2893:
1567:
that has the property that the product of any two of its elements is again in the subspace. In other words, a subalgebra of an algebra is a non-empty subset of elements that is closed under addition, multiplication, and scalar multiplication. In symbols, we say that a subset
2173:-module allows extending this theory as a Gröbner basis theory for submodules of a free module. This extension allows, for computing a Gröbner basis of a submodule, to use, without any modification, any algorithm and any software for computing Gröbner bases of ideals.
2795:
1293:, left distributivity and right distributivity are equivalent, and, in this case, only one distributivity requires a proof. In general, for non-commutative operations left distributivity and right distributivity are not equivalent, and require separate proofs.
3090:
3356:
1638:-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. In symbols, we say that a subset
3746:
3822:
3667:
3591:
3515:
3125:, the structure coefficients are generally written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are
1757:
on its own is usually taken to mean a two-sided ideal. Of course when the algebra is commutative, then all of these notions of ideal are equivalent. Conditions (1) and (2) together are equivalent to
680:
2888:{\displaystyle {\begin{matrix}&&K&&\\&\eta _{A}\swarrow &\,&\eta _{B}\searrow &\\A&&{\begin{matrix}f\\\longrightarrow \end{matrix}}&&B\end{matrix}}}
1517:
3965:
3931:
890:
773:
1914:
of the multiplication operation, which are not required in the broad definition of an algebra. The theories corresponding to the different types of algebras are often very different.
989:
2644:
2511:
2393:. The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited – that is, it means "not necessarily associative".
2391:
1857:
4086:
820:
724:
611:
3426:
3393:
2585:
3273:
There exist two such two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the identity element) and
4058:
3987:
935:
2732:
1981:
in the algebra, not to be confused with the algebra with one element. It is inherently non-unital (except in the case of only one element), associative and commutative.
3258:; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.
3266:
Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by
4024:
2787:
2761:
1892:
3967:, and hence it has the structure of an algebra over its center, which is not a field. Note that the split-biquaternion algebra is also naturally an 8-dimensional
2996:
2143:
These unital zero algebras may be more generally useful, as they allow to translate any general property of the algebras to properties of vector spaces or
1627:
In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra.
3283:
3432:
There exist five such three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element),
358:
464:
is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the
2933:
has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on
4177:
3674:
3134:
3126:
3753:
3598:
3522:
3446:
621:
4294:
4289:. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated by Reid, M. (2nd ed.). Cambridge University Press.
4203:
1772:, in that here we require the condition (2). Of course if the algebra is unital, then condition (3) implies condition (2).
1464:
17:
1906:
Algebras over fields come in many different types. These types are specified by insisting on some further axioms, such as
4088:. On the other hand, not all rings can be given the structure of an algebra over a field (for example the integers). See
351:
4351:
4319:
4233:
3940:
3906:
3130:
495:
is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a
4249:
Study, E. (1890), "Über
Systeme complexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen",
830:
734:
959:
541:, as, for such a space, the result of a product is not in the space, but rather in the field of coefficients.
4094:
for a description of an attempt to give to every ring a structure that behaves like an algebra over a field.
2596:
1813:. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product
1274:. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily
344:
2472:
2364:
1816:
3261:
3118:
Note however that several different sets of structure coefficients can give rise to isomorphic algebras.
2452:-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field
213:
3254:. If it isn't, then the multiplication is still completely determined by its action on a set that spans
4063:
3138:
2952:
1550:
796:
700:
587:
3400:
3367:
2437:
939:
2558:
781:
4041:
3970:
909:
2687:
2329:
422:
304:
4103:
4090:
2922:
2292:
2255:
2243:
2220:
serves as a basis of the vector space and algebra multiplication extends group multiplication.
615:
2277:
2207:
1989:
993:
453:
403:
391:
4003:
2766:
2740:
4123:
3892:
3860:
3122:
2983:
2525:
2407:
2263:
2199:
2144:
1870:
1781:
1123:
461:
255:
250:
241:
198:
140:
8:
4118:
4108:
3879:
3837:
3440:. Taking into account the definition of an identity element, it is sufficient to specify
3085:{\displaystyle \mathbf {e} _{i}\mathbf {e} _{j}=\sum _{k=1}^{n}c_{i,j,k}\mathbf {e} _{k}}
2904:
2427:
2316:
2251:
2217:
2182:
2156:
2097:
1769:
1283:
1017:
538:
418:
407:
395:
309:
299:
150:
50:
42:
33:
824:
4339:
4266:
4219:
4153:
4143:
3900:
3885:
2549:
2457:
515:
430:
115:
106:
64:
3262:
Classification of low-dimensional unital associative algebras over the complex numbers
4370:
4347:
4315:
4290:
4270:
4229:
4199:
4133:
3155:
2464:
2296:
2273:
1240:
399:
4258:
4148:
4138:
4113:
3997:
3845:
2432:
2152:
1033:
526:
480:
135:
2148:
160:
4284:
4191:
3828:
The fourth of these algebras is non-commutative, and the others are commutative.
2400:
2284:
2236:
1787:
1564:
488:
465:
457:
449:
227:
221:
208:
188:
179:
145:
82:
1765:. It follows from condition (3) that every left or right ideal is a subalgebra.
3204:
2677:
is a ring homomorphism that commutes with the scalar multiplication defined by
2422:
2304:
1931:
1143:
580:
269:
3351:{\displaystyle \textstyle 1\cdot 1=1\,,\quad 1\cdot a=a\,,\quad a\cdot 1=a\,.}
483:
with respect to the multiplication. The ring of real square matrices of order
4364:
2288:
2267:
1911:
1907:
1275:
952:
728:
692:
573:
568:
534:
437:
155:
120:
77:
4128:
3267:
3242:
is only a commutative ring and not a field, then the same process works if
2552:. This definition is equivalent to that above, with scalar multiplication
2359:
2300:
1325:
1025:
388:
384:
329:
260:
94:
1239:
These three axioms are another way of saying that the binary operation is
3247:
2948:
2417:
2133:
1529:
1290:
496:
445:
434:
414:
372:
319:
314:
203:
193:
167:
4223:
4262:
3934:
2312:
2308:
2228:
2213:
1351:
902:
789:
69:
2169:
over a field. The construction of the unital zero algebra over a free
2541:
1533:
533:. Algebras are not to be confused with vector spaces equipped with a
324:
130:
87:
55:
2937:, i.e., so the resulting multiplication satisfies the algebra laws.
410:
and satisfying the axioms implied by "vector space" and "bilinear".
2412:
125:
3896:
2090:, the unital zero algebra is the quotient of the polynomial ring
2986:. These structure coefficients determine the multiplication in
59:
4060:-module structure, since one can take the unique homomorphism
456:
since matrix multiplication is associative. Three-dimensional
3741:{\displaystyle \textstyle aa=1\,,\quad bb=0\,,\quad ab=-ba=b}
2945:
413:
The multiplication operation in an algebra may or may not be
4338:
4173:
3817:{\displaystyle \textstyle aa=0\,,\quad bb=0\,,\quad ab=ba=0}
3662:{\displaystyle \textstyle aa=b\,,\quad bb=0\,,\quad ab=ba=0}
3586:{\displaystyle \textstyle aa=a\,,\quad bb=0\,,\quad ab=ba=0}
3510:{\displaystyle \textstyle aa=a\,,\quad bb=b\,,\quad ab=ba=0}
2909:
For algebras over a field, the bilinear multiplication from
1753:
is a subset that is both a left and a right ideal. The term
402:
together with operations of multiplication and addition and
3899:. A classical example of an algebra over its center is the
1805:, then there is a natural way to construct an algebra over
1729:
is closed under left multiplication by arbitrary elements).
4312:
Introduction to
Commutative algebra and algebraic geometry
3840:, it is common to consider the more general concept of an
444:
is an example of an associative algebra over the field of
2140:-algebra built from a one dimensional real vector space.
2004:, and defining the product of every pair of elements of
675:{\displaystyle \left(a+ib\right)\cdot \left(c+id\right)}
530:
3855:. The only part of the definition that changes is that
2536:
is a ring homomorphism, then one must have either that
1768:
This definition is different from the definition of an
3831:
3757:
3678:
3602:
3526:
3450:
3404:
3371:
3287:
3277:. According to the definition of an identity element,
2858:
2800:
1440:
is said to be a unital homomorphism. The space of all
4066:
4044:
4006:
3973:
3943:
3909:
3756:
3677:
3601:
3525:
3449:
3403:
3370:
3286:
3141:. Thus, the structure coefficients are often written
2999:
2798:
2769:
2743:
2690:
2599:
2561:
2475:
2367:
1873:
1819:
1512:{\displaystyle \mathbf {Hom} _{K{\text{-alg}}}(A,B).}
1467:
962:
912:
833:
799:
737:
703:
624:
590:
4342:; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004).
2132:
An example of unital zero algebra is the algebra of
4038:-algebra structure. So a ring comes with a natural
4080:
4052:
4018:
3981:
3959:
3925:
3873:
3816:
3740:
3661:
3585:
3509:
3420:
3387:
3350:
3084:
2944:, any finite-dimensional algebra can be specified
2921:is completely determined by the multiplication of
2887:
2789:. In other words, the following diagram commutes:
2781:
2755:
2726:
2638:
2579:
2505:
2385:
2291:. Here the algebra multiplication is given by the
1886:
1851:
1511:
1104:if the following identities hold for all elements
983:
929:
884:
814:
767:
718:
674:
605:
1901:
4362:
3960:{\displaystyle \mathbb {R} \times \mathbb {R} }
3926:{\displaystyle \mathbb {H} \times \mathbb {H} }
3154:, and their defining rule is written using the
2396:Examples detailed in the main article include:
2307:. If an involution is given as well, we obtain
1266:. The binary operation is often referred to as
1289:When a binary operation on a vector space is
352:
27:Vector space equipped with a bilinear product
2299:; many of them are defined on an underlying
885:{\displaystyle (a+{\vec {v}})(b+{\vec {w}})}
544:
1422:are unital, then a homomorphism satisfying
768:{\displaystyle {\vec {a}}\times {\vec {b}}}
4225:An Introduction to Nonassociative Algebras
4174:Hazewinkel, Gubareni & Kirichenko 2004
3891:is always an associative algebra over its
2323:
2295:of operators. These algebras also carry a
1678:, we have the following three statements.
359:
345:
4282:
4068:
4046:
3975:
3953:
3945:
3919:
3911:
3787:
3770:
3708:
3691:
3632:
3615:
3556:
3539:
3480:
3463:
3343:
3323:
3303:
2898:
2830:
2187:Examples of associative algebras include
1544:
965:
914:
802:
706:
593:
4196:Approximation of Vector Valued Functions
3203:If you apply this to vectors written in
1301:
984:{\displaystyle \mathbb {R} ^{n\times n}}
4218:
2639:{\displaystyle (k,a)\mapsto \eta (k)a.}
1775:
1711:is closed under scalar multiplication),
529:leads to the more general notion of an
14:
4363:
4212:
4189:
3836:In some areas of mathematics, such as
2506:{\displaystyle \eta \colon K\to Z(A),}
2386:{\displaystyle A\times A\rightarrow A}
2266:defined on some fixed open set in the
2206:. Here the multiplication is ordinary
2176:
1992:of a field (or more generally a ring)
549:
520:unital associative commutative algebra
4248:
2443:
1852:{\displaystyle V_{F}:=V\otimes _{K}F}
1278:, although some authors use the term
4309:
4000:, then any unital ring homomorphism
525:Replacing the field of scalars by a
4034:, and this is what is known as the
3832:Generalization: algebra over a ring
2202:over a field (or commutative ring)
24:
2649:Given two such associative unital
25:
4382:
4081:{\displaystyle \mathbb {Z} \to A}
2448:The definition of an associative
2406:with multiplication given by the
1917:
1797:, which is to say a bigger field
1296:
1122:, and all elements (often called
487:forms a unital algebra since the
460:with multiplication given by the
3072:
3014:
3002:
1476:
1473:
1470:
815:{\displaystyle \mathbb {R} ^{4}}
719:{\displaystyle \mathbb {R} ^{3}}
606:{\displaystyle \mathbb {R} ^{2}}
3874:Associative algebras over rings
3791:
3774:
3712:
3695:
3636:
3619:
3560:
3543:
3484:
3467:
3421:{\displaystyle \textstyle aa=0}
3388:{\displaystyle \textstyle aa=1}
3327:
3307:
2929:. Conversely, once a basis for
1957:
1446:-algebra homomorphisms between
514:, or in some subjects such as
4303:
4276:
4242:
4183:
4166:
4072:
4010:
3748: for the fourth algebra,
3593: for the second algebra,
3428: for the second algebra.
2868:
2843:
2825:
2721:
2715:
2703:
2694:
2627:
2621:
2615:
2612:
2600:
2580:{\displaystyle K\times A\to A}
2571:
2497:
2491:
2485:
2377:
1902:Kinds of algebras and examples
1503:
1491:
924:
918:
879:
873:
858:
855:
849:
834:
759:
744:
13:
1:
4332:
3937:. The center of that ring is
3824: for the fifth algebra.
3669: for the third algebra,
3517: for the first algebra,
3395: for the first algebra,
2270:. These are also commutative.
2147:. For example, the theory of
1650:is a left ideal if for every
1580:is a subalgebra if for every
1007:
499:that is also a vector space.
4053:{\displaystyle \mathbb {Z} }
3982:{\displaystyle \mathbb {R} }
3933:, the direct product of two
2250:-algebra of all real-valued
1211:Compatibility with scalars:
1032:equipped with an additional
930:{\displaystyle \mathbb {R} }
417:, leading to the notions of
7:
4344:Algebras, rings and modules
4097:
3992:In commutative algebra, if
3129:indices, and transform via
2727:{\displaystyle f(ka)=kf(a)}
1761:being a linear subspace of
1745:, then this would define a
1559:of an algebra over a field
1247:is sometimes also called a
10:
4387:
4251:Monatshefte für Mathematik
3901:split-biquaternion algebra
3877:
3133:, while upper indices are
2902:
2438:Power-associative algebras
2327:
2180:
2000:-vector space (or module)
1779:
1733:If (3) were replaced with
1697:is closed under addition),
1551:Substructure (mathematics)
1548:
616:product of complex numbers
512:unital associative algebra
502:Many authors use the term
4346:. Vol. 1. Springer.
4190:Prolla, João B. (2011) .
3903:, which is isomorphic to
2681:, which one may write as
2254:functions defined on the
1458:is frequently written as
940:polynomial multiplication
545:Definition and motivation
394:. Thus, an algebra is an
4198:. Elsevier. p. 65.
4159:
2990:via the following rule:
2303:, which turns them into
2223:the commutative algebra
2008:to be zero. That is, if
1062:are any two elements of
423:non-associative algebras
379:(often simply called an
4286:Commutative Ring Theory
2336:non-associative algebra
2330:Non-associative algebra
2324:Non-associative algebra
2315:. These are studied in
1962:An algebra is called a
1541:-algebra homomorphism.
4283:Matsumura, H. (1989).
4104:Algebra over an operad
4091:Field with one element
4082:
4054:
4020:
4019:{\displaystyle R\to A}
3983:
3961:
3927:
3818:
3742:
3663:
3587:
3511:
3422:
3389:
3361:It remains to specify
3352:
3086:
3047:
2964:structure coefficients
2940:Thus, given the field
2899:Structure coefficients
2889:
2783:
2782:{\displaystyle a\in A}
2757:
2756:{\displaystyle k\in K}
2728:
2665:-algebra homomorphism
2640:
2581:
2507:
2387:
2278:partially ordered sets
1888:
1853:
1809:from any algebra over
1545:Subalgebras and ideals
1513:
985:
931:
886:
816:
769:
720:
676:
607:
4083:
4055:
4030:-module structure on
4021:
3984:
3962:
3928:
3819:
3743:
3664:
3588:
3512:
3423:
3390:
3353:
3137:, transforming under
3087:
3027:
2890:
2784:
2758:
2729:
2641:
2582:
2508:
2388:
2276:are built on certain
2264:holomorphic functions
2208:matrix multiplication
2159:in a polynomial ring
1990:direct sum of modules
1889:
1887:{\displaystyle A_{F}}
1854:
1514:
1302:Algebra homomorphisms
1179:Left distributivity:
994:matrix multiplication
986:
932:
887:
817:
770:
721:
677:
608:
454:matrix multiplication
404:scalar multiplication
4310:Kunz, Ernst (1985).
4124:Differential algebra
4064:
4042:
4004:
3971:
3941:
3907:
3859:is assumed to be an
3754:
3675:
3599:
3523:
3447:
3401:
3368:
3284:
3207:, then this becomes
3123:mathematical physics
2997:
2796:
2767:
2741:
2688:
2597:
2559:
2473:
2428:Alternative algebras
2408:vector cross product
2365:
2340:distributive algebra
1934:or identity element
1871:
1817:
1782:Extension of scalars
1776:Extension of scalars
1465:
1342:algebra homomorphism
960:
910:
831:
797:
735:
701:
622:
588:
539:inner product spaces
462:vector cross product
419:associative algebras
377:algebra over a field
256:Group with operators
199:Complemented lattice
34:Algebraic structures
18:Algebra homomorphism
4340:Hazewinkel, Michiel
4220:Schafer, Richard D.
4119:Composition algebra
4109:Alternative algebra
3935:quaternion algebras
3880:Associative algebra
3851:replaces the field
3842:algebra over a ring
3838:commutative algebra
2905:Structure constants
2317:functional analysis
2287:, for example on a
2191:the algebra of all
2183:Associative algebra
2177:Associative algebra
1986:unital zero algebra
1894:is an algebra over
1863:is an algebra over
1284:associative algebra
1080:that is called the
555:
550:Motivating examples
531:algebra over a ring
508:associative algebra
425:. Given an integer
396:algebraic structure
310:Composition algebra
70:Quasigroup and loop
4263:10.1007/BF01692479
4144:Mutation (algebra)
4078:
4050:
4016:
3979:
3957:
3923:
3814:
3813:
3738:
3737:
3659:
3658:
3583:
3582:
3507:
3506:
3418:
3417:
3385:
3384:
3348:
3347:
3082:
2959:), and specifying
2885:
2883:
2873:
2779:
2753:
2724:
2636:
2577:
2503:
2444:Algebras and rings
2383:
2274:Incidence algebras
2151:was introduced by
2136:, the unital zero
1884:
1849:
1509:
1243:. An algebra over
1050:, denoted here by
981:
927:
882:
812:
765:
716:
672:
603:
565:bilinear operator
554:
516:algebraic geometry
4296:978-0-521-36764-6
4205:978-0-08-087136-3
4180:Proposition 1.1.1
4134:Geometric algebra
3156:Einstein notation
2465:ring homomorphism
2433:Flexible algebras
2100:generated by the
1984:One may define a
1487:
1076:is an element of
1005:
1004:
876:
852:
762:
747:
406:by elements of a
369:
368:
16:(Redirected from
4378:
4357:
4326:
4325:
4307:
4301:
4300:
4280:
4274:
4273:
4246:
4240:
4239:
4216:
4210:
4209:
4187:
4181:
4170:
4149:Operator algebra
4139:Max-plus algebra
4114:Clifford algebra
4087:
4085:
4084:
4079:
4071:
4059:
4057:
4056:
4051:
4049:
4025:
4023:
4022:
4017:
3998:commutative ring
3988:
3986:
3985:
3980:
3978:
3966:
3964:
3963:
3958:
3956:
3948:
3932:
3930:
3929:
3924:
3922:
3914:
3870:-vector space).
3846:commutative ring
3823:
3821:
3820:
3815:
3747:
3745:
3744:
3739:
3668:
3666:
3665:
3660:
3592:
3590:
3589:
3584:
3516:
3514:
3513:
3508:
3427:
3425:
3424:
3419:
3394:
3392:
3391:
3386:
3357:
3355:
3354:
3349:
3111:form a basis of
3091:
3089:
3088:
3083:
3081:
3080:
3075:
3069:
3068:
3046:
3041:
3023:
3022:
3017:
3011:
3010:
3005:
2894:
2892:
2891:
2886:
2884:
2876:
2874:
2855:
2847:
2842:
2841:
2824:
2823:
2813:
2810:
2809:
2803:
2802:
2788:
2786:
2785:
2780:
2762:
2760:
2759:
2754:
2733:
2731:
2730:
2725:
2645:
2643:
2642:
2637:
2586:
2584:
2583:
2578:
2512:
2510:
2509:
2504:
2463:together with a
2392:
2390:
2389:
2384:
2354:equipped with a
2285:linear operators
2262:-algebra of all
2168:
2153:Bruno Buchberger
2128:
2095:
2085:
2067:
2035:
2021:
1972:
1954:in the algebra.
1893:
1891:
1890:
1885:
1883:
1882:
1858:
1856:
1855:
1850:
1845:
1844:
1829:
1828:
1540:
1527:
1518:
1516:
1515:
1510:
1490:
1489:
1488:
1485:
1479:
1457:
1451:
1445:
1439:
1421:
1415:
1409:
1403:
1393:
1366:
1349:
1339:
1333:
1323:
1317:
1311:
1273:
1265:
1257:
1251:
1246:
1234:
1208:
1176:
1137:
1133:
1129:
1121:
1117:
1103:
1095:
1091:
1087:
1079:
1075:
1065:
1061:
1057:
1053:
1049:
1045:
1034:binary operation
1031:
1023:
1015:
990:
988:
987:
982:
980:
979:
968:
936:
934:
933:
928:
917:
891:
889:
888:
883:
878:
877:
869:
854:
853:
845:
825:Hamilton product
821:
819:
818:
813:
811:
810:
805:
774:
772:
771:
766:
764:
763:
755:
749:
748:
740:
725:
723:
722:
717:
715:
714:
709:
681:
679:
678:
673:
671:
667:
646:
642:
612:
610:
609:
604:
602:
601:
596:
556:
553:
527:commutative ring
481:identity element
398:consisting of a
387:equipped with a
361:
354:
347:
136:Commutative ring
65:Rack and quandle
30:
29:
21:
4386:
4385:
4381:
4380:
4379:
4377:
4376:
4375:
4361:
4360:
4354:
4335:
4330:
4329:
4322:
4308:
4304:
4297:
4281:
4277:
4247:
4243:
4236:
4217:
4213:
4206:
4188:
4184:
4171:
4167:
4162:
4154:Zariski's lemma
4100:
4067:
4065:
4062:
4061:
4045:
4043:
4040:
4039:
4005:
4002:
4001:
3974:
3972:
3969:
3968:
3952:
3944:
3942:
3939:
3938:
3918:
3910:
3908:
3905:
3904:
3895:, and over the
3882:
3876:
3834:
3755:
3752:
3751:
3676:
3673:
3672:
3600:
3597:
3596:
3524:
3521:
3520:
3448:
3445:
3444:
3402:
3399:
3398:
3369:
3366:
3365:
3285:
3282:
3281:
3264:
3227:
3198:
3190:
3177:
3169:
3153:
3110:
3101:
3076:
3071:
3070:
3052:
3048:
3042:
3031:
3018:
3013:
3012:
3006:
3001:
3000:
2998:
2995:
2994:
2981:
2907:
2901:
2882:
2881:
2875:
2872:
2871:
2865:
2864:
2857:
2854:
2848:
2846:
2837:
2833:
2831:
2828:
2819:
2815:
2811:
2808:
2799:
2797:
2794:
2793:
2768:
2765:
2764:
2742:
2739:
2738:
2689:
2686:
2685:
2598:
2595:
2594:
2560:
2557:
2556:
2474:
2471:
2470:
2446:
2423:Jordan algebras
2401:Euclidean space
2366:
2363:
2362:
2342:) over a field
2332:
2326:
2305:Banach algebras
2237:polynomial ring
2185:
2179:
2160:
2118:
2117:for every pair
2116:
2108:
2091:
2084:
2075:
2069:
2037:
2023:
2009:
1967:
1960:
1920:
1904:
1878:
1874:
1872:
1869:
1868:
1840:
1836:
1824:
1820:
1818:
1815:
1814:
1788:field extension
1784:
1778:
1770:ideal of a ring
1751:two-sided ideal
1600:, we have that
1565:linear subspace
1553:
1547:
1536:
1523:
1484:
1480:
1469:
1468:
1466:
1463:
1462:
1453:
1447:
1441:
1438:
1432:
1423:
1417:
1411:
1405:
1395:
1368:
1354:
1345:
1335:
1329:
1319:
1313:
1307:
1304:
1299:
1282:to refer to an
1271:
1263:
1255:
1249:
1244:
1212:
1180:
1147:
1135:
1131:
1127:
1119:
1105:
1101:
1093:
1089:
1085:
1077:
1067:
1063:
1059:
1055:
1051:
1047:
1037:
1029:
1021:
1013:
1010:
969:
964:
963:
961:
958:
957:
953:square matrices
913:
911:
908:
907:
868:
867:
844:
843:
832:
829:
828:
827:
806:
801:
800:
798:
795:
794:
782:anticommutative
754:
753:
739:
738:
736:
733:
732:
731:
710:
705:
704:
702:
699:
698:
654:
650:
629:
625:
623:
620:
619:
618:
597:
592:
591:
589:
586:
585:
581:complex numbers
552:
547:
489:identity matrix
466:Jacobi identity
458:Euclidean space
450:matrix addition
438:square matrices
365:
336:
335:
334:
305:Non-associative
287:
276:
275:
265:
245:
234:
233:
222:Map of lattices
218:
214:Boolean algebra
209:Heyting algebra
183:
172:
171:
165:
146:Integral domain
110:
99:
98:
92:
46:
28:
23:
22:
15:
12:
11:
5:
4384:
4374:
4373:
4359:
4358:
4352:
4334:
4331:
4328:
4327:
4320:
4314:. Birkhauser.
4302:
4295:
4275:
4257:(1): 283–354,
4241:
4234:
4211:
4204:
4182:
4164:
4163:
4161:
4158:
4157:
4156:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4111:
4106:
4099:
4096:
4077:
4074:
4070:
4048:
4015:
4012:
4009:
3977:
3955:
3951:
3947:
3921:
3917:
3913:
3878:Main article:
3875:
3872:
3866:(instead of a
3833:
3830:
3826:
3825:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3790:
3786:
3783:
3780:
3777:
3773:
3769:
3766:
3763:
3760:
3749:
3736:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3711:
3707:
3704:
3701:
3698:
3694:
3690:
3687:
3684:
3681:
3670:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3635:
3631:
3628:
3625:
3622:
3618:
3614:
3611:
3608:
3605:
3594:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3559:
3555:
3552:
3549:
3546:
3542:
3538:
3535:
3532:
3529:
3518:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3483:
3479:
3476:
3473:
3470:
3466:
3462:
3459:
3456:
3453:
3430:
3429:
3416:
3413:
3410:
3407:
3396:
3383:
3380:
3377:
3374:
3359:
3358:
3346:
3342:
3339:
3336:
3333:
3330:
3326:
3322:
3319:
3316:
3313:
3310:
3306:
3302:
3299:
3296:
3293:
3290:
3263:
3260:
3236:
3235:
3219:
3205:index notation
3201:
3200:
3194:
3182:
3173:
3165:
3145:
3106:
3099:
3093:
3092:
3079:
3074:
3067:
3064:
3061:
3058:
3055:
3051:
3045:
3040:
3037:
3034:
3030:
3026:
3021:
3016:
3009:
3004:
2969:
2951:by giving its
2903:Main article:
2900:
2897:
2896:
2895:
2880:
2877:
2870:
2867:
2866:
2863:
2860:
2859:
2856:
2853:
2850:
2849:
2845:
2840:
2836:
2832:
2829:
2827:
2822:
2818:
2814:
2812:
2807:
2804:
2801:
2778:
2775:
2772:
2752:
2749:
2746:
2735:
2734:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2647:
2646:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2588:
2587:
2576:
2573:
2570:
2567:
2564:
2514:
2513:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2445:
2442:
2441:
2440:
2435:
2430:
2425:
2420:
2415:
2410:
2382:
2379:
2376:
2373:
2370:
2350:-vector space
2328:Main article:
2325:
2322:
2321:
2320:
2281:
2271:
2246:, such as the
2240:
2221:
2214:group algebras
2211:
2181:Main article:
2178:
2175:
2112:
2104:
2086:is a basis of
2080:
2073:
1988:by taking the
1959:
1956:
1922:An algebra is
1919:
1918:Unital algebra
1916:
1903:
1900:
1881:
1877:
1848:
1843:
1839:
1835:
1832:
1827:
1823:
1801:that contains
1780:Main article:
1777:
1774:
1731:
1730:
1712:
1698:
1549:Main article:
1546:
1543:
1520:
1519:
1508:
1505:
1502:
1499:
1496:
1493:
1483:
1478:
1475:
1472:
1434:
1428:
1303:
1300:
1298:
1297:Basic concepts
1295:
1268:multiplication
1258:is called the
1237:
1236:
1209:
1177:
1144:distributivity
1009:
1006:
1003:
1002:
999:
996:
991:
978:
975:
972:
967:
955:
949:
948:
945:
942:
937:
926:
923:
920:
916:
905:
899:
898:
895:
892:
881:
875:
872:
866:
863:
860:
857:
851:
848:
842:
839:
836:
822:
809:
804:
792:
786:
785:
778:
775:
761:
758:
752:
746:
743:
726:
713:
708:
696:
695:of 3D vectors
689:
688:
685:
682:
670:
666:
663:
660:
657:
653:
649:
645:
641:
638:
635:
632:
628:
613:
600:
595:
583:
577:
576:
571:
566:
563:
560:
551:
548:
546:
543:
471:An algebra is
367:
366:
364:
363:
356:
349:
341:
338:
337:
333:
332:
327:
322:
317:
312:
307:
302:
296:
295:
294:
288:
282:
281:
278:
277:
274:
273:
270:Linear algebra
264:
263:
258:
253:
247:
246:
240:
239:
236:
235:
232:
231:
228:Lattice theory
224:
217:
216:
211:
206:
201:
196:
191:
185:
184:
178:
177:
174:
173:
164:
163:
158:
153:
148:
143:
138:
133:
128:
123:
118:
112:
111:
105:
104:
101:
100:
91:
90:
85:
80:
74:
73:
72:
67:
62:
53:
47:
41:
40:
37:
36:
26:
9:
6:
4:
3:
2:
4383:
4372:
4369:
4368:
4366:
4355:
4353:1-4020-2690-0
4349:
4345:
4341:
4337:
4336:
4323:
4321:0-8176-3065-1
4317:
4313:
4306:
4298:
4292:
4288:
4287:
4279:
4272:
4268:
4264:
4260:
4256:
4252:
4245:
4237:
4235:0-486-68813-5
4231:
4227:
4226:
4221:
4215:
4207:
4201:
4197:
4193:
4186:
4179:
4175:
4169:
4165:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4101:
4095:
4093:
4092:
4075:
4037:
4033:
4029:
4013:
4007:
3999:
3995:
3990:
3949:
3936:
3915:
3902:
3898:
3894:
3890:
3887:
3881:
3871:
3869:
3865:
3863:
3858:
3854:
3850:
3847:
3843:
3839:
3829:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3788:
3784:
3781:
3778:
3775:
3771:
3767:
3764:
3761:
3758:
3750:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3709:
3705:
3702:
3699:
3696:
3692:
3688:
3685:
3682:
3679:
3671:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3633:
3629:
3626:
3623:
3620:
3616:
3612:
3609:
3606:
3603:
3595:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3557:
3553:
3550:
3547:
3544:
3540:
3536:
3533:
3530:
3527:
3519:
3503:
3500:
3497:
3494:
3491:
3488:
3485:
3481:
3477:
3474:
3471:
3468:
3464:
3460:
3457:
3454:
3451:
3443:
3442:
3441:
3439:
3435:
3414:
3411:
3408:
3405:
3397:
3381:
3378:
3375:
3372:
3364:
3363:
3362:
3344:
3340:
3337:
3334:
3331:
3328:
3324:
3320:
3317:
3314:
3311:
3308:
3304:
3300:
3297:
3294:
3291:
3288:
3280:
3279:
3278:
3276:
3271:
3269:
3259:
3257:
3253:
3249:
3245:
3241:
3233:
3230:
3226:
3222:
3218:
3214:
3210:
3209:
3208:
3206:
3197:
3193:
3189:
3185:
3181:
3176:
3172:
3168:
3164:
3161:
3160:
3159:
3157:
3152:
3148:
3144:
3140:
3136:
3135:contravariant
3132:
3128:
3124:
3119:
3116:
3114:
3109:
3105:
3098:
3077:
3065:
3062:
3059:
3056:
3053:
3049:
3043:
3038:
3035:
3032:
3028:
3024:
3019:
3007:
2993:
2992:
2991:
2989:
2985:
2980:
2976:
2972:
2968:
2965:
2962:
2958:
2954:
2950:
2947:
2943:
2938:
2936:
2932:
2928:
2924:
2920:
2916:
2912:
2906:
2878:
2861:
2851:
2838:
2834:
2820:
2816:
2805:
2792:
2791:
2790:
2776:
2773:
2770:
2750:
2747:
2744:
2718:
2712:
2709:
2706:
2700:
2697:
2691:
2684:
2683:
2682:
2680:
2676:
2672:
2668:
2664:
2660:
2656:
2652:
2633:
2630:
2624:
2618:
2609:
2606:
2603:
2593:
2592:
2591:
2574:
2568:
2565:
2562:
2555:
2554:
2553:
2551:
2547:
2543:
2539:
2535:
2531:
2527:
2523:
2519:
2500:
2494:
2488:
2482:
2479:
2476:
2469:
2468:
2467:
2466:
2462:
2459:
2455:
2451:
2439:
2436:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2409:
2405:
2402:
2399:
2398:
2397:
2394:
2380:
2374:
2371:
2368:
2361:
2357:
2353:
2349:
2345:
2341:
2337:
2331:
2318:
2314:
2310:
2306:
2302:
2298:
2294:
2290:
2289:Hilbert space
2286:
2282:
2279:
2275:
2272:
2269:
2268:complex plane
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2238:
2234:
2230:
2226:
2222:
2219:
2215:
2212:
2209:
2205:
2201:
2198:
2194:
2190:
2189:
2188:
2184:
2174:
2172:
2167:
2163:
2158:
2154:
2150:
2149:Gröbner bases
2146:
2141:
2139:
2135:
2130:
2126:
2122:
2115:
2111:
2107:
2103:
2099:
2094:
2089:
2083:
2079:
2072:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2034:
2030:
2026:
2020:
2016:
2012:
2007:
2003:
1999:
1995:
1991:
1987:
1982:
1980:
1976:
1970:
1965:
1955:
1953:
1949:
1945:
1941:
1937:
1933:
1929:
1925:
1915:
1913:
1912:associativity
1909:
1908:commutativity
1899:
1897:
1879:
1875:
1866:
1862:
1846:
1841:
1837:
1833:
1830:
1825:
1821:
1812:
1808:
1804:
1800:
1796:
1792:
1789:
1786:If we have a
1783:
1773:
1771:
1766:
1764:
1760:
1756:
1752:
1748:
1744:
1740:
1736:
1728:
1724:
1720:
1716:
1713:
1710:
1706:
1702:
1699:
1696:
1692:
1688:
1684:
1681:
1680:
1679:
1677:
1673:
1669:
1665:
1661:
1657:
1653:
1649:
1645:
1641:
1637:
1633:
1628:
1625:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1579:
1575:
1571:
1566:
1562:
1558:
1552:
1542:
1539:
1535:
1531:
1526:
1506:
1500:
1497:
1494:
1481:
1461:
1460:
1459:
1456:
1450:
1444:
1437:
1431:
1426:
1420:
1414:
1408:
1402:
1398:
1391:
1387:
1383:
1379:
1375:
1371:
1365:
1361:
1357:
1353:
1348:
1343:
1338:
1334:-algebras or
1332:
1327:
1322:
1316:
1310:
1294:
1292:
1287:
1285:
1281:
1277:
1269:
1261:
1253:
1242:
1232:
1228:
1224:
1220:
1216:
1210:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1178:
1175:
1171:
1167:
1163:
1159:
1155:
1151:
1145:
1141:
1140:
1139:
1125:
1116:
1112:
1108:
1099:
1083:
1074:
1070:
1054:(that is, if
1044:
1040:
1035:
1027:
1019:
1000:
997:
995:
992:
976:
973:
970:
956:
954:
951:
950:
946:
943:
941:
938:
921:
906:
904:
901:
900:
896:
893:
870:
864:
861:
846:
840:
837:
826:
823:
807:
793:
791:
788:
787:
783:
779:
776:
756:
750:
741:
730:
729:cross product
727:
711:
697:
694:
693:cross product
691:
690:
686:
683:
668:
664:
661:
658:
655:
651:
647:
643:
639:
636:
633:
630:
626:
617:
614:
598:
584:
582:
579:
578:
575:
574:commutativity
572:
570:
569:associativity
567:
564:
562:vector space
561:
558:
557:
542:
540:
536:
535:bilinear form
532:
528:
523:
521:
517:
513:
509:
505:
500:
498:
497:(unital) ring
494:
490:
486:
482:
479:if it has an
478:
474:
469:
467:
463:
459:
455:
451:
447:
443:
439:
436:
432:
428:
424:
420:
416:
411:
409:
405:
401:
397:
393:
390:
386:
382:
378:
374:
362:
357:
355:
350:
348:
343:
342:
340:
339:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
297:
293:
290:
289:
285:
280:
279:
272:
271:
267:
266:
262:
259:
257:
254:
252:
249:
248:
243:
238:
237:
230:
229:
225:
223:
220:
219:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
181:
176:
175:
170:
169:
162:
159:
157:
156:Division ring
154:
152:
149:
147:
144:
142:
139:
137:
134:
132:
129:
127:
124:
122:
119:
117:
114:
113:
108:
103:
102:
97:
96:
89:
86:
84:
81:
79:
78:Abelian group
76:
75:
71:
68:
66:
63:
61:
57:
54:
52:
49:
48:
44:
39:
38:
35:
32:
31:
19:
4343:
4311:
4305:
4285:
4278:
4254:
4250:
4244:
4224:
4214:
4195:
4192:"Lemma 4.10"
4185:
4168:
4129:Free algebra
4089:
4035:
4031:
4027:
3993:
3991:
3888:
3883:
3867:
3861:
3856:
3852:
3848:
3841:
3835:
3827:
3437:
3433:
3431:
3360:
3274:
3272:
3268:Eduard Study
3265:
3255:
3251:
3243:
3239:
3237:
3231:
3228:
3224:
3220:
3216:
3212:
3202:
3195:
3191:
3187:
3183:
3179:
3174:
3170:
3166:
3162:
3150:
3146:
3142:
3139:pushforwards
3120:
3117:
3112:
3107:
3103:
3096:
3094:
2987:
2982:, which are
2978:
2974:
2970:
2966:
2963:
2960:
2956:
2941:
2939:
2934:
2930:
2926:
2925:elements of
2918:
2914:
2910:
2908:
2736:
2678:
2674:
2670:
2666:
2662:
2658:
2654:
2650:
2648:
2589:
2545:
2537:
2533:
2529:
2521:
2517:
2515:
2460:
2453:
2449:
2447:
2418:Lie algebras
2403:
2395:
2360:bilinear map
2355:
2351:
2347:
2343:
2339:
2335:
2333:
2301:Banach space
2283:algebras of
2259:
2247:
2242:algebras of
2232:
2224:
2203:
2196:
2192:
2186:
2170:
2165:
2161:
2142:
2137:
2134:dual numbers
2131:
2124:
2120:
2113:
2109:
2105:
2101:
2092:
2087:
2081:
2077:
2070:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2032:
2028:
2024:
2018:
2014:
2010:
2005:
2001:
1997:
1993:
1985:
1983:
1978:
1974:
1968:
1964:zero algebra
1963:
1961:
1958:Zero algebra
1951:
1947:
1943:
1939:
1935:
1930:if it has a
1927:
1923:
1921:
1905:
1895:
1864:
1860:
1810:
1806:
1802:
1798:
1794:
1790:
1785:
1767:
1762:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1732:
1726:
1722:
1718:
1714:
1708:
1704:
1700:
1694:
1690:
1686:
1682:
1675:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1643:
1639:
1635:
1631:
1629:
1626:
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1585:
1581:
1577:
1573:
1569:
1560:
1556:
1554:
1537:
1524:
1521:
1454:
1448:
1442:
1435:
1429:
1424:
1418:
1412:
1406:
1400:
1396:
1389:
1385:
1381:
1377:
1373:
1369:
1363:
1359:
1355:
1346:
1341:
1336:
1330:
1326:homomorphism
1320:
1314:
1308:
1305:
1288:
1279:
1267:
1259:
1248:
1238:
1230:
1226:
1222:
1218:
1214:
1205:
1201:
1197:
1193:
1189:
1185:
1181:
1173:
1169:
1165:
1161:
1157:
1153:
1149:
1114:
1110:
1106:
1097:
1081:
1072:
1068:
1042:
1038:
1026:vector space
1011:
524:
519:
511:
507:
503:
501:
492:
484:
476:
472:
470:
446:real numbers
441:
426:
412:
385:vector space
380:
376:
370:
330:Hopf algebra
291:
283:
268:
261:Vector space
226:
166:
95:Group theory
93:
58: /
4026:defines an
3248:free module
2949:isomorphism
2661:, a unital
2313:C*-algebras
2309:B*-algebras
2293:composition
2229:polynomials
1747:right ideal
1620:are all in
1530:isomorphism
1291:commutative
1276:associative
903:polynomials
790:quaternions
415:associative
373:mathematics
315:Lie algebra
300:Associative
204:Total order
194:Semilattice
168:Ring theory
4333:References
4176:, p.
3989:-algebra.
3844:, where a
2653:-algebras
2544:, or that
2252:continuous
2216:, where a
1632:left ideal
1557:subalgebra
1367:such that
1352:linear map
1312:-algebras
1260:base field
1020:, and let
1008:Definition
4271:121426669
4172:See also
4073:→
4011:→
3950:×
3916:×
3723:−
3332:⋅
3312:⋅
3292:⋅
3131:pullbacks
3127:covariant
3029:∑
2953:dimension
2869:⟶
2844:↘
2835:η
2826:↙
2817:η
2774:∈
2748:∈
2619:η
2616:↦
2590:given by
2572:→
2566:×
2550:injective
2542:zero ring
2524:) is the
2486:→
2480::
2477:η
2413:Octonions
2378:→
2372:×
2258:, or the
2244:functions
1838:⊗
1646:-algebra
1576:-algebra
1534:bijective
1528:-algebra
1092:). Then
974:×
874:→
850:→
760:→
751:×
745:→
648:⋅
491:of order
468:instead.
440:of order
325:Bialgebra
131:Near-ring
88:Lie group
56:Semigroup
4371:Algebras
4365:Category
4222:(1996).
4098:See also
3897:integers
2737:for all
2532:. Since
2297:topology
2256:interval
2200:matrices
1973:for all
1950:for all
1859:. So if
1394:for all
1252:-algebra
1241:bilinear
559:Algebra
506:to mean
389:bilinear
161:Lie ring
126:Semiring
3864:-module
2984:scalars
2913:×
2540:is the
2227:of all
2145:modules
2096:by the
2036:, then
1928:unitary
1867:, then
1280:algebra
1124:scalars
1098:algebra
1082:product
1066:, then
537:, like
504:algebra
477:unitary
392:product
383:) is a
381:algebra
292:Algebra
284:Algebra
189:Lattice
180:Lattice
4350:
4318:
4293:
4269:
4232:
4202:
3893:center
3095:where
2526:center
2516:where
2157:ideals
2076:, ...
1996:and a
1924:unital
1741:is in
1721:is in
1703:is in
1689:is in
1616:, and
1306:Given
1254:, and
1142:Right
1096:is an
473:unital
448:under
429:, the
320:Graded
251:Module
242:Module
141:Domain
60:Monoid
4267:S2CID
4160:Notes
3996:is a
3250:over
3246:is a
3102:,...,
2955:(say
2946:up to
2923:basis
2456:is a
2346:is a
2235:(see
2231:over
2218:group
2098:ideal
2068:. If
1938:with
1755:ideal
1642:of a
1634:of a
1572:of a
1563:is a
1532:is a
1433:) = 1
1410:. If
1344:is a
1221:) = (
1217:) · (
1100:over
1036:from
1028:over
1024:be a
1018:field
1016:be a
510:, or
408:field
375:, an
286:-like
244:-like
182:-like
151:Field
109:-like
83:Magma
51:Group
45:-like
43:Group
4348:ISBN
4316:ISBN
4291:ISBN
4230:ISBN
4200:ISBN
3886:ring
3436:and
3215:) =
2763:and
2657:and
2458:ring
2338:(or
2311:and
2195:-by-
2155:for
2054:) =
2022:and
1932:unit
1749:. A
1670:and
1654:and
1592:and
1486:-alg
1452:and
1416:and
1376:) =
1324:, a
1318:and
1192:) =
1156:) ·
1130:and
1088:and
1058:and
1012:Let
998:Yes
947:Yes
944:Yes
894:Yes
780:No (
687:Yes
684:Yes
452:and
435:real
431:ring
421:and
116:Ring
107:Ring
4259:doi
3238:If
3158:as
3121:In
2917:to
2548:is
2528:of
2058:+ (
2046:) (
1971:= 0
1966:if
1926:or
1910:or
1674:in
1666:in
1658:in
1596:in
1588:in
1404:in
1328:of
1270:in
1262:of
1225:) (
1184:· (
1134:in
1118:in
1084:of
1046:to
1001:No
897:No
777:No
475:or
433:of
400:set
371:In
121:Rng
4367::
4265:,
4253:,
4228:.
4194:.
3884:A
3270:.
3213:xy
3178:=
3115:.
2673:→
2669::
2334:A
2239:).
2164:=
2129:.
2123:,
2064:μu
2062:+
2060:λv
2056:λμ
2050:+
2042:+
2031:∈
2027:,
2017:∈
2013:,
1977:,
1969:uv
1948:xI
1946:=
1942:=
1940:Ix
1898:.
1831::=
1737:·
1717:·
1701:cx
1685:+
1662:,
1630:A
1624:.
1618:cx
1612:+
1608:,
1604:·
1584:,
1555:A
1522:A
1427:(1
1399:,
1384:)
1374:xy
1362:→
1358::
1286:.
1229:·
1223:ab
1219:by
1215:ax
1204:·
1200:+
1196:·
1188:+
1172:·
1168:+
1164:·
1160:=
1152:+
1146::
1138::
1126:)
1113:,
1109:,
1071:·
1041:×
784:)
522:.
518:,
4356:.
4324:.
4299:.
4261::
4255:1
4238:.
4208:.
4178:3
4076:A
4069:Z
4047:Z
4036:R
4032:A
4028:R
4014:A
4008:R
3994:A
3976:R
3954:R
3946:R
3920:H
3912:H
3889:A
3868:K
3862:R
3857:A
3853:K
3849:R
3811:0
3808:=
3805:a
3802:b
3799:=
3796:b
3793:a
3789:,
3785:0
3782:=
3779:b
3776:b
3772:,
3768:0
3765:=
3762:a
3759:a
3735:b
3732:=
3729:a
3726:b
3720:=
3717:b
3714:a
3710:,
3706:0
3703:=
3700:b
3697:b
3693:,
3689:1
3686:=
3683:a
3680:a
3656:0
3653:=
3650:a
3647:b
3644:=
3641:b
3638:a
3634:,
3630:0
3627:=
3624:b
3621:b
3617:,
3613:b
3610:=
3607:a
3604:a
3580:0
3577:=
3574:a
3571:b
3568:=
3565:b
3562:a
3558:,
3554:0
3551:=
3548:b
3545:b
3541:,
3537:a
3534:=
3531:a
3528:a
3504:0
3501:=
3498:a
3495:b
3492:=
3489:b
3486:a
3482:,
3478:b
3475:=
3472:b
3469:b
3465:,
3461:a
3458:=
3455:a
3452:a
3438:b
3434:a
3415:0
3412:=
3409:a
3406:a
3382:1
3379:=
3376:a
3373:a
3345:.
3341:a
3338:=
3335:1
3329:a
3325:,
3321:a
3318:=
3315:a
3309:1
3305:,
3301:1
3298:=
3295:1
3289:1
3275:a
3256:A
3252:K
3244:A
3240:K
3234:.
3232:y
3229:x
3225:j
3223:,
3221:i
3217:c
3211:(
3199:.
3196:k
3192:e
3188:j
3186:,
3184:i
3180:c
3175:j
3171:e
3167:i
3163:e
3151:j
3149:,
3147:i
3143:c
3113:A
3108:n
3104:e
3100:1
3097:e
3078:k
3073:e
3066:k
3063:,
3060:j
3057:,
3054:i
3050:c
3044:n
3039:1
3036:=
3033:k
3025:=
3020:j
3015:e
3008:i
3003:e
2988:A
2979:k
2977:,
2975:j
2973:,
2971:i
2967:c
2961:n
2957:n
2942:K
2935:A
2931:A
2927:A
2919:A
2915:A
2911:A
2879:B
2862:f
2852:A
2839:B
2821:A
2806:K
2777:A
2771:a
2751:K
2745:k
2722:)
2719:a
2716:(
2713:f
2710:k
2707:=
2704:)
2701:a
2698:k
2695:(
2692:f
2679:η
2675:B
2671:A
2667:f
2663:K
2659:B
2655:A
2651:K
2634:.
2631:a
2628:)
2625:k
2622:(
2613:)
2610:a
2607:,
2604:k
2601:(
2575:A
2569:A
2563:K
2546:η
2538:A
2534:η
2530:A
2522:A
2520:(
2518:Z
2501:,
2498:)
2495:A
2492:(
2489:Z
2483:K
2461:A
2454:K
2450:K
2404:R
2381:A
2375:A
2369:A
2358:-
2356:K
2352:A
2348:K
2344:K
2319:.
2280:.
2260:C
2248:R
2233:K
2225:K
2210:.
2204:K
2197:n
2193:n
2171:R
2166:K
2162:R
2138:R
2127:)
2125:j
2121:i
2119:(
2114:j
2110:E
2106:i
2102:E
2093:K
2088:V
2082:d
2078:e
2074:1
2071:e
2066:)
2052:v
2048:μ
2044:u
2040:λ
2038:(
2033:V
2029:v
2025:u
2019:K
2015:μ
2011:λ
2006:V
2002:V
1998:K
1994:K
1979:v
1975:u
1952:x
1944:x
1936:I
1896:F
1880:F
1876:A
1865:K
1861:A
1847:F
1842:K
1834:V
1826:F
1822:V
1811:K
1807:F
1803:K
1799:F
1795:K
1793:/
1791:F
1763:A
1759:L
1743:L
1739:z
1735:x
1727:L
1725:(
1723:L
1719:x
1715:z
1709:L
1707:(
1705:L
1695:L
1693:(
1691:L
1687:y
1683:x
1676:K
1672:c
1668:A
1664:z
1660:L
1656:y
1652:x
1648:A
1644:K
1640:L
1636:K
1622:L
1614:y
1610:x
1606:y
1602:x
1598:K
1594:c
1590:L
1586:y
1582:x
1578:A
1574:K
1570:L
1561:K
1538:K
1525:K
1507:.
1504:)
1501:B
1498:,
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1235:.
1233:)
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922:X
919:[
915:R
880:)
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656:c
652:(
644:)
640:b
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634:+
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599:2
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493:n
485:n
442:n
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360:e
353:t
346:v
20:)
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