]] has been formalized in [[universal algebra]]. Abstractly, an "algebraic structure" is the collection of all possible [[model theory|model]]s 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 [[group (mathematics)|group]]s. 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 [[group (mathematics)|group]]s are also [[semigroup]]s and [[magma (algebra)|magma]]s. ==Structures whose axioms are all identities== [[Image:Algebraic-structures.svg|thumb|400px|right|Axioms of [[monoid]]s, [[Group (mathematics)|groups]], [[abelian group]]s, [[rng]]s, [[Ring (mathematics)|rings]], [[commutative ring]]s and [[Field (mathematics)|fields]].]] If the axioms defining a structure are all [[identity (mathematics)|identities]], the structure is a [[variety (universal algebra)|variety]] (not to be confused with [[algebraic variety]] in the sense of [[algebraic geometry]]). Identities are equations formulated using only the operations the structure allows, and variables that are tacitly [[universal quantifier|universally quantified]] over the relevant [[universe (mathematics)|universe]]. Identities contain no [[connective]]s, [[quantification|existentially quantified variables]], or [[Relation (mathematics)|relations]] of any kind other than the allowed operations. The study of varieties is an important part of [[universal algebra]]. All structures in this section are [[variety (universal algebra)|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 [[metamathematics|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 (mathematics)|universe]] ''S'', are listed before ''composite'' ones requiring two sets; *Structures having the same number of required sets are then ordered by the number of [[binary operation]]s (0 to 4) they require. Incidentally, no structure mentioned in this entry requires an operation whose [[arity]] exceeds 2; *Let ''A'' and ''B'' be the two sets that make up a composite structure. Then a composite structure may include 1 or 2 [[function (mathematics)|function]]s 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 operation|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''; the [[converse]] does not hold. Ringoids and [[Lattice (order)|lattice]]s can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the [[distributive law]]; in the case of lattices, they are linked by the [[absorption law]]. Ringoids also tend to have numerical [[model theory|model]]s, while lattices tend to have [[set theory|set-theoretic]] models. '''Simple structures''': '''No''' [[binary operation]]: * [[Set]]: a degenerate algebraic structure having no operations. * [[Pointed set]]: ''S'' has one or more distinguished elements, often 0, 1, or both. * Unary system: ''S'' and a single [[unary operation]] over ''S''. * Pointed unary system: a unary system with ''S'' a pointed set. '''Group-like structures''': '''One''' [[binary operation]], denoted by [[concatenation]]. For [[monoid]]s, [[laws of form|boundary algebra]]s, and sloops, ''S'' is a [[pointed set]]. * [[Magma (algebra)|Magma or groupoid]]: ''S'' and a single [[binary operation]] over ''S''. **Steiner magma: A [[commutative]] magma satisfying ''x''(''xy'') = ''y''. *** Squag: an [[idempotent]] Steiner magma. *** Sloop: a Steiner magma with distinguished element 1, such that ''xx'' = 1. * [[Semigroup]]: an [[associative]] magma. ** [[Monoid]]: a [[unital]] semigroup. *** [[Group (mathematics)|Group]]: a monoid with a [[unary operation]], inverse, giving rise to an [[inverse element]]. **** [[Abelian group]]: a [[commutative]] group. ** [[Band (algebra)|Band]]: a semigroup of [[idempotent]]s. *** [[Semilattice]]: a [[commutative]] band. The binary operation can be called either [[meet (mathematics)|meet]] or [[join (mathematics)|join]]. **** [[laws of form|Boundary algebra]]: a [[unital]] semilattice (equivalently, an [[idempotent]] commutative [[monoid]]) with a [[unary operation]], [[complemented lattice|complementation]], denoted by enclosing its argument in parentheses, giving rise to an [[inverse element]] that is the complement of the [[identity element]]. The identity and inverse elements bound ''S''. Also, ''x''(''xy'') = ''x''(''y'') holds. '''Three''' [[binary operation]]s. Quasigroups are listed here, despite their having 3 binary operations, because they are (nonassociative) [[magma (algebra)|magma]]s. Quasigroups feature 3 binary operations only because establishing the quasigroup [[cancellation property]] by means of identities alone requires two binary operations in addition to the group operation. * [[Quasigroup]]: a [[cancellation property|cancellative]] magma. Equivalently, ∀''x'',''y''∈''S'', ∃!''a'',''b''∈''S'', such that ''xa'' = ''y'' and ''bx'' = ''y''. ** [[Quasigroup|Loop]]: a [[unital]] quasigroup with a unary operation, [[inverse element|inverse]]. ***[[Moufang loop]]: a loop in which a weakened form of associativity, (''zx'')(''yz'') = ''z''(''xy'')''z'', holds. ****[[Group (mathematics)|Group]]: an associative loop. '''[[Lattice (order)|Lattice]]''': '''Two''' or more [[binary operations]], including [[meet (mathematics)|meet]] and [[join (mathematics)|join]], connected by the [[absorption law]]. ''S'' is both a [[meet (mathematics)|meet]] and [[join (mathematics)|join]] semilattice, and is a [[pointed set]] if and only if ''S'' is bounded. Lattices often have no unary operations. Every true statement has a [[duality|dual]], obtained by replacing every instance of meet with join, and vice versa. * [[Bounded lattice]]: ''S'' has two distinguished elements, the [[greatest lower bound]] and the [[least upper bound]]. [[duality|Dualizing]] requires replacing every instance of one bound by the other, and vice versa. ** [[Complemented lattice]]: a lattice with a unary operation, [[complemented lattice|complementation]], denoted by [[reverse Polish notation|postfix]] ', giving rise to an [[inverse element]]. That element and its complement bound the lattice. * [[Modular lattice]]: a lattice in which the modular identity holds. ** [[Distributive lattice]]: a lattice in which each of meet and join [[distributive lattice|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 (structure)|Boolean algebra]]: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation. **** [[Interior algebra]]: a Boolean algebra with an added unary operation, the [[interior operator]], denoted by [[reverse Polish notation|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 [[inverse relation|converse]]. ''S'' is always the [[Cartesian square]] of some set, and is a [[monoid]] under an added residuated binary operation, [[relation composition|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 by [[infix]] " ' ", and governed by the axioms x'x=1, x(x'y) = xy, x'(yz) = (x'y)(x'z), (xy)'z = (x'z)(y'z). '''Ringoids''': '''Two''' [[binary operations]], [[addition]] and [[multiplication]], with multiplication [[distributivity|distributing]] over addition. Semirings are [[pointed set]]s. * [[Semiring]]: a ringoid such that ''S'' is a [[monoid]] under each operation. Each operation has a distinct [[identity element]]. Addition also commutes, and has an [[identity element]] that annihilates multiplication. ** [[Commutative semiring]]: a semiring with commutative multiplication. ** [[Ring (mathematics)|Ring]]: a semiring with a unary operation, additive inverse, giving rise to an [[inverse element]] equal to the additive [[identity element]]. Hence ''S'' is an Abelian group under addition. *** [[Rng (algebra)|Rng]]: a ring lacking a multiplicative identity. *** [[Commutative ring]]: a ring with commutative multiplication. **** [[Boolean ring]]: a commutative ring with [[idempotent]] multiplication, equivalent to a Boolean algebra. ** [[Kleene algebra]]: a semiring with [[idempotent]] addition and a unary operation, the [[Kleene star]], denoted by [[reverse Polish notation|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." '''[[Module (mathematics)|Modules]]: Composite Systems Defined over Two Sets, ''M'' and ''R''''': The members of: #''R'' are [[scalar (mathematics)|scalar]]s, 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 [[vector space|vectors]]), denoted by Latin letters. ''M'' is an [[abelian group]] under addition. There may be other [[binary 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 free [[basis]], {''e''<sub>1</sub>, ... ''e''<sub>''n''</sub>}⊂''M'', where the positive integer ''n'' is the [[dimension]] of the free module. For every ''v''∈''M'', there exist κ<sub>1</sub>, ..., κ<sub>n</sub>∈''R'' such that ''v'' = κ<sub>1</sub>''e''<sub>1</sub> + ... + κ<sub>n</sub>''e''<sub>n</sub>. Let '''0''' and 0 be the respective identity elements for module and scalar addition. If ''r''<sub>1</sub>''e''<sub>1</sub> + ... + ''r''<sub>n</sub>''e''<sub>n</sub> = '''0''', then ''r''<sub>1</sub> = ... = ''r''<sub>n</sub> = 0. * [[Algebra over a ring]] (also ''R-algebra''): a (free) module where ''R'' is a [[commutative ring]]. There is a second binary operation over ''M'', called multiplication and denoted by concatenation, which distributes over module addition and is [[bilinear]]: α(''xy'') = (α''x'')''y'' = ''x''(α''y''). *[[Jordan algebra|Jordan ring]]: an [[algebra over a ring]] whose module multiplication commutes, does not associate, and respects the [[Jordan identity]]. [[Vector space]]s, closely related to modules, are defined in the next section. ==Structures with some axioms that are not identities== The structures in this section are not [[Variety (universal algebra)|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 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. #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., [[Field (mathematics)|field]]s and [[vector space]]s. Moreover, much of theoretical physics can be recast as models of [[multilinear algebra]]s. Although structures with nonidentities retain an undoubted algebraic flavor, they suffer from defects varieties do not have. For example, neither the product of [[integral domain]]s nor a free field over any set exist. '''Arithmetics''': '''Two''' [[binary operation]]s, addition and multiplication. ''S'' is an [[infinite set]]. Arithmetics are pointed unary systems, whose [[unary operation]] is [[injective]] [[successor function|successor]], and with distinguished element 0. *[[Robinson arithmetic]]. Addition and multiplication are [[recursive]]ly defined by means of [[successor function|successor]]. 0 is the [[identity element]] for addition, and annihilates multiplication. Robinson arithmetic is listed here even though it is a variety, because of its closeness to Peano arithmetic. ** [[Peano arithmetic]]. Robinson arithmetic with an [[axiom schema]] of [[induction]]. Most ring and field axioms bearing on the properties of addition and multiplication are theorems of Peano arithmetic or of proper extensions thereof. '''Field-like structures''': '''Two''' [[binary operation]]s, addition and multiplication. ''S'' is nontrivial, i.e., ''S''≠{0}. * [[domain (ring theory)|Domain]]: a ring whose sole [[zero divisor]] is 0. ** [[Integral domain]]: a domain whose multiplication commutes. Also a commutative [[cancellative]] ring. *** [[Euclidean domain]]: an integral domain with a function ''f'': ''S''→'''N''' satisfying the division with remainder property. * [[Division ring]] (or ''sfield'', ''skew field''): a ring in which every member of ''S'' other than 0 has a two-sided multiplicative inverse. The nonzero members of ''S'' form a [[Group (mathematics)|group]] under multiplication. ** [[Field (mathematics)|Field]]: a division ring whose multiplication commutes. The nonzero members of ''S'' form an [[abelian group]] under multiplication. ***[[Ordered field]]: a field whose elements are [[linear order|totally ordered]]. **** [[Real field]]: a [[Dedekind complete]] ordered field. The following structures are not varieties for reasons in addition to ''S''≠{0}: * [[Simple ring]]: a ring having no [[ideal (ring theory)|ideals]] other than 0 and ''S''. **[[Weyl algebra]]: * [[Artinian ring]]: a ring whose [[ideal (ring theory)|ideals]] satisfy the [[descending chain condition]]. '''Composite Systems: Vector Spaces, and Algebras over Fields'''. Two Sets, ''M'' and ''R'', and at least '''three''' binary operations. The members of: #''M'' are [[vector space|vectors]], denoted by lower case letters. ''M'' is at minimum an [[abelian group]] under vector addition, with distinguished member '''0'''. #''R'' are [[scalar (mathematics)|scalar]]s, denoted by Greek letters. ''R'' is a [[field (mathematics)|field]], nearly always the [[real field|real]] or [[complex field]], with 0 and 1 as distinguished members. '''Three''' binary operations. *[[Vector space]]: a [[free module]] of [[dimension]] ''n'' except that ''R'' is a [[field (algebra)|field]]. **[[Normed vector space]]: a vector space with a [[norm (mathematics)|norm]], namely a function ''M''&nbsp;→&nbsp;''R'' that is [[symmetric]], [[linear]], and [[Positive-definite function|positive definite]]. ***[[Inner product space]] (also ''Euclidean'' vector space): a normed vector space such that ''R'' is the [[real field]], whose norm is the square root of the [[inner product]], ''M''×''M''→''R''. Let ''i'',''j'', and ''n'' be positive integers such that 1≤''i'',''j''≤''n''. Then ''M'' has an [[orthonormal basis]] such that ''e''<sub>i</sub>•''e''<sub>j</sub> = 1 if ''i''=''j'' and 0 otherwise; see [[free module]] above. ***[[inner product space|Unitary space]]: Differs from inner product spaces in that ''R'' is the [[complex field]], and the inner product has a different name, the [[inner product|hermitian inner product]], with different properties: [[conjugate]] symmetric, [[bilinear]], and [[positive-definite function|positive definite]]. See Birkhoff and MacLane (1979: 369). **[[Graded vector space]]: a vector space such that the members of ''M'' have a [[direct sum]] decomposition. See [[graded algebra]] below. '''Four''' binary operations. *[[Algebra over a field]]: An [[algebra over a ring]] except that ''R'' is a field instead of a commutative ring. **[[Jordan algebra]]: a [[Jordan algebra|Jordan ring]] except that ''R'' is a field. ** [[Lie algebra]]: an [[algebra over a field]] respecting the [[Jacobi identity]], whose vector multiplication, the [[Lie bracket]] denoted [''u,v''], [[anticommute]]s, does not associate, and is [[nilpotent]]. ** [[Associative algebra]]: an [[algebra over a field]], or a [[Module (mathematics)|module]], whose vector multiplication associates. *** [[Linear algebra]]: an associative [[unital]] algebra with the members of ''M'' being [[matrix (mathematics)|matrices]]. Every matrix has a [[dimension]] ''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 an [[inner product]], denoted 〈''u'',''v''〉. Hence there is an [[orthonormal basis]]; see [[inner product space]] above. There is a unary function, the [[determinant]], from square (''n''x''n'' for any ''n'') matrices to ''R''. *** [[Commutative algebra]]: an associative algebra whose vector multiplication commutes. ****[[Symmetric algebra]]: a commutative algebra with [[unital]] vector multiplication. '''Composite Systems: [[Multilinear algebra]]s'''. Two sets, ''V'' and ''K''. '''Four''' [[binary operation]]s: # The members of ''V'' are [[multivector]]s (including [[vector space|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 [[tensor (intrinsic definition)|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 function|positive definite]]; see [[inner 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 the [[real field]], as multilinear algebras are designed to describe physical phenomena without [[complex 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 (mathematics)|module]]. * [[Graded algebra]]: an associative algebra with [[unital]] outer product. The members of ''V'' have a [[direct sum]] decomposition resulting in their having a "degree," with vectors having degree 1. If ''u'' and ''v'' have degree ''i'' and ''j'', respectively, the outer product of ''u'' and ''v'' is of degree ''i+j''. ''V'' also has a distinguished member '''0''' for each possible degree. Hence all members of ''V'' having the same degree form an [[Abelian group]] under addition. ** [[Exterior algebra]] (also ''Grassmann algebra''): a graded algebra whose [[anticommutative]] outer product, denoted by infix ∧, is called the [[exterior product]]. ''V'' has an [[orthonormal basis]]. ''v''<sub>1</sub> ∧ ''v''<sub>2</sub> ∧ ... ∧ ''v''<sub>k</sub> = 0 if and only if ''v''<sub>1</sub>, ..., ''v''<sub>k</sub> 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''<sub>i</sub>,''e''<sub>i</sub>〉 = -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 of [[Peano arithmetic]] and a commutative [[semiring]]. [[Boolean algebra (structure)|Boolean algebra]]s are at once [[semigroup]]s, [[lattice (order)|lattices]], and [[Ring (mathematics)|rings]]. They would even be [[Abelian group]]s if the identity and inverse elements were identical instead of complements. '''Group-like structures''' *Nonzero '''N''' under [[addition]] (+) is a [[magma (algebra)|magma]]. *'''N''' under addition is a magma with an identity. * '''Z''' under [[subtraction]] (&minus;) is a quasigroup. * Nonzero '''Q''' under [[division (mathematics)|division]] (÷) is a quasigroup. * Every group is a loop, because ''a'' * ''x'' = ''b'' [[if and only if]] ''x'' = ''a''<sup>&minus;1</sup> * ''b'', and ''y'' * ''a'' = ''b'' if and only if ''y'' = ''b'' * ''a''<sup>&minus;1</sup>. * 2x2 [[Matrix (mathematics)|matrices]](of non-zero determinant) with matrix multiplication form a group. * '''Z''' under addition (+) is an Abelian group. * Nonzero '''Q''' under [[multiplication]] (&times;) is an Abelian group. *Every [[cyclic group]] ''G'' is Abelian, because if ''x'', ''y'' are in ''G'', then ''xy'' = ''a''<sup>m</sup>''a''<sup>n</sup> = ''a''<sup>m+n</sup> = ''a''<sup>n+m</sup> = ''a''<sup>n</sup>''a''<sup>m</sup> = ''yx''. In particular, '''Z''' is an Abelian group under addition, as is the [[modular arithmetic|integers modulo ''n'']] '''Z'''/''n'''''Z'''. *A [[monoid]] is a [[category theory|category]] with a single object, in which case the [[function composition|composition of morphisms]] and the [[identity morphism]] interpret monoid multiplication and identity element, respectively. * The [[two-element Boolean algebra|Boolean algebra]] '''2''' is a boundary algebra. *More [[examples of groups]] and [[list of small groups]]. '''[[Lattice]]s''' * The [[normal subgroup]]s of a group, and the [[submodules]] of a module, are modular lattices. * Any [[field of sets]], and the [[connective]]s of [[first-order logic]], are models of Boolean algebra. * The connectives of [[intuitionistic logic]] form a model of [[Heyting algebra]]. * The [[modal logic]] [[S4 (modal logic)|S4]] is a model of [[interior algebra]]. * [[Peano arithmetic]] and most [[axiomatic set theory|axiomatic set theories]], including [[ZFC]], [[von Neumann–Bernays–Gödel set theory|NBG]], and [[New foundations]], can be recast as models of [[relation algebra]]. '''[[ring (mathematics)|Ring]]-like structures''' * The set ''R''[X] of all [[polynomial]]s over some coefficient ring ''R'' is a ring. * 2x2 [[Matrix (mathematics)|matrices]] with matrix addition and multiplication form a ring. * If ''n'' is a positive integer, then the set '''Z'''<sub>''n''</sub> = '''Z'''/n'''Z''' of integers modulo ''n'' (the additive [[cyclic group]] of order ''n'' ) forms a ring having ''n'' elements (see [[modular arithmetic]]). '''[[Integral domain]]s''' * '''Z''' under addition and multiplication is an integral domain. * The [[p-adic number|p-adic integers]]. '''[[field (mathematics)|Fields]]''' * Each of '''Q''', '''R''', and '''C''', under addition and multiplication, is a field. *'''R''' [[linear order|totally ordered]] by "<" in the usual way is an [[ordered field]] and is [[categorical]]. The resulting [[real field]] grounds [[real analysis|real]] and [[functional analysis]]. **'''R''' contains several interesting subfields, the [[algebraic number|algebraic]], the [[computable number|computable]], and the [[definable number]]s. *An [[algebraic number field]] is a finite field extension of '''Q''', that is, a field containing '''Q''' which has finite dimension as a [[vector space]] over '''Q'''. Algebraic number fields are very important in [[number theory]]. *If ''q'' > 1 is a power of a [[prime number]], then there exists ([[up to]] [[isomorphism]]) exactly one [[finite field]] with ''q'' elements, usually denoted '''F'''<sub>''q''</sub>, or in the case that ''q'' is itself prime, by '''Z'''/''q'''''Z'''. Such fields are called [[Galois 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'''<sub>''p''</sub> = '''Z'''/''p'''''Z''' of integers modulo ''p'' is the finite field with ''p'' elements: '''F'''<sub>''p''</sub> = {0, 1, ..., ''p''&nbsp;&minus;&nbsp;1} where the operations are defined by performing the operation in '''Z''', dividing by ''p'' and taking the remainder; see [[modular arithmetic]]. ==Allowing additional structure== Algebraic structures can also be defined on sets with added structure of a non-algebraic nature, such as a [[topology]]. The added structure must be compatible, in some sense, with the algebraic structure. * [[Ordered group]]: a group with a compatible [[partial order]]. I.e., ''S'' is partially ordered. * [[Linearly ordered group]]: a group whose ''S'' is a [[linear order]]. * [[Archimedean group]]: a linearly ordered group for which the [[Archimedean property]] holds. * [[Lie group]]: a group whose ''S'' has a compatible smooth [[manifold]] structure. * [[Topological group]]: a group whose ''S'' has a compatible [[topology]]. * [[Topological vector space]]: a vector space whose ''M'' has a compatible [[topology]]; a superset of [[normed vector space]]s. ==Category theory== The discussion above has been cast in terms of elementary [[abstract algebra|abstract]] and [[universal algebra]]. [[Category theory]] is another way of reasoning about algebraic structures (see, for example, Mac Lane 1998). A category is a collection of ''objects'' with associated ''morphisms.'' Every algebraic structure has its own notion of [[homomorphism]], namely any [[function (mathematics)|function]] compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a [[category theory|category]]. For example, the [[category of groups]] has all [[Group (mathematics)|groups]] as objects and all [[group homomorphism]]s as morphisms. This [[concrete category]] may be seen as a [[category of sets]] with added category-theoretic [[structure (category theory)|structure]]. Likewise, the category of [[topological group]]s (whose morphisms are the continuous [[group homomorphism]]s) is a [[category of topological spaces]] with extra structure. There are various concepts in category theory that try to capture the algebraic character of a context, for instance *[[algebraic category|algebraic]] *[[essentially algebraic category|essentially algebraic]] *[[presentable category|presentable]] *[[locally presentable category|locally presentable]] *[[Monad (category theory)|monadic]] functors and categories *[[universal property]]. ==See also== *[[arity]] *[[category theory]] *[[free object]] *[[list of algebraic structures]] *[[list of first order theories]] *[[signature (logic)|signature]] *[[variety (universal algebra)|variety]] ==References== * {{Citation | last1=MacLane | first1=Saunders | author1-link=Saunders MacLane | last2=Birkhoff | first2=Garrett | author2-link=Garrett Birkhoff | title=Algebra | publisher=AMS Chelsea | edition=2nd | isbn=978-0-8218-1646-2 | year=1999}} * {{Citation | last1=Michel | first1=Anthony N. | last2=Herget | first2=Charles J. | title=Applied Algebra and Functional Analysis | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-67598-5 | year=1993}} A monograph available free online: * {{Citation | last1=Burris | first1=Stanley N. | last2=Sankappanavar | first2=H. P. | title=A Course in Universal Algebra | url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-90578-3 | year=1981}} Category theory: * {{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | title=[[Categories for the Working Mathematician|Categories for the Working Mathematician]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-98403-2 | year=1998}} * {{Citation | last1=Taylor | first1=Paul | title=Practical foundations of mathematics | publisher=[[Cambridge University Press]] | isbn=978-0-521-63107-5 | year=1999}} ==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] by [[Vaughan Pratt]]. [[Category:Abstract algebra]] [[Category:Mathematics-related lists|Algebraic structures]] [[ar:بنية جبرية]] [[ca:Estructura algebraica]] [[cs:Algebraická struktura]] [[de:Algebraische Struktur]] [[es:Estructura algebraica]] [[eu:Egitura aljebraiko]] [[fr:Structure algébrique]] [[ko:대수적 구조]] [[it:Struttura algebrica]] [[he:מבנה אלגברי]] [[nl:Algebraïsche structuur]] [[ja:代数的構造]] [[no:Algebraisk struktur]] [[nn:Algebraisk struktur]] [[oc:Estructura algebrica]] [[pms:Strutura algébrica]] [[pt:Estrutura algébrica]] [[ru:Алгебраическая система]] [[simple:Algebraic structure]] [[sk:Algebrická štruktúra]] [[sr:Алгебарска структура]] [[sv:Algebraisk struktur]] [[uk:Алгебраїчна система]] [[zh:代数结构]]