36:
3015:
3770:
676:. Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, the possible moves of an object in three-dimensional space can be combined by performing a first move of the object, and then a second move from its new position. Such moves, formally called
1516:
The introduction of such auxiliary operation complicates slightly the statement of an axiom, but has some advantages. Given a specific algebraic structure, the proof that an existential axiom is satisfied consists generally of the definition of the auxiliary function, completed with straightforward
1567:
is a class of algebraic structures that share the same operations, and the same axioms, with the condition that all axioms are identities. What precedes shows that existential axioms of the above form are accepted in the definition of a variety.
1820:
1989:
1305:. In general, such a clause can be avoided by introducing further operations, and replacing the existential clause by an identity involving the new operation. More precisely, let us consider an axiom of the form
687:. When a new problem involves the same laws as such an algebraic structure, all the results that have been proved using only the laws of the structure can be directly applied to the new problem.
1659:
1512:
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910:
1371:
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1555:
2869:, the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself. For example, the sentence, "We have defined a ring
808:
2280:: a module over a field, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and
2060:
binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers.
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950:
844:
766:
739:. If the variables in the identity are replaced by arbitrary elements of the algebraic structure, the equality must remain true. Here are some common examples.
2837:
690:
In full generality, algebraic structures may involve an arbitrary collection of operations, including operations that combine more than two elements (higher
1517:
verifications. Also, when computing in an algebraic structure, one generally uses explicitly the auxiliary operations. For example, in the case of
3079:
can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the
2018:
420:
2523:
is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator
515:
is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in
485:
An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a
1826:
is an algebraic structure with a binary operation that is associative, has an identity element, and for which all elements are invertible.
596:
that combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example,
1610:
706:). The examples listed below are by no means a complete list, but include the most common structures taught in undergraduate courses.
3192:
3223:
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3171:
3147:
3125:
17:
2145:: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element).
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1137:
979:
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2205:: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.
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There are various concepts in category theory that try to capture the algebraic character of a context, for instance
2411:: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure.
853:
79:
57:
2500:") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given
50:
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is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of
2535:, a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables
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Identities are equations formulated using only the operations the structure allows, and variables that are tacitly
505:
2669:
Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g.,
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compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a
3043:
2813:
2325:
3799:
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and structures that are not. If all axioms defining a class of algebras are identities, then this class is a
1712:
775:
573:. In category theory, the collection of all structures of a given type and homomorphisms between them form a
406:
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2449:
abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by
2254:, which satisfies several axioms. Counting the ring operations these systems have at least three operations.
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2511:. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure
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1892:. This axiom cannot be reduced to axioms of preceding types. (it follows that fields do not form a
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2595:))) would be an element of the term algebra. One of the axioms defining a group is the identity
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1815:{\displaystyle \operatorname {inv} (x)*x=e\quad {\text{and}}\quad x*\operatorname {inv} (x)=e.}
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of any kind other than the allowed operations. The study of varieties is an important part of
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The collection of all structures of a given type (same operations and same laws) is called a
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Algebraic structures can also coexist with added structure of non-algebraic nature, such as
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1984:{\displaystyle \forall x,\quad x=0\quad {\text{or}}\quad x\cdot \operatorname {inv} (x)=1.}
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in the structure. Here, the auxiliary operation is the operation of arity zero that has
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on the free algebra; the quotient algebra then has the algebraic structure of a group.
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2332:. The added structure must be compatible, in some sense, with the algebraic structure.
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2677:. Structures with nonidentities present challenges varieties do not. For example, the
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1999:
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703:
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is commutative, left and right distributivity are both equivalent to distributivity.
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in universal algebra; this term is also used with a completely different meaning in
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2657:. For an algebraic structure to be a variety, its operations must be defined for
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Structures such as fields have some axioms that hold only for nonzero members of
2178:
2072:
2009:; or as an ordinary function whose value at 0 is arbitrary and must not be used.
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between categories of algebraic structures "forgets" a part of a structure.
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and 1 being a multiplicative identity element, but this is a nonidentity;
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if it is both left distributive and right distributive. If the operation
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Algebraic structures are defined through different configurations of
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680:, obey the associative law, but fail to satisfy the commutative law.
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192:
149:
117:
2019:
Outline of algebraic structures § Types of algebraic structures
1082:, because the second operation is addition in many common examples).
1062:
in the algebraic structure (the second operation is denoted here as
683:
Sets with one or more operations that obey specific laws are called
3463:
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3374:
2808:
with added category-theoretic structure. Likewise, the category of
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2329:
2101:
585:
187:
1709:
if it has an inverse element, that is, if there exists an element
3295:
2422:: a *-algebra of operators on a Hilbert space equipped with the
2492:. An algebraic structure in a variety may be understood as the
2068:
1518:
121:
2860:
2812:(whose morphisms are the continuous group homomorphisms) is a
2339:: a group with a topology compatible with the group operation.
2139:: a ring in which the multiplication operation is commutative.
2442:
1431:
1384:
691:
1885:) that apply to elements (not to subsets) of the structure.
719:
An axiom of an algebraic structure often has the form of an
2646:
It is necessary that 0 ≠ 1, 0 being the additive
3264:
2547:, etc. the term algebra is the collection of all possible
2199:
under union and intersection forms a distributive lattice.
538:
In universal algebra, an algebraic structure is called an
472:
such as addition and multiplication), and a finite set of
546:
is an algebraic structure that is a vector space over a
542:; this term may be ambiguous, since, in other contexts,
2642:
Some structures do not form varieties, because either:
2156:
or more binary operations, including operations called
2119:: a semiring whose additive monoid is an abelian group.
735:
that involve operations of the algebraic structure and
2989:
2960:
2926:
2906:
2879:
2691:
1927:
1865:
1747:
1715:
1683:
1613:
1589:
1571:
Here are some of the most common existential axioms.
1531:
1443:
1401:
1323:
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1258:
1234:
1140:
1120:
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1068:
982:
962:
938:
856:
832:
778:
754:
3010:
3196:(2nd ed.), Berlin, New York: Springer-Verlag,
3155:
3083:; in the case of lattices, they are linked by the
2995:
2968:
2946:
2912:
2885:
2745:
1983:
1877:
1814:
1733:
1689:
1654:{\displaystyle x*e=x\quad {\text{and}}\quad e*x=x}
1653:
1595:
1549:
1507:{\displaystyle f(X,\varphi (X))=g(X,\varphi (X)).}
1506:
1422:
1365:
1284:
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1240:
1200:
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1102:
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1042:
968:
944:
904:
838:
802:
760:
523:is another formalization that includes also other
3156:Burris, Stanley N.; Sankappanavar, H. P. (1981),
2527:, taking two arguments, and the inverse operator
2071:with a unary operation (inverse), giving rise to
3786:
3108:
2776:Every algebraic structure has its own notion of
2531:, taking one argument, and the identity element
1835:The axioms of an algebraic structure can be any
3134:Michel, Anthony N.; Herget, Charles J. (1993),
2345:: a topological group with a compatible smooth
1603:has an identity element if there is an element
2012:
694:operations) and operations that take only one
3280:
414:
3241:Includes many structures not mentioned here.
3133:
503:), and elements of the vector space (called
2363:: each type of structure with a compatible
2209:
2191:: a lattice in which each of meet and join
1387:of variables. Choosing a specific value of
3287:
3273:
2023:
421:
407:
2962:
2931:
2496:of term algebra (also called "absolutely
2373:: a linearly ordered group for which the
1525:is provided by the unary minus operation
80:Learn how and when to remove this message
3193:Categories for the Working Mathematician
3186:
2401:(as a metric space) then it is called a
1830:
43:This article includes a list of general
27:Set with operations obeying given axioms
3136:Applied Algebra and Functional Analysis
3087:. Ringoids also tend to have numerical
1734:{\displaystyle \operatorname {inv} (x)}
1430:which can be viewed as an operation of
14:
3787:
3209:
2746:{\displaystyle (1,0)\cdot (0,1)=(0,0)}
497:between elements of the field (called
482:) that these operations must satisfy.
3268:
2665:; there can be no partial operations.
1902:"Every nonzero element of a field is
1888:Such a typical axiom is inversion in
1437:, and the axiom becomes the identity
1296:
531:between structures of the same type (
489:involves a second structure called a
3212:Practical foundations of mathematics
2430:
2319:
2081:: a group whose binary operation is
1423:{\displaystyle \varphi :X\mapsto y,}
714:
29:
3251:Stanford Encyclopedia of Philosophy
2631:. The axioms can be represented as
2563:and the variables; so for example,
2393:: a vector space with a compatible
2045:: a degenerate algebraic structure
1201:{\displaystyle (y+z)*x=(y*x)+(z*x)}
1043:{\displaystyle x*(y+z)=(x*y)+(x*z)}
24:
2760:
2482:existentially quantified variables
1928:
1872:
1866:
1252:with respect to another operation
1114:with respect to another operation
956:with respect to another operation
49:it lacks sufficient corresponding
25:
3816:
3232:
2920:. For another example, the group
2861:Different meanings of "structure"
1878:{\displaystyle \forall ,\exists }
3769:
3768:
3013:
2947:{\displaystyle (\mathbb {Z} ,+)}
2100:binary operations, often called
709:
34:
2284:with respect to multiplication.
2171:: a lattice in which arbitrary
2133:by nonzero elements is defined.
1953:
1947:
1937:
1781:
1775:
1635:
1629:
905:{\displaystyle (x*y)*z=x*(y*z)}
727:such that the two sides of the
580:
3091:, while lattices tend to have
3069:
3056:
3044:Structure (mathematical logic)
2941:
2927:
2893:", means that we have defined
2833:essentially algebraic category
2814:category of topological spaces
2740:
2728:
2722:
2710:
2704:
2692:
2238:, and the binary operation of
1972:
1966:
1839:, that is a formula involving
1800:
1794:
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1728:
1722:
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1495:
1489:
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1468:
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1339:
1327:
1301:Some common axioms contain an
1195:
1183:
1177:
1165:
1153:
1141:
1037:
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1019:
1007:
1001:
989:
899:
887:
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857:
13:
1:
3159:A Course in Universal Algebra
3120:(2nd ed.), AMS Chelsea,
3102:
2504:generate a free algebra, the
1697:that has an identity element
1366:{\displaystyle f(X,y)=g(X,y)}
592:are prototypical examples of
3573:Eigenvalues and eigenvectors
3239:Jipsen's algebra structures.
2969:{\displaystyle \mathbb {Z} }
2843:locally presentable category
1550:{\displaystyle x\mapsto -x.}
7:
3294:
3006:
2013:Common algebraic structures
1220:in the algebraic structure.
924:in the algebraic structure.
818:in the algebraic structure.
10:
3821:
3216:Cambridge University Press
2434:
2387:has a compatible topology.
2260:: a module where the ring
2016:
1998:can be viewed either as a
702:) or even zero arguments (
493:, and an operation called
3764:
3733:
3717:
3646:
3553:
3492:
3413:
3320:
3302:
3247:page on abstract algebra.
2976:that is equipped with an
2753:, but fields do not have
2635:. These equations induce
2457:(not to be confused with
1677:Given a binary operation
3049:
2816:with extra structure. A
2685:is not a field, because
2515:. The quotient algebra
2476:. Identities contain no
2381:Topological vector space
2268:or, in some contexts, a
2210:Two sets with operations
2002:that is not defined for
569:, as an abbreviation of
3805:Mathematical structures
3647:Algebraic constructions
3350:Algebraic number theory
2850:functors and categories
2415:Vertex operator algebra
2383:: a vector space whose
2226:acting as operators on
2024:One set with operations
803:{\displaystyle x*y=y*x}
525:mathematical structures
441:consists of a nonempty
64:more precise citations.
3390:Noncommutative algebra
3034:Mathematical structure
2997:
2970:
2948:
2914:
2887:
2747:
2470:universally quantified
2424:weak operator topology
2300:definite bilinear form
2108:, with multiplication
1985:
1879:
1816:
1735:
1691:
1655:
1597:
1551:
1508:
1424:
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1044:
970:
946:
906:
840:
804:
762:
3627:Orthogonal complement
3400:Representation theory
3210:Taylor, Paul (1999),
2998:
2971:
2954:can be seen as a set
2949:
2915:
2888:
2748:
2397:. If such a space is
2240:scalar multiplication
2234:are sometimes called
2054:Group-like structures
2049:having no operations.
1986:
1900:.) It can be stated:
1880:
1831:Non-equational axioms
1817:
1736:
1692:
1656:
1598:
1552:
1509:
1425:
1368:
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1267:
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1203:
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1045:
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907:
841:
805:
763:
495:scalar multiplication
3800:Algebraic structures
3725:Algebraic structures
3493:Algebraic structures
3478:Equivalence relation
3421:Algebraic expression
3162:, Berlin, New York:
2987:
2978:algebraic structure,
2958:
2924:
2904:
2877:
2838:presentable category
2689:
2375:Archimedean property
2278:Algebra over a field
2189:Distributive lattice
2090:Ring-like structures
1925:
1910:the structure has a
1863:
1745:
1713:
1681:
1611:
1587:
1529:
1441:
1399:
1321:
1276:
1256:
1232:
1138:
1118:
1094:
1086:Right distributivity
1066:
980:
960:
936:
854:
830:
776:
752:
685:algebraic structures
318:Group with operators
261:Complemented lattice
96:Algebraic structures
18:Algebraic structures
3654:Composition algebra
3578:Inner product space
3556:multilinear algebra
3444:Polynomial function
3385:Multilinear algebra
3370:Homological algebra
3360:Commutative algebra
2800:as morphisms. This
2798:group homomorphisms
2796:as objects and all
2788:. For example, the
2637:equivalence classes
2459:algebraic varieties
2420:Von Neumann algebra
2391:Normed vector space
2288:Inner product space
2218:: an abelian group
2181:: a lattice with a
2160:, connected by the
1857:logical quantifiers
1841:logical connectives
1837:first-order formula
1395:defines a function
928:Left distributivity
460:), a collection of
439:algebraic structure
372:Composition algebra
132:Quasigroup and loop
3434:Quadratic equation
3365:Elementary algebra
3333:Algebraic geometry
3188:Mac Lane, Saunders
3140:Dover Publications
3110:Mac Lane, Saunders
3066:, Springer, p. 41.
3062:P.M. Cohn. (1981)
3021:Mathematics portal
2993:
2966:
2944:
2910:
2883:
2854:universal property
2828:algebraic category
2810:topological groups
2790:category of groups
2743:
2472:over the relevant
2463:algebraic geometry
2195:over the other. A
2185:and least element.
2150:Lattice structures
1981:
1908:or, equivalently:
1875:
1812:
1731:
1687:
1651:
1593:
1547:
1504:
1420:
1391:for each value of
1363:
1303:existential clause
1297:Existential axioms
1282:
1262:
1238:
1198:
1124:
1112:right distributive
1100:
1072:
1040:
966:
942:
902:
836:
800:
758:
704:nullary operations
567:algebraic geometry
3782:
3781:
3704:Symmetric algebra
3674:Geometric algebra
3454:Linear inequality
3405:Universal algebra
3338:Algebraic variety
3225:978-0-521-63107-5
3203:978-0-387-98403-2
3173:978-3-540-90578-3
3149:978-0-486-67598-5
3127:978-0-8218-1646-2
3114:Birkhoff, Garrett
3064:Universal Algebra
3039:Signature (logic)
2996:{\displaystyle +}
2913:{\displaystyle A}
2886:{\displaystyle A}
2867:abuse of notation
2818:forgetful functor
2804:may be seen as a
2802:concrete category
2490:universal algebra
2447:Universal algebra
2437:Universal algebra
2431:Universal algebra
2371:Archimedean group
2337:Topological group
2320:Hybrid structures
2230:. The members of
2029:Simple structures
2000:partial operation
1951:
1898:universal algebra
1779:
1690:{\displaystyle *}
1633:
1596:{\displaystyle *}
1561:universal algebra
1285:{\displaystyle *}
1265:{\displaystyle +}
1241:{\displaystyle *}
1127:{\displaystyle +}
1103:{\displaystyle *}
1075:{\displaystyle +}
969:{\displaystyle +}
954:left distributive
945:{\displaystyle *}
839:{\displaystyle *}
761:{\displaystyle *}
715:Equational axioms
575:concrete category
571:algebraic variety
517:universal algebra
470:binary operations
431:
430:
90:
89:
82:
16:(Redirected from
3812:
3795:Abstract algebra
3772:
3771:
3659:Exterior algebra
3328:Abstract algebra
3289:
3282:
3275:
3266:
3265:
3228:
3206:
3176:
3152:
3130:
3096:
3081:distributive law
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2892:
2890:
2889:
2884:
2806:category of sets
2772:with associated
2752:
2750:
2749:
2744:
2648:identity element
2494:quotient algebra
2314:
2183:greatest element
2169:Complete lattice
2137:Commutative ring
2073:inverse elements
2036:binary operation
2008:
1997:
1990:
1988:
1987:
1982:
1952:
1949:
1916:
1896:in the sense of
1884:
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1594:
1582:binary operation
1576:Identity element
1556:
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1553:
1548:
1523:additive inverse
1513:
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700:unary operations
674:commutative laws
671:
661:
642:associative laws
639:
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556:commutative ring
513:Abstract algebra
423:
416:
409:
198:Commutative ring
127:Rack and quandle
92:
91:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
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3694:Quotient object
3684:Polynomial ring
3642:
3603:Linear subspace
3555:
3549:
3488:
3430:Linear equation
3409:
3355:Category theory
3316:
3298:
3293:
3235:
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3181:Category theory
3174:
3164:Springer-Verlag
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2766:Category theory
2763:
2761:Category theory
2690:
2687:
2686:
2439:
2433:
2322:
2302:
2212:
2203:Boolean algebra
2179:Bounded lattice
2112:over addition.
2026:
2021:
2015:
2003:
1995:
1948:
1926:
1923:
1922:
1914:
1912:unary operation
1864:
1861:
1860:
1833:
1822:For example, a
1776:
1746:
1743:
1742:
1714:
1711:
1710:
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1673:Inverse element
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521:Category theory
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398:
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396:
367:Non-associative
349:
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337:
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307:
296:
295:
284:Map of lattices
280:
276:Boolean algebra
271:Heyting algebra
245:
234:
233:
227:
208:Integral domain
172:
161:
160:
154:
108:
86:
75:
69:
66:
56:Please help to
55:
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35:
28:
23:
22:
15:
12:
11:
5:
3818:
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3746:Linear algebra
3743:
3737:
3735:
3731:
3730:
3728:
3727:
3721:
3719:
3715:
3714:
3712:
3711:
3709:Tensor algebra
3706:
3701:
3698:Quotient group
3691:
3681:
3671:
3661:
3656:
3650:
3648:
3644:
3643:
3641:
3640:
3635:
3630:
3620:
3617:Euclidean norm
3610:
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3585:
3575:
3570:
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3557:
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3527:
3517:
3507:
3496:
3494:
3490:
3489:
3487:
3486:
3481:
3471:
3468:Multiplication
3457:
3447:
3437:
3423:
3417:
3415:
3414:Basic concepts
3411:
3410:
3408:
3407:
3402:
3397:
3392:
3387:
3382:
3380:Linear algebra
3377:
3372:
3367:
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3233:External links
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3085:absorption law
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2679:direct product
2675:division rings
2667:
2666:
2651:
2435:Main article:
2432:
2429:
2428:
2427:
2417:
2412:
2406:
2388:
2378:
2368:
2361:ordered fields
2353:Ordered groups
2350:
2340:
2321:
2318:
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2316:
2285:
2274:
2273:
2255:
2242:is a function
2211:
2208:
2207:
2206:
2200:
2186:
2176:
2173:meet and joins
2162:absorption law
2147:
2146:
2140:
2134:
2129:ring in which
2120:
2106:multiplication
2087:
2086:
2076:
2051:
2050:
2025:
2022:
2017:Main article:
2014:
2011:
1994:The operation
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1980:
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1669:as its result.
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1224:Distributivity
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784:
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757:
746:
723:, that is, an
716:
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582:
579:
450:underlying set
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3623:Orthogonality
3621:
3618:
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3591:
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3588:Hilbert space
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3270:
3267:
3260:
3259:Vaughan Pratt
3256:
3252:
3249:
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3227:
3221:
3217:
3213:
3208:
3205:
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3194:
3189:
3185:
3184:
3180:
3179:
3175:
3169:
3165:
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3160:
3154:
3151:
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3141:
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3123:
3119:
3115:
3111:
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3106:
3094:
3093:set-theoretic
3090:
3086:
3082:
3078:
3075:Ringoids and
3072:
3065:
3059:
3055:
3045:
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3040:
3037:
3035:
3032:
3030:
3027:
3026:
3022:
3016:
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2983:
2979:
2938:
2935:
2907:
2899:
2896:
2880:
2872:
2868:
2855:
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2849:
2846:
2844:
2841:
2839:
2836:
2834:
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2829:
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2824:
2821:
2819:
2815:
2811:
2807:
2803:
2799:
2795:
2791:
2787:
2783:
2780:, namely any
2779:
2775:
2771:
2767:
2758:
2756:
2755:zero divisors
2737:
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2731:
2725:
2719:
2716:
2713:
2707:
2701:
2698:
2695:
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2649:
2645:
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2640:
2638:
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2618:
2615:; another is
2614:
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2487:
2483:
2479:
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2456:
2452:
2448:
2444:
2438:
2425:
2421:
2418:
2416:
2413:
2410:
2409:Hilbert space
2407:
2404:
2400:
2396:
2392:
2389:
2386:
2382:
2379:
2376:
2372:
2369:
2366:
2365:partial order
2362:
2358:
2357:ordered rings
2354:
2351:
2348:
2344:
2341:
2338:
2335:
2334:
2333:
2331:
2327:
2326:partial order
2313:
2309:
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2301:
2297:
2294:vector space
2293:
2289:
2286:
2283:
2279:
2276:
2275:
2271:
2270:division ring
2267:
2263:
2259:
2256:
2253:
2249:
2246: ×
2245:
2241:
2237:
2233:
2229:
2225:
2221:
2217:
2214:
2213:
2204:
2201:
2198:
2194:
2190:
2187:
2184:
2180:
2177:
2174:
2170:
2167:
2166:
2165:
2163:
2159:
2158:meet and join
2155:
2151:
2144:
2141:
2138:
2135:
2132:
2128:
2124:
2123:Division ring
2121:
2118:
2115:
2114:
2113:
2111:
2107:
2103:
2099:
2095:
2091:
2084:
2080:
2079:Abelian group
2077:
2074:
2070:
2066:
2063:
2062:
2061:
2059:
2055:
2048:
2044:
2041:
2040:
2039:
2037:
2034:
2030:
2020:
2010:
2006:
2001:
1978:
1975:
1969:
1963:
1960:
1957:
1954:
1944:
1941:
1938:
1934:
1931:
1921:
1920:
1919:
1918:
1913:
1907:
1905:
1899:
1895:
1891:
1886:
1869:
1858:
1854:
1850:
1846:
1842:
1838:
1825:
1809:
1806:
1803:
1797:
1791:
1788:
1785:
1782:
1772:
1769:
1766:
1763:
1757:
1751:
1748:
1725:
1719:
1716:
1708:
1701:, an element
1684:
1676:
1674:
1671:
1648:
1645:
1642:
1639:
1636:
1626:
1623:
1620:
1617:
1614:
1590:
1583:
1579:
1577:
1574:
1573:
1572:
1569:
1566:
1562:
1557:
1544:
1541:
1538:
1532:
1524:
1520:
1514:
1501:
1492:
1486:
1483:
1480:
1474:
1471:
1462:
1456:
1453:
1450:
1444:
1433:
1417:
1414:
1408:
1405:
1402:
1386:
1357:
1354:
1351:
1345:
1342:
1336:
1333:
1330:
1324:
1316:
1304:
1279:
1259:
1251:
1235:
1228:An operation
1227:
1225:
1222:
1192:
1189:
1186:
1180:
1174:
1171:
1168:
1162:
1159:
1156:
1150:
1147:
1144:
1121:
1113:
1097:
1090:An operation
1089:
1087:
1084:
1069:
1034:
1031:
1028:
1022:
1016:
1013:
1010:
1004:
998:
995:
992:
986:
983:
963:
955:
939:
932:An operation
931:
929:
926:
896:
893:
890:
884:
881:
878:
875:
872:
866:
863:
860:
849:
833:
826:An operation
825:
823:
822:Associativity
820:
797:
794:
791:
788:
785:
782:
779:
771:
755:
748:An operation
747:
745:
744:Commutativity
742:
741:
740:
738:
734:
730:
726:
722:
710:Common axioms
707:
705:
701:
697:
693:
688:
686:
681:
679:
678:rigid motions
675:
670:
666:
660:
656:
652:
648:
643:
638:
634:
630:
626:
620:
616:
612:
608:
604:
600:
595:
591:
587:
578:
576:
572:
568:
564:
559:
557:
553:
549:
545:
541:
536:
534:
533:homomorphisms
530:
526:
522:
518:
514:
510:
508:
507:
502:
501:
496:
492:
488:
483:
481:
480:
475:
471:
467:
463:
459:
455:
451:
447:
444:
440:
436:
424:
419:
417:
412:
410:
405:
404:
402:
401:
393:
390:
388:
385:
383:
380:
378:
375:
373:
370:
368:
365:
363:
360:
359:
355:
352:
351:
347:
342:
341:
334:
333:
329:
328:
324:
321:
319:
316:
314:
311:
310:
305:
300:
299:
292:
291:
287:
285:
282:
281:
277:
274:
272:
269:
267:
264:
262:
259:
257:
254:
252:
249:
248:
243:
238:
237:
232:
231:
224:
221:
219:
218:Division ring
216:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
175:
170:
165:
164:
159:
158:
151:
148:
146:
143:
141:
140:Abelian group
138:
137:
133:
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128:
125:
123:
119:
116:
114:
111:
110:
106:
101:
100:
97:
94:
93:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
3751:Order theory
3741:Field theory
3607:Affine space
3540:Vector space
3395:Order theory
3211:
3191:
3158:
3138:, New York:
3135:
3117:
3071:
3063:
3058:
2981:
2977:
2897:
2870:
2865:In a slight
2864:
2822:
2778:homomorphism
2773:
2769:
2764:
2668:
2662:
2658:
2654:
2641:
2628:
2624:
2620:
2616:
2612:
2608:
2604:
2600:
2596:
2592:
2588:
2584:
2580:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2544:
2540:
2536:
2532:
2528:
2524:
2520:
2516:
2512:
2508:
2506:term algebra
2498:free algebra
2467:
2450:
2440:
2403:Banach space
2384:
2323:
2311:
2307:
2303:
2295:
2291:
2261:
2258:Vector space
2251:
2247:
2243:
2239:
2231:
2227:
2223:
2219:
2153:
2149:
2148:
2110:distributing
2097:
2093:
2089:
2088:
2057:
2053:
2052:
2046:
2032:
2028:
2027:
2004:
1993:
1909:
1901:
1887:
1852:
1848:
1844:
1834:
1706:
1570:
1558:
1515:
1306:
1300:
1250:distributive
1249:
1111:
953:
847:
769:
718:
689:
684:
682:
668:
664:
658:
654:
650:
646:
636:
632:
628:
624:
618:
614:
610:
606:
602:
598:
584:
581:Introduction
560:
539:
537:
511:
504:
498:
494:
487:vector space
484:
478:
465:
457:
453:
449:
448:(called the
445:
438:
432:
392:Hopf algebra
330:
323:Vector space
288:
228:
157:Group theory
155:
120: /
95:
76:
67:
48:
3756:Ring theory
3718:Topic lists
3678:Multivector
3664:Free object
3582:Dot product
3568:Determinant
3554:Linear and
3029:Free object
2980:namely the
2900:on the set
2873:on the set
2661:members of
2478:connectives
2222:and a ring
2193:distributes
2083:commutative
848:associative
770:commutative
733:expressions
729:equals sign
468:(typically
454:carrier set
435:mathematics
377:Lie algebra
362:Associative
266:Total order
256:Semilattice
230:Ring theory
70:August 2024
62:introducing
3789:Categories
3734:Glossaries
3688:Polynomial
3668:Free group
3593:Linear map
3450:Inequality
3103:References
2898:operations
2774:morphisms.
2551:involving
2502:signatures
2451:identities
2349:structure.
2127:nontrivial
1904:invertible
1741:such that
1707:invertible
1607:such that
1208:for every
1050:for every
912:for every
810:for every
594:operations
544:an algebra
476:(known as
474:identities
462:operations
45:references
3460:Operation
3245:Mathworld
2982:operation
2871:structure
2708:⋅
2486:relations
2343:Lie group
2282:linearity
2197:power set
1964:
1958:⋅
1929:∀
1917:such that
1873:∃
1867:∀
1843:(such as
1792:
1786:∗
1764:∗
1752:
1720:
1685:∗
1640:∗
1618:∗
1591:∗
1559:Also, in
1539:−
1536:↦
1487:φ
1457:φ
1412:↦
1403:φ
1315:such that
1311:there is
1307:"for all
1280:∗
1236:∗
1190:∗
1172:∗
1157:∗
1098:∗
1032:∗
1014:∗
987:∗
940:∗
894:∗
885:∗
873:∗
864:∗
834:∗
795:∗
783:∗
756:∗
737:variables
529:functions
387:Bialgebra
193:Near-ring
150:Lie group
118:Semigroup
3774:Category
3484:Variable
3474:Relation
3464:Addition
3440:Function
3426:Equation
3375:K-theory
3190:(1998),
3116:(1999),
3077:lattices
3007:See also
2792:has all
2786:category
2782:function
2474:universe
2399:complete
2347:manifold
2330:topology
2131:division
2102:addition
2094:Ringoids
1661:for all
725:equation
721:identity
696:argument
586:Addition
223:Lie ring
188:Semiring
3686: (
3676: (
3542: (
3532: (
3522: (
3512: (
3502: (
3312:History
3307:Outline
3296:Algebra
3255:Algebra
3118:Algebra
3095:models.
2848:monadic
2770:objects
2681:of two
2455:variety
2298:with a
2236:scalars
1894:variety
1855:), and
1565:variety
1519:numbers
563:variety
554:over a
540:algebra
506:vectors
500:scalars
354:Algebra
346:Algebra
251:Lattice
242:Lattice
58:improve
3700:, ...)
3670:, ...)
3597:Matrix
3544:Vector
3534:theory
3524:theory
3520:Module
3514:theory
3504:theory
3343:Scheme
3222:
3200:
3170:
3146:
3124:
3089:models
2794:groups
2683:fields
2671:fields
2443:axioms
2377:holds.
2216:Module
2175:exist.
2069:monoid
1890:fields
1521:, the
1375:where
644:, and
552:module
479:axioms
458:domain
382:Graded
313:Module
304:Module
203:Domain
122:Monoid
47:, but
3638:Trace
3563:Basis
3510:Group
3500:Field
3321:Areas
3050:Notes
2633:trees
2611:)) =
2549:terms
2484:, or
2328:or a
2290:: an
2266:field
2264:is a
2143:Field
2065:Group
1853:"not"
1845:"and"
1824:group
1432:arity
1385:tuple
1379:is a
692:arity
631:) = (
609:) = (
550:or a
548:field
491:field
437:, an
348:-like
306:-like
244:-like
213:Field
171:-like
145:Magma
113:Group
107:-like
105:Group
3633:Rank
3613:Norm
3530:Ring
3220:ISBN
3198:ISBN
3168:ISBN
3144:ISBN
3122:ISBN
2895:ring
2673:and
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2359:and
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2117:Ring
2104:and
2067:: a
1851:and
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1216:and
1058:and
920:and
814:and
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672:are
662:and
640:are
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588:and
527:and
178:Ring
169:Ring
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1961:inv
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464:on
456:or
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433:In
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2306:×
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1979:1.
1950:or
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1970:x
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1008:(
1005:=
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999:z
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627:(
625:a
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466:A
446:A
422:e
415:t
408:v
83:)
77:(
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68:(
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20:)
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