1083:
2481:
1484:
1673:
871:
An algebraic structure may have more than one operation, and a homomorphism is required to preserve each operation. Thus a map that preserves only some of the operations is not a homomorphism of the structure, but only a homomorphism of the substructure obtained by considering only the preserved
6473:
2675:
1295:
1495:
3323:
674:
3613:
5229:
1995:
9233:"Ich will nach einem Vorschlage von Hrn. Prof. Klein statt der umständlichen und nicht immer ausreichenden Bezeichnungen: "holoedrisch, bezw. hemiedrisch u.s.w. isomorph" die Benennung "isomorph" auf den Fall des
9239:(Following a suggestion of Prof. Klein, instead of the cumbersome and not always satisfactory designations "holohedric, or hemihedric, etc. isomorphic", I will limit the denomination "isomorphic" to the case of a
1272:
9293:"*) Im Anschluss an einen von Hrn. Klein bei seinen neueren Vorlesungen eingeführten Brauch schreibe ich an Stelle der bisherigen Bezeichnung "meroedrischer Isomorphismus" die sinngemässere "Homomorphismus"."
7833:. The kernels of homomorphisms of a given type of algebraic structure are naturally equipped with some structure. This structure type of the kernels is the same as the considered structure, in the case of
6433:
8579:
8178:
5085:
4988:
2106:
9295:(Following a usage that has been introduced by Mr. Klein during his more recent lectures, I write in place of the earlier designation "merohedral isomorphism" the more logical "homomorphism".)
6418:
4304:
2524:
that is also a morphism. In the specific case of algebraic structures, the two definitions are equivalent, although they may differ for non-algebraic structures, which have an underlying set.
3840:
330:
2213:
2298:
1069:
8462:
803:
that preserves the group operation. This implies that the group homomorphism maps the identity element of the first group to the identity element of the second group, and maps the
2932:
981:
872:
operations. For example, a map between monoids that preserves the monoid operation and not the identity element, is not a monoid homomorphism, but only a semigroup homomorphism.
8488:
8276:
6857:
5773:
4367:
2609:
2173:
1479:{\displaystyle f(r+s)={\begin{pmatrix}r+s&0\\0&r+s\end{pmatrix}}={\begin{pmatrix}r&0\\0&r\end{pmatrix}}+{\begin{pmatrix}s&0\\0&s\end{pmatrix}}=f(r)+f(s)}
7145:
6204:
4093:
8790:
8689:
6633:
1668:{\displaystyle f(rs)={\begin{pmatrix}rs&0\\0&rs\end{pmatrix}}={\begin{pmatrix}r&0\\0&r\end{pmatrix}}{\begin{pmatrix}s&0\\0&s\end{pmatrix}}=f(r)\,f(s).}
8611:
8520:
8364:
8312:
5732:
4429:
729:
8237:
7015:
6939:
6523:
6376:
6344:
5396:
4262:
4230:
916:
5290:
8424:
8384:
8868:
8122:
8095:
6478:
the last implication is an equivalence for sets, vector spaces, modules, abelian groups, and groups; the first implication is an equivalence for sets and vector spaces.
1169:
7659:
7450:
7349:
6983:
6286:, which is a homomorphism of rings and of multiplicative semigroups. For both structures it is a monomorphism and a non-surjective epimorphism, but not an isomorphism.
5500:
4573:
3040:
2858:
1759:
8757:
8656:
7236:
6086:
4816:
3975:
3147:
3115:
2750:
2598:
2560:
2419:
448:
195:
8912:
8888:
7305:
7087:
7061:
6816:
6721:
7526:
6692:
5670:
5635:
4851:
3256:
3251:
2333:
258:
7807:
7546:
7369:
7259:
4897:
6750:
6595:
3176:
468:
7725:
6230:
5845:
5819:
4599:
4119:
3659:
9005:
8985:
8830:
8810:
8709:
8404:
8332:
7831:
7787:
7767:
7745:
7693:
7626:
7606:
7586:
7566:
7470:
7417:
7393:
7279:
7185:
7165:
7107:
7035:
6907:
6887:
6790:
6770:
6653:
6566:
6546:
6166:
6146:
6126:
6106:
6014:
5994:
5962:
5938:
5918:
5889:
5865:
5793:
5600:
5580:
5560:
5540:
5520:
5456:
5436:
5416:
5362:
5338:
5318:
5253:
5109:
5012:
4921:
4871:
4784:
4764:
4744:
4720:
4700:
4680:
4660:
4635:
4529:
4509:
4489:
4469:
4449:
4055:
4035:
4015:
3995:
3887:
3864:
3683:
3636:
3409:
3389:
3369:
3349:
3216:
3196:
3080:
3060:
2996:
2976:
2952:
2810:
2790:
2770:
2718:
2698:
2463:
2443:
2379:
2359:
2233:
2133:
1826:
1806:
1782:
1708:
1107:
807:
of an element of the first group to the inverse of the image of this element. Thus a semigroup homomorphism between groups is necessarily a group homomorphism.
762:
is required by the type of structure, the identity element of the first structure must be mapped to the corresponding identity element of the second structure.
749:
531:
511:
491:
413:
393:
373:
353:
238:
218:
539:
5972:, if the identities are not subject to conditions, that is if one works with a variety. Then the operations of the variety are well defined on the set of
4140:, that is algebraic structures for which operations and axioms (identities) are defined without any restriction (the fields do not form a variety, as the
3417:
5117:
6235:
A surjective homomorphism is always right cancelable, but the converse is not always true for algebraic structures. However, the two definitions of
1834:
4310:. For sets and vector spaces, every monomorphism is a split monomorphism, but this property does not hold for most common algebraic structures.
9291:
is based on the group of modulo n incongruent substitutions with rational whole coefficients of the determinant 1.) From footnote on p. 466:
6424:. For sets and vector spaces, every epimorphism is a split epimorphism, but this property does not hold for most common algebraic structures.
788:
that preserves the monoid operation and maps the identity element of the first monoid to that of the second monoid (the identity element is a
6293:
by a multiplicative set. Every localization is a ring epimorphism, which is not, in general, surjective. As localizations are fundamental in
4605:
whose objects are sets and arrows are maps between these sets. For example, an injective continuous map is a monomorphism in the category of
9571:
Developments in language theory : 10th international conference, DLT 2006, Santa
Barbara, CA, USA, June 26-29, 2006 : proceedings
7699:
of this operation suffices to characterize the equivalence relation. In this case, the quotient by the equivalence relation is denoted by
5964:
and the operations of the structure. Two such formulas are said equivalent if one may pass from one to the other by applying the axioms (
1686:
under the operation of multiplication, as do the nonzero real numbers. (Zero must be excluded from both groups since it does not have a
1209:
127:, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of
6468:{\displaystyle {\text{split epimorphism}}\implies {\text{epimorphism (surjective)}}\implies {\text{epimorphism (right cancelable)}};}
9625:
9397:
2006:
cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition:
9020:
operation, while Σ denotes the set of words formed from the alphabet Σ, including the empty word. Juxtaposition of terms denotes
9569:
Krieger, Dalia (2006). "On critical exponents in fixed points of non-erasing morphisms". In Ibarra, Oscar H.; Dang, Zhe (eds.).
8528:
8127:
6301:, this may explain why in these areas, the definition of epimorphisms as right cancelable homomorphisms is generally preferred.
8918:
and the identity element is the empty word. From this perspective, a language homomorphism is precisely a monoid homomorphism.
875:
The notation for the operations does not need to be the same in the source and the target of a homomorphism. For example, the
9668:
9610:
9578:
9496:
9371:
9329:
3696:, and the inverse of a bijective continuous map is not necessarily continuous. An isomorphism of topological spaces, called
93:) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German
5017:
9285:
auf die Gruppe der mod. n incongruenten
Substitutionen mit rationalen ganzen Coefficienten der Determinante 1 begründet."
4926:
2012:
6381:
4267:
105:
meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician
9642:
9540:
9519:
9414:
9277:
9248:
9251:[On the arithmetic character of the triangle functions belonging to the branch points (2,3,7) and (2,4,7)].
9228:
9199:
7877:, the notion of an algebraic structure is generalized to structures involving both operations and relations. Let
9200:"Die eindeutigen automorphen Formen vom Geschlecht Null, eine Revision und Erweiterung der Poincaré'schen Sätze"
3806:
270:
9460:
9249:"Ueber den arithmetischen Charakter der zu den Verzweigungen (2,3,7) und (2,4,7) gehörenden Dreiecksfunctionen"
2178:
9630:
9402:
9321:
2215:. Due to the different names of corresponding operations, the structure preservation properties satisfied by
146:, etc. (see below). Each of those can be defined in a way that may be generalized to any class of morphisms.
83:
3800:
Many groups that have received a name are automorphism groups of some algebraic structure. For example, the
2238:
1000:
8432:
6311:
4197:
2670:{\displaystyle f\circ g=\operatorname {Id} _{B}\qquad {\text{and}}\qquad g\circ f=\operatorname {Id} _{A}.}
9597:
9358:
5892:
4639:
4133:
2863:
927:
71:
9320:, American Mathematical Society Colloquium Publications, vol. 25 (3rd ed.), Providence, R.I.:
9171:
8467:
8242:
6821:
5737:
4331:
2138:
8937:
7112:
6171:
4060:
9202:[The unique automorphic forms of genus zero, a revision and extension of Poincaré's theorem].
8762:
8661:
6600:
4612:
For proving that, conversely, a left cancelable homomorphism is injective, it is useful to consider a
4148:
or as a property of the multiplication, which are, in both cases, defined only for nonzero elements).
8584:
8493:
8337:
8285:
5675:
4372:
841:; that is, a group homomorphism between vector spaces that preserves the abelian group structure and
682:
8183:
6988:
6912:
6496:
6349:
6317:
5369:
4235:
4203:
888:
5261:
3759:
986:
and is thus a homomorphism between these two groups. It is even an isomorphism (see below), as its
8409:
8369:
8846:
8100:
8073:
819:
159:
1132:
8932:
7635:
7425:
7310:
6944:
6861:
In the case of vector spaces, abelian groups and modules, the proof relies on the existence of
6290:
5996:
for this relation. It is straightforward to show that the resulting object is a free object on
5965:
5461:
4602:
4534:
4141:
3732:
3318:{\displaystyle f\circ g=\operatorname {Id} _{B}{\text{ and }}g\circ f=\operatorname {Id} _{A},}
3001:
2819:
1716:
1687:
770:
66:
8717:
8616:
7209:
6525:
be a homomorphism. We want to prove that if it is not surjective, it is not right cancelable.
6059:
4990:
which, as, a semigroup, is isomorphic to the additive semigroup of the positive integers; for
4789:
3948:
3120:
3088:
2723:
2571:
2533:
2392:
879:
form a group for addition, and the positive real numbers form a group for multiplication. The
421:
168:
8897:
8873:
7284:
7066:
7040:
6795:
6697:
5112:
3941:
3701:
1200:
842:
9483:
For a detailed discussion of relational homomorphisms and isomorphisms see
Section 17.3, in
7475:
6662:
5640:
5605:
4821:
4128:
are equivalent for all common algebraic structures. More precisely, they are equivalent for
3221:
2303:
1082:
243:
9339:
7842:
7792:
7531:
7354:
7244:
7239:
6252:
5969:
5297:
5087:
which, as, a monoid, is isomorphic to the additive monoid of the nonnegative integers; for
4876:
4184:
3901:
3801:
3763:
3747:
3717:
2422:
1192:
880:
864:
860:
853:
9652:
9470:
9424:
6726:
6571:
4200:
and thus it is itself a right inverse of that other homomorphism. That is, a homomorphism
3152:
453:
8:
9455:. Pure and Applied Mathematics. Vol. 235. New York, NY: Marcel Dekker. p. 363.
7854:
7850:
7747:
7702:
7669:
7662:
7396:
6314:
and thus it is itself a left inverse of that other homomorphism. That is, a homomorphism
6294:
6264:
6256:
6209:
5941:
5824:
5798:
5088:
4578:
4176:
4168:
4129:
4098:
3893:
3867:
3790:
2502:
2339:
1683:
1176:
849:
800:
781:
155:
50:
46:
9237:
Isomorphismus zweier
Gruppen einschränken, sonst aber von "Homomorphismus" sprechen, … "
3641:
822:. Whether the multiplicative identity is to be preserved depends upon the definition of
9686:
9268:
9219:
8990:
8970:
8815:
8795:
8694:
8389:
8317:
8051:
7816:
7772:
7752:
7730:
7678:
7611:
7591:
7571:
7551:
7455:
7402:
7378:
7264:
7170:
7150:
7092:
7020:
6892:
6872:
6775:
6755:
6638:
6551:
6531:
6298:
6275:
6151:
6131:
6111:
6091:
5999:
5979:
5947:
5923:
5903:
5874:
5850:
5778:
5585:
5565:
5545:
5525:
5505:
5441:
5421:
5401:
5347:
5323:
5303:
5238:
5232:
5094:
4997:
4906:
4856:
4769:
4749:
4729:
4723:
4705:
4685:
4665:
4645:
4620:
4514:
4494:
4474:
4454:
4434:
4192:
4172:
4040:
4020:
4000:
3980:
3872:
3849:
3794:
3751:
3668:
3621:
3394:
3374:
3354:
3334:
3201:
3181:
3065:
3045:
2981:
2961:
2937:
2795:
2775:
2755:
2703:
2683:
2485:
2473:
Several kinds of homomorphisms have a specific name, which is also defined for general
2448:
2428:
2364:
2344:
2218:
2118:
1811:
1791:
1767:
1693:
1188:
1172:
1092:
815:
796:
734:
516:
496:
476:
398:
378:
358:
338:
223:
203:
54:
3897:
158:
of the same type (e.g. two groups, two fields, two vector spaces), that preserves the
95:
9664:
9638:
9620:
9606:
9584:
9574:
9536:
9515:
9492:
9456:
9410:
9392:
9367:
9325:
9272:
9223:
9177:
7858:
7673:
6656:
6306:
6240:
5973:
4606:
4152:
4137:
3689:
991:
811:
669:{\displaystyle f(\mu _{A}(a_{1},\ldots ,a_{k}))=\mu _{B}(f(a_{1}),\ldots ,f(a_{k})),}
198:
163:
42:
3608:{\displaystyle g(x*_{B}y)=g(f(g(x))*_{B}f(g(y)))=g(f(g(x)*_{A}g(y)))=g(x)*_{A}g(y),}
9648:
9507:
9466:
9420:
9281:"Hierdurch ist, wie man sofort überblickt, eine homomorphe*) Beziehung der Gruppe Γ
9260:
9211:
8967:
As it is often the case, but not always, the same symbol for the operation of both
7810:
7696:
7201:
5897:
3767:
2521:
987:
759:
261:
5224:{\displaystyle \{\ldots ,x^{-n},\ldots ,x^{-1},1,x,x^{2},\ldots ,x^{n},\ldots \},}
9634:
9484:
9406:
9335:
8063:
7862:
7846:
6283:
6042:
5256:
4156:
4145:
3932:
3789:
The automorphisms of an algebraic structure or of an object of a category form a
3771:
2513:
1196:
804:
128:
2480:
8927:
6270:
Algebraic structures for which there exist non-surjective epimorphisms include
6260:
3693:
2382:
1785:
1679:
789:
755:
117:
9243:
isomorphism of two groups; otherwise, however, speak of a "homomorphism", … )
9680:
9588:
9021:
8948:
8915:
8050:
In the special case with just one binary relation, we obtain the notion of a
7834:
6248:
5231:
which, as, a group, is isomorphic to the additive group of the integers; for
3909:
3697:
827:
260:
is an operation of the structure (supposed here, for simplification, to be a
24:
9181:
7845:, but is different and has received a specific name in other cases, such as
7874:
7838:
7420:
6244:
5293:
4180:
3921:
3843:
3783:
3743:
3713:
1990:{\displaystyle f(z_{1}z_{2})=|z_{1}z_{2}|=|z_{1}||z_{2}|=f(z_{1})f(z_{2}).}
838:
143:
139:
58:
20:
16:
Structure-preserving map between two algebraic structures of the same type
9017:
8891:
6046:
4614:
3755:
2498:
1184:
876:
135:
106:
4151:
In particular, the two definitions of a monomorphism are equivalent for
3704:, is thus a bijective continuous map, whose inverse is also continuous.
3688:
This proof does not work for non-algebraic structures. For example, for
9264:
9215:
7629:
6259:. The importance of these structures in all mathematics, especially in
6038:
4601:. This proof works not only for algebraic structures, but also for any
3905:
2955:
834:
113:
7472:, in a natural way, by defining the operations of the quotient set by
2565:
is a (homo)morphism, it has an inverse if there exists a homomorphism
9287:(Thus, as one immediately sees, a homomorphic relation of the group Γ
6271:
4900:
4160:
3925:
2813:
2506:
774:
88:
76:
4306:
A split monomorphism is always a monomorphism, for both meanings of
123:
The concept of homomorphism has been generalized, under the name of
9595:
9356:
8943:
6866:
6862:
6420:
A split epimorphism is always an epimorphism, for both meanings of
6053:
3936:
3725:
3721:
2517:
2474:
1267:{\displaystyle f(r)={\begin{pmatrix}r&0\\0&r\end{pmatrix}}}
826:
in use. If the multiplicative identity is not preserved, one has a
818:
that preserves the ring addition, the ring multiplication, and the
124:
39:
9451:
Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Șerban (2001).
3731:
The endomorphisms of an algebraic structure, or of an object of a
6279:
6267:, may explain the coexistence of two non-equivalent definitions.
6030:
1690:, which is required for elements of a group.) Define a function
31:
9535:, Volume I, edited by G. Rozenberg, A. Salomaa, Springer, 1997,
7881:
be a signature consisting of function and relation symbols, and
5775:
by the uniqueness in the definition of a universal property. As
4318:
Proof of the equivalence of the two definitions of monomorphisms
1828:
is a homomorphism of groups, since it preserves multiplication:
1710:
from the nonzero complex numbers to the nonzero real numbers by
754:
The operations that must be preserved by a homomorphism include
9512:
Algebraic and automata theoretic properties of formal languages
4991:
4164:
3736:
2112:
1191:, having both addition and multiplication. The set of all 2×2
1086:
785:
3117:
is a bijective homomorphism between algebraic structures, let
9437:
Linderholm, C. E. (1970). A group epimorphism is surjective.
3662:
471:
415:
preserves the operation or is compatible with the operation.
100:
5364:
exists, then every left cancelable homomorphism is injective
9352:
9350:
9348:
7861:(in the case of non-commutative rings, the kernels are the
1203:. If we define a function between these rings as follows:
9450:
8574:{\displaystyle h\colon \Sigma _{1}^{*}\to \Sigma _{2}^{*}}
8173:{\displaystyle h\colon \Sigma _{1}^{*}\to \Sigma _{2}^{*}}
4132:, for which every homomorphism is a monomorphism, and for
9345:
9173:
Vorlesungen über die
Theorie der automorphen Functionen
4232:
is a split monomorphism if there exists a homomorphism
3351:
is a binary operation of the structure, for every pair
6346:
is a split epimorphism if there exists a homomorphism
1603:
1567:
1522:
1415:
1376:
1325:
1233:
8993:
8973:
8900:
8876:
8849:
8818:
8798:
8765:
8720:
8697:
8664:
8619:
8587:
8531:
8496:
8470:
8435:
8412:
8392:
8372:
8340:
8320:
8288:
8245:
8186:
8130:
8103:
8076:
7819:
7795:
7775:
7755:
7733:
7705:
7681:
7638:
7614:
7594:
7574:
7554:
7534:
7478:
7458:
7428:
7405:
7381:
7357:
7313:
7287:
7267:
7247:
7212:
7173:
7153:
7115:
7095:
7069:
7043:
7023:
6991:
6947:
6915:
6895:
6875:
6824:
6798:
6778:
6758:
6729:
6700:
6665:
6641:
6603:
6574:
6554:
6534:
6499:
6436:
6384:
6352:
6320:
6212:
6174:
6154:
6134:
6114:
6094:
6062:
6002:
5982:
5950:
5926:
5906:
5877:
5853:
5827:
5801:
5781:
5740:
5678:
5643:
5608:
5588:
5568:
5548:
5528:
5508:
5464:
5444:
5424:
5404:
5372:
5350:
5326:
5306:
5264:
5241:
5120:
5097:
5020:
5000:
4929:
4909:
4879:
4859:
4824:
4792:
4772:
4752:
4732:
4708:
4688:
4668:
4648:
4623:
4581:
4537:
4517:
4497:
4477:
4457:
4437:
4375:
4334:
4270:
4238:
4206:
4101:
4063:
4043:
4023:
4003:
3983:
3951:
3875:
3852:
3809:
3671:
3644:
3624:
3420:
3397:
3377:
3357:
3337:
3259:
3224:
3204:
3184:
3155:
3123:
3091:
3068:
3048:
3004:
2984:
2964:
2940:
2866:
2822:
2798:
2778:
2758:
2726:
2706:
2686:
2612:
2574:
2536:
2451:
2431:
2395:
2367:
2347:
2306:
2241:
2221:
2181:
2141:
2121:
2015:
1837:
1814:
1794:
1770:
1719:
1696:
1498:
1298:
1212:
1135:
1095:
1003:
930:
891:
737:
685:
542:
519:
499:
479:
456:
424:
401:
381:
361:
341:
273:
246:
226:
206:
171:
5080:{\displaystyle \{1,x,x^{2},\ldots ,x^{n},\ldots \},}
8999:
8979:
8906:
8882:
8862:
8824:
8804:
8784:
8751:
8703:
8683:
8650:
8605:
8573:
8514:
8482:
8456:
8418:
8398:
8378:
8358:
8326:
8306:
8270:
8231:
8172:
8116:
8089:
7825:
7801:
7781:
7761:
7739:
7719:
7687:
7653:
7620:
7600:
7580:
7560:
7540:
7520:
7464:
7452:can then be given a structure of the same type as
7444:
7411:
7387:
7363:
7343:
7299:
7273:
7253:
7230:
7179:
7159:
7139:
7101:
7089:(one is a zero map, while the other is not). Thus
7081:
7055:
7029:
7009:
6977:
6933:
6901:
6881:
6851:
6810:
6784:
6764:
6744:
6715:
6686:
6647:
6627:
6589:
6560:
6540:
6517:
6467:
6412:
6370:
6338:
6224:
6198:
6160:
6140:
6120:
6100:
6080:
6008:
5988:
5956:
5932:
5912:
5883:
5859:
5839:
5813:
5787:
5767:
5726:
5664:
5629:
5594:
5574:
5554:
5534:
5514:
5494:
5450:
5430:
5410:
5390:
5356:
5332:
5312:
5284:
5247:
5223:
5103:
5079:
5006:
4983:{\displaystyle \{x,x^{2},\ldots ,x^{n},\ldots \},}
4982:
4915:
4891:
4865:
4845:
4810:
4778:
4758:
4738:
4714:
4694:
4674:
4654:
4629:
4593:
4567:
4523:
4503:
4483:
4463:
4443:
4423:
4361:
4298:
4256:
4224:
4113:
4087:
4049:
4029:
4009:
3989:
3969:
3881:
3858:
3834:
3770:between the ring of endomorphisms and the ring of
3677:
3653:
3630:
3607:
3403:
3383:
3363:
3343:
3317:
3245:
3210:
3190:
3170:
3141:
3109:
3074:
3054:
3034:
2990:
2970:
2946:
2926:
2852:
2804:
2784:
2764:
2744:
2712:
2692:
2669:
2592:
2554:
2457:
2437:
2413:
2373:
2353:
2327:
2292:
2227:
2207:
2167:
2127:
2101:{\displaystyle |z_{1}+z_{2}|\neq |z_{1}|+|z_{2}|.}
2100:
1989:
1820:
1800:
1776:
1753:
1702:
1667:
1478:
1266:
1163:
1101:
1063:
975:
910:
743:
723:
668:
525:
505:
485:
462:
442:
407:
387:
367:
347:
324:
252:
232:
212:
189:
7647:
6486:Equivalence of the two definitions of epimorphism
6413:{\displaystyle f\circ g=\operatorname {Id} _{B}.}
4299:{\displaystyle g\circ f=\operatorname {Id} _{A}.}
2484:General relationship of homomorphisms (Including
9678:
3786:is an endomorphism that is also an isomorphism.
758:, that is the constants. In particular, when an
9311:
9309:
9307:
9305:
9303:
4662:is a pair consisting of an algebraic structure
240:equipped with the same structure such that, if
112:Homomorphisms of vector spaces are also called
9596:Stanley N. Burris; H.P. Sankappanavar (2012).
9357:Stanley N. Burris; H.P. Sankappanavar (2012).
6289:A wide generalization of this example is the
6278:. The most basic example is the inclusion of
9658:
9300:
9065:We are assured that a language homomorphism
8940:– a simplistic decentralized voting protocol
8062:Homomorphisms are also used in the study of
5320:is the vector space or free module that has
5215:
5121:
5071:
5021:
4974:
4930:
4886:
4880:
4853:. For example, for sets, the free object on
4326:An injective homomorphism is left cancelable
9659:Fraleigh, John B.; Katz, Victor J. (2003),
9169:
9387:
9385:
9383:
6456:
6452:
6446:
6442:
3835:{\displaystyle \operatorname {GL} _{n}(k)}
2505:of the same type is commonly defined as a
325:{\displaystyle f(x\cdot y)=f(x)\cdot f(y)}
8057:
5266:
4642:of algebraic structures a free object on
2421:, which is a group homomorphism from the
2186:
2146:
1649:
9626:Categories for the Working Mathematician
9619:
9398:Categories for the Working Mathematician
9391:
9315:
7868:
2479:
2468:
2208:{\displaystyle (\mathbb {N} ,\times ,1)}
2111:As another example, the diagram shows a
1081:
9568:
9552:
9380:
5398:be a left cancelable homomorphism, and
3793:under composition, which is called the
777:that preserves the semigroup operation.
9679:
9573:. Berlin: Springer. pp. 280–291.
9246:
9197:
2293:{\displaystyle f(x+y)=f(x)\times f(y)}
1064:{\displaystyle \ln(xy)=\ln(x)+\ln(y),}
9531:T. Harju, J. Karhumӓki, Morphisms in
8457:{\displaystyle h(x)\neq \varepsilon }
8066:and are often briefly referred to as
6041:homomorphisms. On the other hand, in
3754:. In the case of a vector space or a
3728:whose source is equal to its target.
9605:. S. Burris and H.P. Sankappanavar.
9366:. S. Burris and H.P. Sankappanavar.
154:A homomorphism is a map between two
116:, and their study is the subject of
7668:When the algebraic structure is a
6088:is an epimorphism if, for any pair
6056:. This means that a (homo)morphism
5968:of the structure). This defines an
5900:): For building a free object over
5502:. By definition of the free object
4017:of morphisms from any other object
3977:is a monomorphism if, for any pair
3945:. This means that a (homo)morphism
2927:{\displaystyle x=g(f(x))=g(f(y))=y}
1788:(or modulus) of the complex number
976:{\displaystyle e^{x+y}=e^{x}e^{y},}
852:, also called a linear map between
13:
9661:A First Course in Abstract Algebra
8901:
8877:
8870:of words formed from the alphabet
8851:
8773:
8672:
8589:
8557:
8539:
8498:
8483:{\displaystyle x\neq \varepsilon }
8342:
8290:
8271:{\displaystyle u,v\in \Sigma _{1}}
8259:
8156:
8138:
8105:
8078:
6852:{\displaystyle g\circ f=h\circ f.}
5768:{\displaystyle f\circ g=f\circ h,}
4746:of the variety, and every element
4362:{\displaystyle f\circ g=f\circ h,}
2168:{\displaystyle (\mathbb {N} ,+,0)}
1285:is a homomorphism of rings, since
1074:and is also a group homomorphism.
14:
9698:
9439:The American Mathematical Monthly
7140:{\displaystyle g\circ f=h\circ f}
6199:{\displaystyle g\circ f=h\circ f}
4786:, there is a unique homomorphism
4088:{\displaystyle f\circ g=f\circ h}
3935:, a monomorphism is defined as a
3325:and it remains only to show that
2516:, an isomorphism is defined as a
1678:For another example, the nonzero
8785:{\displaystyle a\in \Sigma _{1}}
8684:{\displaystyle a\in \Sigma _{1}}
6941:be the canonical map, such that
6628:{\displaystyle g,h\colon B\to B}
3661:As the proof is similar for any
162:of the structures. This means a
9546:
9525:
9501:
9059:
9040:) denotes the concatenation of
9010:
8914:. Here the monoid operation is
8606:{\displaystyle \Sigma _{1}^{*}}
8515:{\displaystyle \Sigma _{1}^{*}}
8386:denotes the empty string, then
8359:{\displaystyle \Sigma _{1}^{*}}
8307:{\displaystyle \Sigma _{1}^{*}}
5727:{\displaystyle f(g(x))=f(h(x))}
4682:of this variety and an element
4424:{\displaystyle f(g(x))=f(h(x))}
3931:In the more general context of
3915:
3842:is the automorphism group of a
3777:
3707:
2641:
2635:
2512:In the more general context of
2445:to the multiplicative group of
724:{\displaystyle a_{1},...,a_{k}}
9491:. Cambridge University Press,
9477:
9444:
9431:
9188:
9163:
8961:
8739:
8735:
8729:
8722:
8638:
8634:
8628:
8621:
8553:
8445:
8439:
8232:{\displaystyle h(uv)=h(u)h(v)}
8226:
8220:
8214:
8208:
8199:
8190:
8152:
7515:
7503:
7497:
7491:
7485:
7479:
7338:
7332:
7323:
7317:
7222:
7010:{\displaystyle h\colon B\to C}
7001:
6966:
6963:
6957:
6951:
6934:{\displaystyle g\colon B\to C}
6925:
6739:
6733:
6675:
6669:
6619:
6584:
6578:
6518:{\displaystyle f\colon A\to B}
6509:
6459:epimorphism (right cancelable)
6453:
6443:
6371:{\displaystyle g\colon B\to A}
6362:
6339:{\displaystyle f\colon A\to B}
6330:
6072:
6024:
5871:Existence of a free object on
5721:
5718:
5712:
5706:
5697:
5694:
5688:
5682:
5653:
5647:
5618:
5612:
5489:
5483:
5474:
5468:
5391:{\displaystyle f\colon A\to B}
5382:
5276:
5270:
4834:
4828:
4802:
4562:
4556:
4547:
4541:
4418:
4415:
4409:
4403:
4394:
4391:
4385:
4379:
4257:{\displaystyle g\colon B\to A}
4248:
4225:{\displaystyle f\colon A\to B}
4216:
3961:
3829:
3823:
3599:
3593:
3577:
3571:
3562:
3559:
3556:
3550:
3534:
3528:
3522:
3516:
3507:
3504:
3501:
3495:
3489:
3473:
3470:
3464:
3458:
3452:
3443:
3424:
3234:
3228:
3165:
3159:
3133:
3101:
3062:is the image of an element of
3029:
3026:
3020:
3014:
2915:
2912:
2906:
2900:
2891:
2888:
2882:
2876:
2847:
2841:
2832:
2826:
2736:
2584:
2546:
2492:
2405:
2316:
2310:
2287:
2281:
2272:
2266:
2257:
2245:
2202:
2182:
2162:
2142:
2091:
2076:
2068:
2053:
2045:
2017:
1981:
1968:
1962:
1949:
1939:
1924:
1919:
1904:
1896:
1871:
1864:
1841:
1744:
1736:
1729:
1723:
1659:
1653:
1646:
1640:
1511:
1502:
1473:
1467:
1458:
1452:
1314:
1302:
1222:
1216:
1145:
1139:
1055:
1049:
1037:
1031:
1019:
1010:
911:{\displaystyle x\mapsto e^{x}}
895:
660:
657:
644:
629:
616:
610:
594:
591:
559:
546:
434:
319:
313:
304:
298:
289:
277:
181:
134:A homomorphism may also be an
49:of the same type (such as two
1:
9631:Graduate Texts in Mathematics
9599:A Course in Universal Algebra
9562:
9453:Hopf Algebra: An Introduction
9409:. Exercise 4 in section I.5.
9403:Graduate Texts in Mathematics
9360:A Course in Universal Algebra
9322:American Mathematical Society
7769:"). Also in this case, it is
6310:is a homomorphism that has a
6255:(see below for a proof), and
5285:{\displaystyle \mathbb {Z} ;}
4196:is a homomorphism that has a
149:
9533:Handbook of Formal Languages
9170:Fricke, Robert (1897–1912).
9156:
8419:{\displaystyle \varepsilon }
8379:{\displaystyle \varepsilon }
7568:. In that case the image of
7147:(both are the zero map from
6792:is not right cancelable, as
5898:Free object § Existence
5795:is left cancelable, one has
5522:, there exist homomorphisms
863:is a map that preserves the
89:
77:
7:
9316:Birkhoff, Garrett (1967) ,
8921:
8863:{\displaystyle \Sigma ^{*}}
8117:{\displaystyle \Sigma _{2}}
8090:{\displaystyle \Sigma _{1}}
3892:The automorphism groups of
1077:
10:
9703:
8938:Homomorphic secret sharing
7661:; this fact is one of the
7199:
3920:For algebraic structures,
2720:have underlying sets, and
1164:{\displaystyle f(x)=2^{x}}
101:
82:
70:
18:
9073:to the empty word. Since
8890:may be thought of as the
7654:{\displaystyle X/\!\sim }
7445:{\displaystyle X/{\sim }}
7344:{\displaystyle f(a)=f(b)}
7195:
6978:{\displaystyle g(f(A))=0}
6865:and on the fact that the
6528:In the case of sets, let
5495:{\displaystyle f(a)=f(b)}
4722:satisfying the following
4568:{\displaystyle g(x)=h(x)}
4124:These two definitions of
3035:{\displaystyle x=f(g(x))}
2853:{\displaystyle f(x)=f(y)}
1754:{\displaystyle f(z)=|z|.}
1289:preserves both addition:
8954:
8752:{\displaystyle |h(a)|=1}
8651:{\displaystyle |h(a)|=k}
7672:for some operation, the
7231:{\displaystyle f:X\to Y}
6752:is any other element of
6449:epimorphism (surjective)
6081:{\displaystyle f:A\to B}
4811:{\displaystyle f:L\to S}
3970:{\displaystyle f:A\to B}
3924:are commonly defined as
3724:, or, more generally, a
3716:is a homomorphism whose
3638:is thus compatible with
3142:{\displaystyle g:B\to A}
3110:{\displaystyle f:A\to B}
2745:{\displaystyle f:A\to B}
2593:{\displaystyle g:B\to A}
2555:{\displaystyle f:A\to B}
2414:{\displaystyle N:A\to F}
443:{\displaystyle f:A\to B}
190:{\displaystyle f:A\to B}
19:Not to be confused with
9514:, North-Holland, 1975,
9247:Fricke, Robert (1892).
8907:{\displaystyle \Sigma }
8883:{\displaystyle \Sigma }
7608:under the homomorphism
7300:{\displaystyle a\sim b}
7082:{\displaystyle g\neq h}
7056:{\displaystyle C\neq 0}
6869:are homomorphisms: let
6811:{\displaystyle g\neq h}
6716:{\displaystyle x\in B,}
4471:, the common source of
4144:is defined either as a
3908:, and are the basis of
3774:of the same dimension.
3742:The endomorphisms of a
2792:is bijective. In fact,
2488:, labelled as “Inner”).
1281:is a real number, then
856:, is defined similarly.
820:multiplicative identity
450:preserves an operation
9489:Relational Mathematics
9229:From footnote on p. 22
9198:Ritter, Ernst (1892).
9001:
8981:
8933:Homomorphic encryption
8908:
8884:
8864:
8826:
8806:
8786:
8753:
8705:
8685:
8652:
8607:
8575:
8516:
8484:
8458:
8420:
8400:
8380:
8360:
8328:
8308:
8272:
8233:
8174:
8118:
8091:
8058:Formal language theory
7827:
7803:
7783:
7763:
7741:
7721:
7689:
7655:
7622:
7602:
7582:
7562:
7542:
7522:
7521:{\displaystyle \ast =}
7466:
7446:
7413:
7389:
7365:
7345:
7301:
7275:
7255:
7232:
7181:
7161:
7141:
7109:is not cancelable, as
7103:
7083:
7057:
7031:
7011:
6979:
6935:
6903:
6883:
6853:
6812:
6786:
6766:
6746:
6717:
6688:
6687:{\displaystyle h(x)=x}
6649:
6629:
6591:
6562:
6542:
6519:
6469:
6414:
6372:
6340:
6291:localization of a ring
6226:
6200:
6162:
6142:
6122:
6102:
6082:
6010:
5990:
5958:
5934:
5914:
5885:
5861:
5841:
5815:
5789:
5769:
5728:
5666:
5665:{\displaystyle h(x)=b}
5631:
5630:{\displaystyle g(x)=a}
5596:
5576:
5556:
5536:
5516:
5496:
5452:
5432:
5412:
5392:
5358:
5344:If a free object over
5334:
5314:
5286:
5249:
5225:
5105:
5081:
5008:
4984:
4917:
4893:
4867:
4847:
4846:{\displaystyle f(x)=s}
4812:
4780:
4760:
4740:
4726:: for every structure
4716:
4696:
4676:
4656:
4631:
4595:
4569:
4525:
4505:
4485:
4465:
4445:
4425:
4363:
4300:
4258:
4226:
4142:multiplicative inverse
4115:
4089:
4051:
4031:
4011:
3991:
3971:
3883:
3860:
3836:
3679:
3655:
3632:
3609:
3405:
3385:
3365:
3345:
3331:is a homomorphism. If
3319:
3247:
3246:{\displaystyle f(x)=y}
3212:
3192:
3178:is the unique element
3172:
3143:
3111:
3076:
3056:
3036:
2992:
2972:
2948:
2928:
2854:
2806:
2786:
2766:
2746:
2714:
2694:
2671:
2594:
2556:
2489:
2459:
2439:
2415:
2375:
2355:
2329:
2328:{\displaystyle f(0)=1}
2294:
2229:
2209:
2169:
2129:
2102:
1991:
1822:
1802:
1778:
1755:
1704:
1688:multiplicative inverse
1669:
1480:
1268:
1195:is also a ring, under
1180:
1165:
1103:
1065:
977:
912:
771:semigroup homomorphism
745:
725:
670:
527:
507:
487:
464:
444:
409:
395:. One says often that
389:
369:
349:
326:
254:
253:{\displaystyle \cdot }
234:
214:
191:
94:
67:Ancient Greek language
9253:Mathematische Annalen
9204:Mathematische Annalen
9117:) equals the number 2
9002:
8982:
8909:
8885:
8865:
8827:
8807:
8787:
8754:
8706:
8686:
8653:
8608:
8576:
8517:
8485:
8459:
8421:
8401:
8381:
8361:
8334:is a homomorphism on
8329:
8309:
8273:
8234:
8175:
8119:
8092:
8038:-ary relation symbol
7976:-ary function symbol
7869:Relational structures
7828:
7809:, that is called the
7804:
7802:{\displaystyle \sim }
7784:
7764:
7742:
7722:
7690:
7656:
7623:
7603:
7583:
7563:
7543:
7541:{\displaystyle \ast }
7528:, for each operation
7523:
7467:
7447:
7414:
7390:
7366:
7364:{\displaystyle \sim }
7346:
7302:
7276:
7256:
7254:{\displaystyle \sim }
7233:
7182:
7162:
7142:
7104:
7084:
7058:
7032:
7012:
6980:
6936:
6904:
6884:
6854:
6813:
6787:
6767:
6747:
6718:
6689:
6650:
6630:
6592:
6563:
6543:
6520:
6470:
6427:In summary, one has
6415:
6373:
6341:
6227:
6201:
6163:
6143:
6123:
6103:
6083:
6037:are often defined as
6011:
5991:
5959:
5935:
5915:
5886:
5862:
5842:
5816:
5790:
5770:
5729:
5667:
5632:
5597:
5577:
5557:
5537:
5517:
5497:
5453:
5433:
5413:
5393:
5359:
5335:
5315:
5300:, the free object on
5287:
5250:
5235:, the free object on
5226:
5113:infinite cyclic group
5106:
5091:, the free object on
5082:
5009:
4994:, the free object on
4985:
4918:
4903:, the free object on
4894:
4892:{\displaystyle \{x\}}
4868:
4848:
4813:
4781:
4761:
4741:
4717:
4697:
4677:
4657:
4632:
4596:
4570:
4526:
4506:
4486:
4466:
4446:
4426:
4364:
4301:
4259:
4227:
4116:
4090:
4052:
4032:
4012:
3992:
3972:
3884:
3861:
3837:
3680:
3656:
3633:
3610:
3406:
3386:
3366:
3346:
3320:
3248:
3213:
3193:
3173:
3149:be the map such that
3144:
3112:
3077:
3057:
3037:
2993:
2973:
2949:
2929:
2855:
2807:
2787:
2767:
2747:
2715:
2695:
2672:
2595:
2557:
2483:
2469:Special homomorphisms
2460:
2440:
2416:
2376:
2356:
2330:
2295:
2230:
2210:
2170:
2130:
2103:
1992:
1823:
1803:
1779:
1756:
1705:
1670:
1481:
1269:
1201:matrix multiplication
1166:
1104:
1085:
1066:
978:
913:
843:scalar multiplication
837:is a homomorphism of
746:
726:
671:
528:
508:
488:
465:
445:
410:
390:
370:
350:
327:
255:
235:
215:
192:
99:meaning "similar" to
81:) meaning "same" and
9069:maps the empty word
8991:
8971:
8898:
8874:
8847:
8816:
8812:is 1-uniform), then
8796:
8763:
8718:
8695:
8662:
8617:
8585:
8529:
8494:
8468:
8433:
8410:
8390:
8370:
8338:
8318:
8286:
8243:
8184:
8128:
8101:
8074:
7893:-structures. Then a
7817:
7793:
7773:
7753:
7731:
7703:
7679:
7663:isomorphism theorems
7636:
7612:
7592:
7572:
7552:
7532:
7476:
7456:
7426:
7403:
7379:
7355:
7311:
7285:
7265:
7245:
7240:equivalence relation
7210:
7171:
7151:
7113:
7093:
7067:
7041:
7021:
7017:be the zero map. If
6989:
6945:
6913:
6893:
6873:
6822:
6796:
6776:
6756:
6745:{\displaystyle h(b)}
6727:
6698:
6663:
6639:
6601:
6590:{\displaystyle f(A)}
6572:
6568:that not belongs to
6552:
6532:
6497:
6434:
6382:
6350:
6318:
6210:
6172:
6152:
6148:to any other object
6132:
6112:
6092:
6060:
6000:
5980:
5970:equivalence relation
5948:
5942:well-formed formulas
5924:
5904:
5875:
5851:
5825:
5799:
5779:
5738:
5676:
5641:
5606:
5586:
5566:
5546:
5526:
5506:
5462:
5442:
5422:
5402:
5370:
5348:
5324:
5304:
5262:
5239:
5118:
5095:
5018:
4998:
4927:
4907:
4877:
4857:
4822:
4790:
4770:
4750:
4730:
4706:
4686:
4666:
4646:
4621:
4579:
4535:
4515:
4495:
4475:
4455:
4435:
4373:
4332:
4268:
4236:
4204:
4099:
4061:
4041:
4021:
4001:
3981:
3949:
3873:
3850:
3807:
3802:general linear group
3669:
3642:
3622:
3418:
3395:
3375:
3355:
3335:
3257:
3222:
3202:
3182:
3171:{\displaystyle g(y)}
3153:
3121:
3089:
3066:
3046:
3002:
2982:
2962:
2938:
2864:
2820:
2796:
2776:
2756:
2724:
2704:
2684:
2610:
2572:
2534:
2527:More precisely, if
2503:algebraic structures
2449:
2429:
2423:multiplicative group
2393:
2365:
2345:
2304:
2239:
2219:
2179:
2139:
2119:
2013:
1835:
1812:
1792:
1768:
1717:
1694:
1496:
1489:and multiplication:
1296:
1210:
1133:
1093:
1001:
928:
889:
881:exponential function
861:algebra homomorphism
735:
683:
540:
517:
497:
477:
463:{\displaystyle \mu }
454:
422:
399:
379:
359:
339:
271:
244:
224:
204:
169:
156:algebraic structures
47:algebraic structures
40:structure-preserving
8602:
8570:
8552:
8511:
8355:
8303:
8169:
8151:
7909:from the domain of
7851:group homomorphisms
7720:{\displaystyle X/K}
7397:congruence relation
7037:is not surjective,
6889:be the cokernel of
6295:commutative algebra
6265:homological algebra
6239:are equivalent for
6225:{\displaystyle g=h}
5974:equivalence classes
5920:, consider the set
5840:{\displaystyle a=b}
5814:{\displaystyle g=h}
5438:be two elements of
4594:{\displaystyle g=h}
4531:is injective, then
4114:{\displaystyle g=h}
3896:were introduced by
3739:under composition.
3685:is a homomorphism.
2486:Inner Automorphisms
2340:composition algebra
850:module homomorphism
782:monoid homomorphism
9663:, Addison-Wesley,
9621:Mac Lane, Saunders
9393:Mac Lane, Saunders
9265:10.1007/BF01443421
9216:10.1007/BF01443449
9149:) has null length.
9016:The ∗ denotes the
8997:
8977:
8904:
8880:
8860:
8822:
8802:
8782:
8749:
8701:
8681:
8648:
8603:
8588:
8571:
8556:
8538:
8512:
8497:
8480:
8454:
8427:-free homomorphism
8416:
8396:
8376:
8356:
8341:
8324:
8304:
8289:
8268:
8229:
8170:
8155:
8137:
8114:
8087:
8070:. Given alphabets
8052:graph homomorphism
7859:ring homomorphisms
7823:
7799:
7779:
7759:
7737:
7727:(usually read as "
7717:
7685:
7651:
7618:
7598:
7578:
7558:
7538:
7518:
7462:
7442:
7409:
7385:
7361:
7341:
7297:
7271:
7251:
7228:
7177:
7157:
7137:
7099:
7079:
7053:
7027:
7007:
6975:
6931:
6899:
6879:
6849:
6808:
6782:
6762:
6742:
6713:
6684:
6645:
6625:
6587:
6558:
6538:
6515:
6465:
6410:
6368:
6336:
6299:algebraic geometry
6222:
6196:
6158:
6138:
6128:of morphisms from
6118:
6098:
6078:
6006:
5986:
5954:
5930:
5910:
5881:
5857:
5837:
5811:
5785:
5765:
5724:
5662:
5627:
5592:
5572:
5552:
5532:
5512:
5492:
5448:
5428:
5408:
5388:
5354:
5330:
5310:
5282:
5245:
5221:
5101:
5077:
5004:
4980:
4913:
4889:
4863:
4843:
4808:
4776:
4756:
4736:
4724:universal property
4712:
4692:
4672:
4652:
4627:
4607:topological spaces
4591:
4565:
4521:
4501:
4481:
4461:
4441:
4421:
4359:
4296:
4254:
4222:
4193:split monomorphism
4111:
4085:
4047:
4027:
4007:
3987:
3967:
3879:
3856:
3832:
3797:of the structure.
3795:automorphism group
3762:, the choice of a
3692:, a morphism is a
3690:topological spaces
3675:
3665:, this shows that
3654:{\displaystyle *.}
3651:
3628:
3605:
3401:
3381:
3361:
3341:
3315:
3243:
3208:
3188:
3168:
3139:
3107:
3072:
3052:
3032:
2988:
2968:
2944:
2924:
2850:
2802:
2782:
2762:
2742:
2710:
2690:
2667:
2590:
2552:
2490:
2455:
2435:
2411:
2371:
2351:
2325:
2290:
2225:
2205:
2165:
2125:
2098:
1987:
1818:
1798:
1774:
1751:
1700:
1665:
1628:
1592:
1553:
1476:
1440:
1401:
1362:
1264:
1258:
1181:
1161:
1099:
1061:
973:
908:
797:group homomorphism
741:
721:
666:
523:
503:
493:, defined on both
483:
460:
440:
405:
385:
365:
345:
322:
250:
230:
210:
187:
9670:978-1-292-02496-7
9612:978-0-9880552-0-9
9580:978-3-540-35430-7
9497:978-0-521-76268-7
9441:, 77(2), 176-177.
9373:978-0-9880552-0-9
9331:978-0-8218-1025-5
9121:of characters in
9109:of characters in
9000:{\displaystyle B}
8980:{\displaystyle A}
8832:is also called a
8825:{\displaystyle h}
8805:{\displaystyle h}
8714:homomorphism. If
8704:{\displaystyle k}
8399:{\displaystyle h}
8327:{\displaystyle h}
7913:to the domain of
7826:{\displaystyle f}
7782:{\displaystyle K}
7762:{\displaystyle K}
7740:{\displaystyle X}
7688:{\displaystyle K}
7674:equivalence class
7621:{\displaystyle f}
7601:{\displaystyle Y}
7581:{\displaystyle X}
7561:{\displaystyle X}
7465:{\displaystyle X}
7412:{\displaystyle X}
7388:{\displaystyle f}
7274:{\displaystyle X}
7206:Any homomorphism
7192:
7191:
7180:{\displaystyle C}
7160:{\displaystyle A}
7102:{\displaystyle f}
7030:{\displaystyle f}
6902:{\displaystyle f}
6882:{\displaystyle C}
6785:{\displaystyle f}
6765:{\displaystyle B}
6657:identity function
6648:{\displaystyle g}
6561:{\displaystyle B}
6548:be an element of
6541:{\displaystyle b}
6460:
6450:
6440:
6439:split epimorphism
6307:split epimorphism
6161:{\displaystyle C}
6141:{\displaystyle B}
6121:{\displaystyle h}
6101:{\displaystyle g}
6021:
6020:
6009:{\displaystyle x}
5989:{\displaystyle W}
5957:{\displaystyle x}
5933:{\displaystyle W}
5913:{\displaystyle x}
5884:{\displaystyle x}
5860:{\displaystyle f}
5788:{\displaystyle f}
5595:{\displaystyle A}
5575:{\displaystyle F}
5555:{\displaystyle h}
5535:{\displaystyle g}
5515:{\displaystyle F}
5451:{\displaystyle A}
5431:{\displaystyle b}
5411:{\displaystyle a}
5357:{\displaystyle x}
5333:{\displaystyle x}
5313:{\displaystyle x}
5248:{\displaystyle x}
5104:{\displaystyle x}
5007:{\displaystyle x}
4916:{\displaystyle x}
4866:{\displaystyle x}
4779:{\displaystyle S}
4759:{\displaystyle s}
4739:{\displaystyle S}
4715:{\displaystyle L}
4695:{\displaystyle x}
4675:{\displaystyle L}
4655:{\displaystyle x}
4630:{\displaystyle x}
4524:{\displaystyle f}
4504:{\displaystyle h}
4484:{\displaystyle g}
4464:{\displaystyle C}
4444:{\displaystyle x}
4138:universal algebra
4050:{\displaystyle A}
4030:{\displaystyle C}
4010:{\displaystyle h}
3990:{\displaystyle g}
3900:for studying the
3882:{\displaystyle k}
3859:{\displaystyle n}
3678:{\displaystyle g}
3631:{\displaystyle g}
3404:{\displaystyle B}
3384:{\displaystyle y}
3364:{\displaystyle x}
3344:{\displaystyle *}
3288:
3211:{\displaystyle A}
3191:{\displaystyle x}
3075:{\displaystyle A}
3055:{\displaystyle x}
2991:{\displaystyle B}
2971:{\displaystyle x}
2947:{\displaystyle f}
2805:{\displaystyle f}
2785:{\displaystyle f}
2765:{\displaystyle g}
2713:{\displaystyle B}
2693:{\displaystyle A}
2639:
2458:{\displaystyle F}
2438:{\displaystyle A}
2374:{\displaystyle F}
2354:{\displaystyle A}
2228:{\displaystyle f}
2128:{\displaystyle f}
1821:{\displaystyle f}
1801:{\displaystyle z}
1777:{\displaystyle f}
1703:{\displaystyle f}
1102:{\displaystyle f}
992:natural logarithm
814:is a map between
812:ring homomorphism
799:is a map between
784:is a map between
773:is a map between
744:{\displaystyle A}
679:for all elements
526:{\displaystyle B}
506:{\displaystyle A}
486:{\displaystyle k}
408:{\displaystyle f}
388:{\displaystyle A}
368:{\displaystyle y}
348:{\displaystyle x}
233:{\displaystyle B}
213:{\displaystyle A}
9694:
9673:
9655:
9616:
9604:
9592:
9556:
9550:
9544:
9529:
9523:
9508:Seymour Ginsburg
9505:
9499:
9481:
9475:
9474:
9448:
9442:
9435:
9429:
9428:
9389:
9378:
9377:
9365:
9354:
9343:
9342:
9313:
9298:
9276:
9227:
9192:
9186:
9185:
9176:. B.G. Teubner.
9167:
9150:
9063:
9057:
9014:
9008:
9006:
9004:
9003:
8998:
8986:
8984:
8983:
8978:
8965:
8913:
8911:
8910:
8905:
8889:
8887:
8886:
8881:
8869:
8867:
8866:
8861:
8859:
8858:
8831:
8829:
8828:
8823:
8811:
8809:
8808:
8803:
8791:
8789:
8788:
8783:
8781:
8780:
8758:
8756:
8755:
8750:
8742:
8725:
8710:
8708:
8707:
8702:
8690:
8688:
8687:
8682:
8680:
8679:
8657:
8655:
8654:
8649:
8641:
8624:
8612:
8610:
8609:
8604:
8601:
8596:
8580:
8578:
8577:
8572:
8569:
8564:
8551:
8546:
8521:
8519:
8518:
8513:
8510:
8505:
8489:
8487:
8486:
8481:
8463:
8461:
8460:
8455:
8425:
8423:
8422:
8417:
8405:
8403:
8402:
8397:
8385:
8383:
8382:
8377:
8365:
8363:
8362:
8357:
8354:
8349:
8333:
8331:
8330:
8325:
8313:
8311:
8310:
8305:
8302:
8297:
8277:
8275:
8274:
8269:
8267:
8266:
8238:
8236:
8235:
8230:
8179:
8177:
8176:
8171:
8168:
8163:
8150:
8145:
8123:
8121:
8120:
8115:
8113:
8112:
8096:
8094:
8093:
8088:
8086:
8085:
8064:formal languages
7863:two-sided ideals
7832:
7830:
7829:
7824:
7808:
7806:
7805:
7800:
7788:
7786:
7785:
7780:
7768:
7766:
7765:
7760:
7746:
7744:
7743:
7738:
7726:
7724:
7723:
7718:
7713:
7697:identity element
7694:
7692:
7691:
7686:
7660:
7658:
7657:
7652:
7646:
7627:
7625:
7624:
7619:
7607:
7605:
7604:
7599:
7587:
7585:
7584:
7579:
7567:
7565:
7564:
7559:
7547:
7545:
7544:
7539:
7527:
7525:
7524:
7519:
7471:
7469:
7468:
7463:
7451:
7449:
7448:
7443:
7441:
7436:
7418:
7416:
7415:
7410:
7394:
7392:
7391:
7386:
7370:
7368:
7367:
7362:
7350:
7348:
7347:
7342:
7306:
7304:
7303:
7298:
7280:
7278:
7277:
7272:
7260:
7258:
7257:
7252:
7237:
7235:
7234:
7229:
7202:Kernel (algebra)
7186:
7184:
7183:
7178:
7166:
7164:
7163:
7158:
7146:
7144:
7143:
7138:
7108:
7106:
7105:
7100:
7088:
7086:
7085:
7080:
7062:
7060:
7059:
7054:
7036:
7034:
7033:
7028:
7016:
7014:
7013:
7008:
6984:
6982:
6981:
6976:
6940:
6938:
6937:
6932:
6908:
6906:
6905:
6900:
6888:
6886:
6885:
6880:
6858:
6856:
6855:
6850:
6817:
6815:
6814:
6809:
6791:
6789:
6788:
6783:
6771:
6769:
6768:
6763:
6751:
6749:
6748:
6743:
6722:
6720:
6719:
6714:
6693:
6691:
6690:
6685:
6654:
6652:
6651:
6646:
6634:
6632:
6631:
6626:
6596:
6594:
6593:
6588:
6567:
6565:
6564:
6559:
6547:
6545:
6544:
6539:
6524:
6522:
6521:
6516:
6482:
6481:
6474:
6472:
6471:
6466:
6461:
6458:
6451:
6448:
6441:
6438:
6419:
6417:
6416:
6411:
6406:
6405:
6377:
6375:
6374:
6369:
6345:
6343:
6342:
6337:
6284:rational numbers
6231:
6229:
6228:
6223:
6205:
6203:
6202:
6197:
6167:
6165:
6164:
6159:
6147:
6145:
6144:
6139:
6127:
6125:
6124:
6119:
6107:
6105:
6104:
6099:
6087:
6085:
6084:
6079:
6051:right cancelable
6015:
6013:
6012:
6007:
5995:
5993:
5992:
5987:
5963:
5961:
5960:
5955:
5939:
5937:
5936:
5931:
5919:
5917:
5916:
5911:
5890:
5888:
5887:
5882:
5866:
5864:
5863:
5858:
5846:
5844:
5843:
5838:
5820:
5818:
5817:
5812:
5794:
5792:
5791:
5786:
5774:
5772:
5771:
5766:
5733:
5731:
5730:
5725:
5671:
5669:
5668:
5663:
5636:
5634:
5633:
5628:
5601:
5599:
5598:
5593:
5581:
5579:
5578:
5573:
5561:
5559:
5558:
5553:
5541:
5539:
5538:
5533:
5521:
5519:
5518:
5513:
5501:
5499:
5498:
5493:
5457:
5455:
5454:
5449:
5437:
5435:
5434:
5429:
5417:
5415:
5414:
5409:
5397:
5395:
5394:
5389:
5363:
5361:
5360:
5355:
5339:
5337:
5336:
5331:
5319:
5317:
5316:
5311:
5291:
5289:
5288:
5283:
5269:
5254:
5252:
5251:
5246:
5230:
5228:
5227:
5222:
5208:
5207:
5189:
5188:
5164:
5163:
5142:
5141:
5110:
5108:
5107:
5102:
5086:
5084:
5083:
5078:
5064:
5063:
5045:
5044:
5013:
5011:
5010:
5005:
4989:
4987:
4986:
4981:
4967:
4966:
4948:
4947:
4922:
4920:
4919:
4914:
4898:
4896:
4895:
4890:
4872:
4870:
4869:
4864:
4852:
4850:
4849:
4844:
4817:
4815:
4814:
4809:
4785:
4783:
4782:
4777:
4765:
4763:
4762:
4757:
4745:
4743:
4742:
4737:
4721:
4719:
4718:
4713:
4701:
4699:
4698:
4693:
4681:
4679:
4678:
4673:
4661:
4659:
4658:
4653:
4636:
4634:
4633:
4628:
4600:
4598:
4597:
4592:
4574:
4572:
4571:
4566:
4530:
4528:
4527:
4522:
4510:
4508:
4507:
4502:
4490:
4488:
4487:
4482:
4470:
4468:
4467:
4462:
4450:
4448:
4447:
4442:
4430:
4428:
4427:
4422:
4368:
4366:
4365:
4360:
4314:
4313:
4305:
4303:
4302:
4297:
4292:
4291:
4263:
4261:
4260:
4255:
4231:
4229:
4228:
4223:
4120:
4118:
4117:
4112:
4094:
4092:
4091:
4086:
4056:
4054:
4053:
4048:
4036:
4034:
4033:
4028:
4016:
4014:
4013:
4008:
3996:
3994:
3993:
3988:
3976:
3974:
3973:
3968:
3928:homomorphisms.
3888:
3886:
3885:
3880:
3865:
3863:
3862:
3857:
3841:
3839:
3838:
3833:
3819:
3818:
3768:ring isomorphism
3702:bicontinuous map
3684:
3682:
3681:
3676:
3660:
3658:
3657:
3652:
3637:
3635:
3634:
3629:
3614:
3612:
3611:
3606:
3589:
3588:
3546:
3545:
3485:
3484:
3439:
3438:
3410:
3408:
3407:
3402:
3390:
3388:
3387:
3382:
3370:
3368:
3367:
3362:
3350:
3348:
3347:
3342:
3330:
3324:
3322:
3321:
3316:
3311:
3310:
3289:
3286:
3281:
3280:
3252:
3250:
3249:
3244:
3217:
3215:
3214:
3209:
3197:
3195:
3194:
3189:
3177:
3175:
3174:
3169:
3148:
3146:
3145:
3140:
3116:
3114:
3113:
3108:
3081:
3079:
3078:
3073:
3061:
3059:
3058:
3053:
3041:
3039:
3038:
3033:
2997:
2995:
2994:
2989:
2977:
2975:
2974:
2969:
2953:
2951:
2950:
2945:
2933:
2931:
2930:
2925:
2859:
2857:
2856:
2851:
2811:
2809:
2808:
2803:
2791:
2789:
2788:
2783:
2771:
2769:
2768:
2763:
2751:
2749:
2748:
2743:
2719:
2717:
2716:
2711:
2699:
2697:
2696:
2691:
2676:
2674:
2673:
2668:
2663:
2662:
2640:
2637:
2634:
2633:
2599:
2597:
2596:
2591:
2561:
2559:
2558:
2553:
2464:
2462:
2461:
2456:
2444:
2442:
2441:
2436:
2420:
2418:
2417:
2412:
2380:
2378:
2377:
2372:
2360:
2358:
2357:
2352:
2334:
2332:
2331:
2326:
2299:
2297:
2296:
2291:
2234:
2232:
2231:
2226:
2214:
2212:
2211:
2206:
2189:
2174:
2172:
2171:
2166:
2149:
2135:from the monoid
2134:
2132:
2131:
2126:
2107:
2105:
2104:
2099:
2094:
2089:
2088:
2079:
2071:
2066:
2065:
2056:
2048:
2043:
2042:
2030:
2029:
2020:
2005:
1996:
1994:
1993:
1988:
1980:
1979:
1961:
1960:
1942:
1937:
1936:
1927:
1922:
1917:
1916:
1907:
1899:
1894:
1893:
1884:
1883:
1874:
1863:
1862:
1853:
1852:
1827:
1825:
1824:
1819:
1807:
1805:
1804:
1799:
1783:
1781:
1780:
1775:
1760:
1758:
1757:
1752:
1747:
1739:
1709:
1707:
1706:
1701:
1674:
1672:
1671:
1666:
1633:
1632:
1597:
1596:
1558:
1557:
1485:
1483:
1482:
1477:
1445:
1444:
1406:
1405:
1367:
1366:
1288:
1284:
1280:
1273:
1271:
1270:
1265:
1263:
1262:
1170:
1168:
1167:
1162:
1160:
1159:
1128:
1127:
1118:
1117:
1109:from the monoid
1108:
1106:
1105:
1100:
1070:
1068:
1067:
1062:
988:inverse function
982:
980:
979:
974:
969:
968:
959:
958:
946:
945:
917:
915:
914:
909:
907:
906:
760:identity element
756:0-ary operations
750:
748:
747:
742:
730:
728:
727:
722:
720:
719:
695:
694:
675:
673:
672:
667:
656:
655:
628:
627:
609:
608:
590:
589:
571:
570:
558:
557:
532:
530:
529:
524:
512:
510:
509:
504:
492:
490:
489:
484:
469:
467:
466:
461:
449:
447:
446:
441:
418:Formally, a map
414:
412:
411:
406:
394:
392:
391:
386:
374:
372:
371:
366:
354:
352:
351:
346:
331:
329:
328:
323:
262:binary operation
259:
257:
256:
251:
239:
237:
236:
231:
219:
217:
216:
211:
196:
194:
193:
188:
104:
103:
92:
86:
80:
74:
9702:
9701:
9697:
9696:
9695:
9693:
9692:
9691:
9677:
9676:
9671:
9645:
9635:Springer-Verlag
9633:, vol. 5,
9613:
9602:
9581:
9565:
9560:
9559:
9551:
9547:
9530:
9526:
9506:
9502:
9485:Gunther Schmidt
9482:
9478:
9463:
9449:
9445:
9436:
9432:
9417:
9407:Springer-Verlag
9405:. Vol. 5.
9390:
9381:
9374:
9363:
9355:
9346:
9332:
9314:
9301:
9290:
9284:
9193:
9189:
9168:
9164:
9159:
9154:
9153:
9064:
9060:
9024:. For example,
9015:
9011:
8992:
8989:
8988:
8972:
8969:
8968:
8966:
8962:
8957:
8924:
8899:
8896:
8895:
8875:
8872:
8871:
8854:
8850:
8848:
8845:
8844:
8817:
8814:
8813:
8797:
8794:
8793:
8776:
8772:
8764:
8761:
8760:
8738:
8721:
8719:
8716:
8715:
8696:
8693:
8692:
8675:
8671:
8663:
8660:
8659:
8637:
8620:
8618:
8615:
8614:
8613:that satisfies
8597:
8592:
8586:
8583:
8582:
8565:
8560:
8547:
8542:
8530:
8527:
8526:
8525:A homomorphism
8506:
8501:
8495:
8492:
8491:
8469:
8466:
8465:
8434:
8431:
8430:
8411:
8408:
8407:
8391:
8388:
8387:
8371:
8368:
8367:
8350:
8345:
8339:
8336:
8335:
8319:
8316:
8315:
8298:
8293:
8287:
8284:
8283:
8262:
8258:
8244:
8241:
8240:
8185:
8182:
8181:
8164:
8159:
8146:
8141:
8129:
8126:
8125:
8108:
8104:
8102:
8099:
8098:
8081:
8077:
8075:
8072:
8071:
8060:
8033:
8020:
8005:
7996:
7971:
7958:
7943:
7934:
7871:
7857:for kernels of
7849:for kernels of
7847:normal subgroup
7818:
7815:
7814:
7794:
7791:
7790:
7774:
7771:
7770:
7754:
7751:
7750:
7732:
7729:
7728:
7709:
7704:
7701:
7700:
7680:
7677:
7676:
7642:
7637:
7634:
7633:
7628:is necessarily
7613:
7610:
7609:
7593:
7590:
7589:
7573:
7570:
7569:
7553:
7550:
7549:
7533:
7530:
7529:
7477:
7474:
7473:
7457:
7454:
7453:
7437:
7432:
7427:
7424:
7423:
7404:
7401:
7400:
7380:
7377:
7376:
7356:
7353:
7352:
7351:. The relation
7312:
7309:
7308:
7307:if and only if
7286:
7283:
7282:
7266:
7263:
7262:
7246:
7243:
7242:
7211:
7208:
7207:
7204:
7198:
7193:
7172:
7169:
7168:
7152:
7149:
7148:
7114:
7111:
7110:
7094:
7091:
7090:
7068:
7065:
7064:
7042:
7039:
7038:
7022:
7019:
7018:
6990:
6987:
6986:
6946:
6943:
6942:
6914:
6911:
6910:
6894:
6891:
6890:
6874:
6871:
6870:
6823:
6820:
6819:
6797:
6794:
6793:
6777:
6774:
6773:
6757:
6754:
6753:
6728:
6725:
6724:
6699:
6696:
6695:
6664:
6661:
6660:
6640:
6637:
6636:
6602:
6599:
6598:
6573:
6570:
6569:
6553:
6550:
6549:
6533:
6530:
6529:
6498:
6495:
6494:
6487:
6457:
6447:
6437:
6435:
6432:
6431:
6401:
6397:
6383:
6380:
6379:
6351:
6348:
6347:
6319:
6316:
6315:
6211:
6208:
6207:
6173:
6170:
6169:
6168:, the equality
6153:
6150:
6149:
6133:
6130:
6129:
6113:
6110:
6109:
6093:
6090:
6089:
6061:
6058:
6057:
6049:are defined as
6043:category theory
6027:
6022:
6001:
5998:
5997:
5981:
5978:
5977:
5949:
5946:
5945:
5925:
5922:
5921:
5905:
5902:
5901:
5876:
5873:
5872:
5852:
5849:
5848:
5826:
5823:
5822:
5800:
5797:
5796:
5780:
5777:
5776:
5739:
5736:
5735:
5677:
5674:
5673:
5642:
5639:
5638:
5607:
5604:
5603:
5587:
5584:
5583:
5567:
5564:
5563:
5547:
5544:
5543:
5527:
5524:
5523:
5507:
5504:
5503:
5463:
5460:
5459:
5443:
5440:
5439:
5423:
5420:
5419:
5403:
5400:
5399:
5371:
5368:
5367:
5349:
5346:
5345:
5325:
5322:
5321:
5305:
5302:
5301:
5265:
5263:
5260:
5259:
5257:polynomial ring
5240:
5237:
5236:
5203:
5199:
5184:
5180:
5156:
5152:
5134:
5130:
5119:
5116:
5115:
5096:
5093:
5092:
5059:
5055:
5040:
5036:
5019:
5016:
5015:
4999:
4996:
4995:
4962:
4958:
4943:
4939:
4928:
4925:
4924:
4908:
4905:
4904:
4878:
4875:
4874:
4858:
4855:
4854:
4823:
4820:
4819:
4791:
4788:
4787:
4771:
4768:
4767:
4751:
4748:
4747:
4731:
4728:
4727:
4707:
4704:
4703:
4687:
4684:
4683:
4667:
4664:
4663:
4647:
4644:
4643:
4622:
4619:
4618:
4580:
4577:
4576:
4536:
4533:
4532:
4516:
4513:
4512:
4496:
4493:
4492:
4476:
4473:
4472:
4456:
4453:
4452:
4436:
4433:
4432:
4374:
4371:
4370:
4333:
4330:
4329:
4319:
4287:
4283:
4269:
4266:
4265:
4237:
4234:
4233:
4205:
4202:
4201:
4146:unary operation
4100:
4097:
4096:
4062:
4059:
4058:
4042:
4039:
4038:
4022:
4019:
4018:
4002:
3999:
3998:
3982:
3979:
3978:
3950:
3947:
3946:
3942:left cancelable
3933:category theory
3918:
3898:Évariste Galois
3874:
3871:
3870:
3851:
3848:
3847:
3814:
3810:
3808:
3805:
3804:
3780:
3772:square matrices
3710:
3670:
3667:
3666:
3643:
3640:
3639:
3623:
3620:
3619:
3584:
3580:
3541:
3537:
3480:
3476:
3434:
3430:
3419:
3416:
3415:
3396:
3393:
3392:
3391:of elements of
3376:
3373:
3372:
3356:
3353:
3352:
3336:
3333:
3332:
3326:
3306:
3302:
3287: and
3285:
3276:
3272:
3258:
3255:
3254:
3223:
3220:
3219:
3203:
3200:
3199:
3183:
3180:
3179:
3154:
3151:
3150:
3122:
3119:
3118:
3090:
3087:
3086:
3085:Conversely, if
3067:
3064:
3063:
3047:
3044:
3043:
3003:
3000:
2999:
2983:
2980:
2979:
2963:
2960:
2959:
2939:
2936:
2935:
2865:
2862:
2861:
2821:
2818:
2817:
2797:
2794:
2793:
2777:
2774:
2773:
2757:
2754:
2753:
2752:has an inverse
2725:
2722:
2721:
2705:
2702:
2701:
2685:
2682:
2681:
2658:
2654:
2636:
2629:
2625:
2611:
2608:
2607:
2573:
2570:
2569:
2535:
2532:
2531:
2514:category theory
2509:homomorphism.
2495:
2471:
2450:
2447:
2446:
2430:
2427:
2426:
2394:
2391:
2390:
2366:
2363:
2362:
2346:
2343:
2342:
2305:
2302:
2301:
2240:
2237:
2236:
2220:
2217:
2216:
2185:
2180:
2177:
2176:
2145:
2140:
2137:
2136:
2120:
2117:
2116:
2090:
2084:
2080:
2075:
2067:
2061:
2057:
2052:
2044:
2038:
2034:
2025:
2021:
2016:
2014:
2011:
2010:
2001:
1975:
1971:
1956:
1952:
1938:
1932:
1928:
1923:
1918:
1912:
1908:
1903:
1895:
1889:
1885:
1879:
1875:
1870:
1858:
1854:
1848:
1844:
1836:
1833:
1832:
1813:
1810:
1809:
1793:
1790:
1789:
1769:
1766:
1765:
1743:
1735:
1718:
1715:
1714:
1695:
1692:
1691:
1680:complex numbers
1627:
1626:
1621:
1615:
1614:
1609:
1599:
1598:
1591:
1590:
1585:
1579:
1578:
1573:
1563:
1562:
1552:
1551:
1543:
1537:
1536:
1531:
1518:
1517:
1497:
1494:
1493:
1439:
1438:
1433:
1427:
1426:
1421:
1411:
1410:
1400:
1399:
1394:
1388:
1387:
1382:
1372:
1371:
1361:
1360:
1349:
1343:
1342:
1337:
1321:
1320:
1297:
1294:
1293:
1286:
1282:
1278:
1257:
1256:
1251:
1245:
1244:
1239:
1229:
1228:
1211:
1208:
1207:
1197:matrix addition
1155:
1151:
1134:
1131:
1130:
1121:
1120:
1111:
1110:
1094:
1091:
1090:
1080:
1002:
999:
998:
964:
960:
954:
950:
935:
931:
929:
926:
925:
902:
898:
890:
887:
886:
790:0-ary operation
736:
733:
732:
715:
711:
690:
686:
684:
681:
680:
651:
647:
623:
619:
604:
600:
585:
581:
566:
562:
553:
549:
541:
538:
537:
518:
515:
514:
498:
495:
494:
478:
475:
474:
455:
452:
451:
423:
420:
419:
400:
397:
396:
380:
377:
376:
375:of elements of
360:
357:
356:
340:
337:
336:
335:for every pair
272:
269:
268:
245:
242:
241:
225:
222:
221:
205:
202:
201:
170:
167:
166:
152:
129:category theory
65:comes from the
28:
17:
12:
11:
5:
9700:
9690:
9689:
9675:
9674:
9669:
9656:
9643:
9617:
9611:
9593:
9579:
9564:
9561:
9558:
9557:
9553:Krieger (2006)
9545:
9524:
9500:
9476:
9461:
9443:
9430:
9415:
9379:
9372:
9344:
9330:
9318:Lattice theory
9299:
9297:
9296:
9288:
9282:
9259:(3): 443–468.
9244:
9187:
9161:
9160:
9158:
9155:
9152:
9151:
9105:), the number
9058:
9009:
9007:was used here.
8996:
8976:
8959:
8958:
8956:
8953:
8952:
8951:
8946:
8941:
8935:
8930:
8928:Diffeomorphism
8923:
8920:
8903:
8879:
8857:
8853:
8821:
8801:
8779:
8775:
8771:
8768:
8748:
8745:
8741:
8737:
8734:
8731:
8728:
8724:
8700:
8678:
8674:
8670:
8667:
8647:
8644:
8640:
8636:
8633:
8630:
8627:
8623:
8600:
8595:
8591:
8568:
8563:
8559:
8555:
8550:
8545:
8541:
8537:
8534:
8509:
8504:
8500:
8479:
8476:
8473:
8453:
8450:
8447:
8444:
8441:
8438:
8415:
8395:
8375:
8353:
8348:
8344:
8323:
8301:
8296:
8292:
8265:
8261:
8257:
8254:
8251:
8248:
8228:
8225:
8222:
8219:
8216:
8213:
8210:
8207:
8204:
8201:
8198:
8195:
8192:
8189:
8167:
8162:
8158:
8154:
8149:
8144:
8140:
8136:
8133:
8111:
8107:
8084:
8080:
8059:
8056:
8048:
8047:
8029:
8018:
8001:
7994:
7985:
7967:
7956:
7939:
7932:
7870:
7867:
7835:abelian groups
7822:
7798:
7789:, rather than
7778:
7758:
7736:
7716:
7712:
7708:
7684:
7650:
7645:
7641:
7617:
7597:
7577:
7557:
7537:
7517:
7514:
7511:
7508:
7505:
7502:
7499:
7496:
7493:
7490:
7487:
7484:
7481:
7461:
7440:
7435:
7431:
7408:
7384:
7371:is called the
7360:
7340:
7337:
7334:
7331:
7328:
7325:
7322:
7319:
7316:
7296:
7293:
7290:
7270:
7250:
7227:
7224:
7221:
7218:
7215:
7200:Main article:
7197:
7194:
7190:
7189:
7176:
7156:
7136:
7133:
7130:
7127:
7124:
7121:
7118:
7098:
7078:
7075:
7072:
7052:
7049:
7046:
7026:
7006:
7003:
7000:
6997:
6994:
6974:
6971:
6968:
6965:
6962:
6959:
6956:
6953:
6950:
6930:
6927:
6924:
6921:
6918:
6898:
6878:
6848:
6845:
6842:
6839:
6836:
6833:
6830:
6827:
6807:
6804:
6801:
6781:
6761:
6741:
6738:
6735:
6732:
6712:
6709:
6706:
6703:
6683:
6680:
6677:
6674:
6671:
6668:
6644:
6624:
6621:
6618:
6615:
6612:
6609:
6606:
6586:
6583:
6580:
6577:
6557:
6537:
6514:
6511:
6508:
6505:
6502:
6489:
6488:
6485:
6480:
6476:
6475:
6464:
6455:
6445:
6409:
6404:
6400:
6396:
6393:
6390:
6387:
6367:
6364:
6361:
6358:
6355:
6335:
6332:
6329:
6326:
6323:
6261:linear algebra
6249:abelian groups
6221:
6218:
6215:
6195:
6192:
6189:
6186:
6183:
6180:
6177:
6157:
6137:
6117:
6097:
6077:
6074:
6071:
6068:
6065:
6026:
6023:
6019:
6018:
6005:
5985:
5953:
5944:built up from
5929:
5909:
5880:
5867:is injective.
5856:
5836:
5833:
5830:
5810:
5807:
5804:
5784:
5764:
5761:
5758:
5755:
5752:
5749:
5746:
5743:
5723:
5720:
5717:
5714:
5711:
5708:
5705:
5702:
5699:
5696:
5693:
5690:
5687:
5684:
5681:
5661:
5658:
5655:
5652:
5649:
5646:
5626:
5623:
5620:
5617:
5614:
5611:
5591:
5571:
5551:
5531:
5511:
5491:
5488:
5485:
5482:
5479:
5476:
5473:
5470:
5467:
5447:
5427:
5407:
5387:
5384:
5381:
5378:
5375:
5353:
5329:
5309:
5281:
5278:
5275:
5272:
5268:
5244:
5220:
5217:
5214:
5211:
5206:
5202:
5198:
5195:
5192:
5187:
5183:
5179:
5176:
5173:
5170:
5167:
5162:
5159:
5155:
5151:
5148:
5145:
5140:
5137:
5133:
5129:
5126:
5123:
5100:
5076:
5073:
5070:
5067:
5062:
5058:
5054:
5051:
5048:
5043:
5039:
5035:
5032:
5029:
5026:
5023:
5003:
4979:
4976:
4973:
4970:
4965:
4961:
4957:
4954:
4951:
4946:
4942:
4938:
4935:
4932:
4912:
4888:
4885:
4882:
4862:
4842:
4839:
4836:
4833:
4830:
4827:
4807:
4804:
4801:
4798:
4795:
4775:
4755:
4735:
4711:
4691:
4671:
4651:
4626:
4590:
4587:
4584:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4520:
4500:
4480:
4460:
4440:
4420:
4417:
4414:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4321:
4320:
4317:
4312:
4295:
4290:
4286:
4282:
4279:
4276:
4273:
4253:
4250:
4247:
4244:
4241:
4221:
4218:
4215:
4212:
4209:
4110:
4107:
4104:
4084:
4081:
4078:
4075:
4072:
4069:
4066:
4046:
4026:
4006:
3986:
3966:
3963:
3960:
3957:
3954:
3917:
3914:
3878:
3855:
3831:
3828:
3825:
3822:
3817:
3813:
3779:
3776:
3709:
3706:
3694:continuous map
3674:
3650:
3647:
3627:
3616:
3615:
3604:
3601:
3598:
3595:
3592:
3587:
3583:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3544:
3540:
3536:
3533:
3530:
3527:
3524:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3491:
3488:
3483:
3479:
3475:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3448:
3445:
3442:
3437:
3433:
3429:
3426:
3423:
3400:
3380:
3360:
3340:
3314:
3309:
3305:
3301:
3298:
3295:
3292:
3284:
3279:
3275:
3271:
3268:
3265:
3262:
3242:
3239:
3236:
3233:
3230:
3227:
3207:
3187:
3167:
3164:
3161:
3158:
3138:
3135:
3132:
3129:
3126:
3106:
3103:
3100:
3097:
3094:
3071:
3051:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
2987:
2967:
2958:, as, for any
2943:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2801:
2781:
2761:
2741:
2738:
2735:
2732:
2729:
2709:
2689:
2678:
2677:
2666:
2661:
2657:
2653:
2650:
2647:
2644:
2632:
2628:
2624:
2621:
2618:
2615:
2601:
2600:
2589:
2586:
2583:
2580:
2577:
2563:
2562:
2551:
2548:
2545:
2542:
2539:
2494:
2491:
2470:
2467:
2454:
2434:
2410:
2407:
2404:
2401:
2398:
2383:quadratic form
2370:
2350:
2324:
2321:
2318:
2315:
2312:
2309:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2224:
2204:
2201:
2198:
2195:
2192:
2188:
2184:
2175:to the monoid
2164:
2161:
2158:
2155:
2152:
2148:
2144:
2124:
2109:
2108:
2097:
2093:
2087:
2083:
2078:
2074:
2070:
2064:
2060:
2055:
2051:
2047:
2041:
2037:
2033:
2028:
2024:
2019:
1998:
1997:
1986:
1983:
1978:
1974:
1970:
1967:
1964:
1959:
1955:
1951:
1948:
1945:
1941:
1935:
1931:
1926:
1921:
1915:
1911:
1906:
1902:
1898:
1892:
1888:
1882:
1878:
1873:
1869:
1866:
1861:
1857:
1851:
1847:
1843:
1840:
1817:
1797:
1786:absolute value
1773:
1762:
1761:
1750:
1746:
1742:
1738:
1734:
1731:
1728:
1725:
1722:
1699:
1676:
1675:
1664:
1661:
1658:
1655:
1652:
1648:
1645:
1642:
1639:
1636:
1631:
1625:
1622:
1620:
1617:
1616:
1613:
1610:
1608:
1605:
1604:
1602:
1595:
1589:
1586:
1584:
1581:
1580:
1577:
1574:
1572:
1569:
1568:
1566:
1561:
1556:
1550:
1547:
1544:
1542:
1539:
1538:
1535:
1532:
1530:
1527:
1524:
1523:
1521:
1516:
1513:
1510:
1507:
1504:
1501:
1487:
1486:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1443:
1437:
1434:
1432:
1429:
1428:
1425:
1422:
1420:
1417:
1416:
1414:
1409:
1404:
1398:
1395:
1393:
1390:
1389:
1386:
1383:
1381:
1378:
1377:
1375:
1370:
1365:
1359:
1356:
1353:
1350:
1348:
1345:
1344:
1341:
1338:
1336:
1333:
1330:
1327:
1326:
1324:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1275:
1274:
1261:
1255:
1252:
1250:
1247:
1246:
1243:
1240:
1238:
1235:
1234:
1232:
1227:
1224:
1221:
1218:
1215:
1158:
1154:
1150:
1147:
1144:
1141:
1138:
1119:to the monoid
1098:
1079:
1076:
1072:
1071:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
984:
983:
972:
967:
963:
957:
953:
949:
944:
941:
938:
934:
919:
918:
905:
901:
897:
894:
869:
868:
857:
846:
831:
808:
793:
778:
740:
718:
714:
710:
707:
704:
701:
698:
693:
689:
677:
676:
665:
662:
659:
654:
650:
646:
643:
640:
637:
634:
631:
626:
622:
618:
615:
612:
607:
603:
599:
596:
593:
588:
584:
580:
577:
574:
569:
565:
561:
556:
552:
548:
545:
522:
502:
482:
459:
439:
436:
433:
430:
427:
404:
384:
364:
344:
333:
332:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
249:
229:
209:
186:
183:
180:
177:
174:
151:
148:
118:linear algebra
15:
9:
6:
4:
3:
2:
9699:
9688:
9685:
9684:
9682:
9672:
9666:
9662:
9657:
9654:
9650:
9646:
9644:0-387-90036-5
9640:
9636:
9632:
9628:
9627:
9622:
9618:
9614:
9608:
9601:
9600:
9594:
9590:
9586:
9582:
9576:
9572:
9567:
9566:
9554:
9549:
9542:
9541:3-540-61486-9
9538:
9534:
9528:
9521:
9520:0-7204-2506-9
9517:
9513:
9509:
9504:
9498:
9494:
9490:
9486:
9480:
9472:
9468:
9464:
9458:
9454:
9447:
9440:
9434:
9426:
9422:
9418:
9416:0-387-90036-5
9412:
9408:
9404:
9400:
9399:
9394:
9388:
9386:
9384:
9375:
9369:
9362:
9361:
9353:
9351:
9349:
9341:
9337:
9333:
9327:
9323:
9319:
9312:
9310:
9308:
9306:
9304:
9294:
9286:
9279:
9274:
9270:
9266:
9262:
9258:
9255:(in German).
9254:
9250:
9245:
9242:
9238:
9235:holoedrischen
9234:
9230:
9225:
9221:
9217:
9213:
9209:
9206:(in German).
9205:
9201:
9196:
9195:
9191:
9183:
9179:
9175:
9174:
9166:
9162:
9148:
9144:
9140:
9136:
9132:
9128:
9124:
9120:
9116:
9112:
9108:
9104:
9100:
9096:
9092:
9088:
9084:
9080:
9076:
9072:
9068:
9062:
9055:
9051:
9047:
9043:
9039:
9035:
9031:
9027:
9023:
9022:concatenation
9019:
9013:
8994:
8974:
8964:
8960:
8950:
8949:Quasimorphism
8947:
8945:
8942:
8939:
8936:
8934:
8931:
8929:
8926:
8925:
8919:
8917:
8916:concatenation
8894:generated by
8893:
8855:
8841:
8839:
8835:
8819:
8799:
8777:
8769:
8766:
8746:
8743:
8732:
8726:
8713:
8698:
8676:
8668:
8665:
8645:
8642:
8631:
8625:
8598:
8593:
8566:
8561:
8548:
8543:
8535:
8532:
8523:
8507:
8502:
8477:
8474:
8471:
8451:
8448:
8442:
8436:
8428:
8413:
8406:is called an
8393:
8373:
8351:
8346:
8321:
8299:
8294:
8281:
8263:
8255:
8252:
8249:
8246:
8223:
8217:
8211:
8205:
8202:
8196:
8193:
8187:
8165:
8160:
8147:
8142:
8134:
8131:
8124:, a function
8109:
8082:
8069:
8065:
8055:
8053:
8045:
8041:
8037:
8032:
8028:
8024:
8017:
8013:
8009:
8004:
8000:
7993:
7989:
7986:
7983:
7979:
7975:
7970:
7966:
7962:
7955:
7951:
7947:
7942:
7938:
7931:
7927:
7923:
7920:
7919:
7918:
7916:
7912:
7908:
7905:is a mapping
7904:
7900:
7896:
7892:
7888:
7884:
7880:
7876:
7866:
7864:
7860:
7856:
7852:
7848:
7844:
7840:
7839:vector spaces
7836:
7820:
7812:
7796:
7776:
7756:
7749:
7734:
7714:
7710:
7706:
7698:
7682:
7675:
7671:
7666:
7664:
7648:
7643:
7639:
7631:
7615:
7595:
7575:
7555:
7535:
7512:
7509:
7506:
7500:
7494:
7488:
7482:
7459:
7438:
7433:
7429:
7422:
7406:
7398:
7382:
7374:
7358:
7335:
7329:
7326:
7320:
7314:
7294:
7291:
7288:
7268:
7248:
7241:
7225:
7219:
7216:
7213:
7203:
7188:
7174:
7154:
7134:
7131:
7128:
7125:
7122:
7119:
7116:
7096:
7076:
7073:
7070:
7050:
7047:
7044:
7024:
7004:
6998:
6995:
6992:
6972:
6969:
6960:
6954:
6948:
6928:
6922:
6919:
6916:
6896:
6876:
6868:
6864:
6859:
6846:
6843:
6840:
6837:
6834:
6831:
6828:
6825:
6805:
6802:
6799:
6779:
6759:
6736:
6730:
6710:
6707:
6704:
6701:
6681:
6678:
6672:
6666:
6658:
6642:
6622:
6616:
6613:
6610:
6607:
6604:
6597:, and define
6581:
6575:
6555:
6535:
6526:
6512:
6506:
6503:
6500:
6491:
6490:
6484:
6483:
6479:
6462:
6430:
6429:
6428:
6425:
6423:
6407:
6402:
6398:
6394:
6391:
6388:
6385:
6365:
6359:
6356:
6353:
6333:
6327:
6324:
6321:
6313:
6312:right inverse
6309:
6308:
6302:
6300:
6296:
6292:
6287:
6285:
6281:
6277:
6273:
6268:
6266:
6262:
6258:
6254:
6250:
6246:
6245:vector spaces
6242:
6238:
6233:
6219:
6216:
6213:
6193:
6190:
6187:
6184:
6181:
6178:
6175:
6155:
6135:
6115:
6095:
6075:
6069:
6066:
6063:
6055:
6052:
6048:
6044:
6040:
6036:
6032:
6017:
6003:
5983:
5975:
5971:
5967:
5951:
5943:
5927:
5907:
5899:
5895:
5894:
5878:
5868:
5854:
5847:. Therefore,
5834:
5831:
5828:
5808:
5805:
5802:
5782:
5762:
5759:
5756:
5753:
5750:
5747:
5744:
5741:
5715:
5709:
5703:
5700:
5691:
5685:
5679:
5659:
5656:
5650:
5644:
5624:
5621:
5615:
5609:
5589:
5569:
5549:
5529:
5509:
5486:
5480:
5477:
5471:
5465:
5445:
5425:
5405:
5385:
5379:
5376:
5373:
5365:
5351:
5341:
5327:
5307:
5299:
5295:
5294:vector spaces
5279:
5273:
5258:
5242:
5234:
5218:
5212:
5209:
5204:
5200:
5196:
5193:
5190:
5185:
5181:
5177:
5174:
5171:
5168:
5165:
5160:
5157:
5153:
5149:
5146:
5143:
5138:
5135:
5131:
5127:
5124:
5114:
5098:
5090:
5074:
5068:
5065:
5060:
5056:
5052:
5049:
5046:
5041:
5037:
5033:
5030:
5027:
5024:
5001:
4993:
4977:
4971:
4968:
4963:
4959:
4955:
4952:
4949:
4944:
4940:
4936:
4933:
4910:
4902:
4883:
4860:
4840:
4837:
4831:
4825:
4805:
4799:
4796:
4793:
4773:
4753:
4733:
4725:
4709:
4689:
4669:
4649:
4641:
4637:
4624:
4616:
4610:
4608:
4604:
4588:
4585:
4582:
4559:
4553:
4550:
4544:
4538:
4518:
4498:
4478:
4458:
4438:
4412:
4406:
4400:
4397:
4388:
4382:
4376:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4327:
4323:
4322:
4316:
4315:
4311:
4309:
4293:
4288:
4284:
4280:
4277:
4274:
4271:
4251:
4245:
4242:
4239:
4219:
4213:
4210:
4207:
4199:
4195:
4194:
4188:
4186:
4182:
4181:vector spaces
4178:
4174:
4170:
4166:
4162:
4158:
4154:
4149:
4147:
4143:
4139:
4135:
4131:
4127:
4122:
4108:
4105:
4102:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4044:
4024:
4004:
3984:
3964:
3958:
3955:
3952:
3944:
3943:
3938:
3934:
3929:
3927:
3923:
3922:monomorphisms
3913:
3911:
3910:Galois theory
3907:
3903:
3899:
3895:
3890:
3876:
3869:
3853:
3846:of dimension
3845:
3826:
3820:
3815:
3811:
3803:
3798:
3796:
3792:
3787:
3785:
3775:
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3745:
3740:
3738:
3734:
3729:
3727:
3723:
3719:
3715:
3705:
3703:
3699:
3698:homeomorphism
3695:
3691:
3686:
3672:
3664:
3648:
3645:
3625:
3602:
3596:
3590:
3585:
3581:
3574:
3568:
3565:
3553:
3547:
3542:
3538:
3531:
3525:
3519:
3513:
3510:
3498:
3492:
3486:
3481:
3477:
3467:
3461:
3455:
3449:
3446:
3440:
3435:
3431:
3427:
3421:
3414:
3413:
3412:
3398:
3378:
3358:
3338:
3329:
3312:
3307:
3303:
3299:
3296:
3293:
3290:
3282:
3277:
3273:
3269:
3266:
3263:
3260:
3240:
3237:
3231:
3225:
3205:
3185:
3162:
3156:
3136:
3130:
3127:
3124:
3104:
3098:
3095:
3092:
3083:
3069:
3049:
3023:
3017:
3011:
3008:
3005:
2985:
2965:
2957:
2941:
2921:
2918:
2909:
2903:
2897:
2894:
2885:
2879:
2873:
2870:
2867:
2844:
2838:
2835:
2829:
2823:
2815:
2799:
2779:
2759:
2739:
2733:
2730:
2727:
2707:
2687:
2664:
2659:
2655:
2651:
2648:
2645:
2642:
2630:
2626:
2622:
2619:
2616:
2613:
2606:
2605:
2604:
2587:
2581:
2578:
2575:
2568:
2567:
2566:
2549:
2543:
2540:
2537:
2530:
2529:
2528:
2525:
2523:
2519:
2515:
2510:
2508:
2504:
2500:
2487:
2482:
2478:
2476:
2466:
2452:
2432:
2424:
2408:
2402:
2399:
2396:
2388:
2384:
2368:
2361:over a field
2348:
2341:
2336:
2322:
2319:
2313:
2307:
2284:
2278:
2275:
2269:
2263:
2260:
2254:
2251:
2248:
2242:
2222:
2199:
2196:
2193:
2190:
2159:
2156:
2153:
2150:
2122:
2115:homomorphism
2114:
2095:
2085:
2081:
2072:
2062:
2058:
2049:
2039:
2035:
2031:
2026:
2022:
2009:
2008:
2007:
2004:
1984:
1976:
1972:
1965:
1957:
1953:
1946:
1943:
1933:
1929:
1913:
1909:
1900:
1890:
1886:
1880:
1876:
1867:
1859:
1855:
1849:
1845:
1838:
1831:
1830:
1829:
1815:
1795:
1787:
1771:
1748:
1740:
1732:
1726:
1720:
1713:
1712:
1711:
1697:
1689:
1685:
1681:
1662:
1656:
1650:
1643:
1637:
1634:
1629:
1623:
1618:
1611:
1606:
1600:
1593:
1587:
1582:
1575:
1570:
1564:
1559:
1554:
1548:
1545:
1540:
1533:
1528:
1525:
1519:
1514:
1508:
1505:
1499:
1492:
1491:
1490:
1470:
1464:
1461:
1455:
1449:
1446:
1441:
1435:
1430:
1423:
1418:
1412:
1407:
1402:
1396:
1391:
1384:
1379:
1373:
1368:
1363:
1357:
1354:
1351:
1346:
1339:
1334:
1331:
1328:
1322:
1317:
1311:
1308:
1305:
1299:
1292:
1291:
1290:
1259:
1253:
1248:
1241:
1236:
1230:
1225:
1219:
1213:
1206:
1205:
1204:
1202:
1198:
1194:
1190:
1186:
1178:
1174:
1156:
1152:
1148:
1142:
1136:
1129:, defined by
1125:
1115:
1096:
1089:homomorphism
1088:
1084:
1075:
1058:
1052:
1046:
1043:
1040:
1034:
1028:
1025:
1022:
1016:
1013:
1007:
1004:
997:
996:
995:
994:, satisfies
993:
989:
970:
965:
961:
955:
951:
947:
942:
939:
936:
932:
924:
923:
922:
903:
899:
892:
885:
884:
883:
882:
878:
873:
866:
862:
858:
855:
851:
847:
844:
840:
839:vector spaces
836:
832:
830:homomorphism.
829:
825:
821:
817:
813:
809:
806:
802:
798:
794:
791:
787:
783:
779:
776:
772:
768:
767:
766:
765:For example:
763:
761:
757:
752:
738:
716:
712:
708:
705:
702:
699:
696:
691:
687:
663:
652:
648:
641:
638:
635:
632:
624:
620:
613:
605:
601:
597:
586:
582:
578:
575:
572:
567:
563:
554:
550:
543:
536:
535:
534:
520:
500:
480:
473:
457:
437:
431:
428:
425:
416:
402:
382:
362:
342:
316:
310:
307:
301:
295:
292:
286:
283:
280:
274:
267:
266:
265:
263:
247:
227:
207:
200:
184:
178:
175:
172:
165:
161:
157:
147:
145:
141:
137:
132:
130:
126:
121:
119:
115:
110:
109:(1849–1925).
108:
98:
97:
91:
85:
79:
73:
68:
64:
60:
59:vector spaces
56:
52:
48:
44:
41:
37:
33:
26:
25:homeomorphism
22:
9660:
9624:
9598:
9570:
9548:
9532:
9527:
9511:
9503:
9488:
9479:
9452:
9446:
9438:
9433:
9396:
9359:
9317:
9292:
9280:
9278:From p. 466:
9256:
9252:
9240:
9236:
9232:
9207:
9203:
9190:
9172:
9165:
9146:
9142:
9138:
9134:
9130:
9126:
9122:
9118:
9114:
9110:
9106:
9102:
9098:
9094:
9090:
9086:
9082:
9078:
9074:
9070:
9066:
9061:
9053:
9049:
9045:
9041:
9037:
9033:
9029:
9025:
9012:
8963:
8842:
8837:
8833:
8711:
8691:is called a
8524:
8426:
8280:homomorphism
8279:
8278:is called a
8067:
8061:
8049:
8043:
8039:
8035:
8034:)) for each
8030:
8026:
8022:
8015:
8011:
8007:
8002:
7998:
7991:
7987:
7981:
7977:
7973:
7972:)) for each
7968:
7964:
7960:
7953:
7949:
7945:
7940:
7936:
7929:
7925:
7921:
7914:
7910:
7906:
7902:
7898:
7895:homomorphism
7894:
7890:
7886:
7882:
7878:
7875:model theory
7872:
7667:
7421:quotient set
7372:
7205:
6860:
6723:except that
6527:
6492:
6477:
6426:
6421:
6305:
6303:
6288:
6269:
6236:
6234:
6050:
6047:epimorphisms
6035:epimorphisms
6034:
6028:
5870:
5869:
5343:
5342:
5340:as a basis.
4613:
4611:
4325:
4324:
4308:monomorphism
4307:
4198:left inverse
4191:
4189:
4150:
4126:monomorphism
4125:
4123:
3940:
3930:
3919:
3916:Monomorphism
3891:
3844:vector space
3799:
3788:
3784:automorphism
3781:
3778:Automorphism
3744:vector space
3741:
3730:
3714:endomorphism
3711:
3708:Endomorphism
3687:
3617:
3327:
3084:
2679:
2602:
2564:
2526:
2520:that has an
2511:
2496:
2472:
2386:
2337:
2110:
2002:
1999:
1763:
1677:
1488:
1276:
1185:real numbers
1182:
1123:
1113:
1073:
985:
920:
877:real numbers
874:
870:
823:
764:
753:
678:
417:
334:
197:between two
153:
144:automorphism
140:endomorphism
133:
122:
111:
63:homomorphism
62:
61:). The word
45:between two
36:homomorphism
35:
29:
21:holomorphism
9018:Kleene star
8892:free monoid
7238:defines an
7063:, and thus
6659:, and that
6422:epimorphism
6237:epimorphism
6025:Epimorphism
5821:, and thus
4615:free object
4575:, and thus
3906:polynomials
3756:free module
3720:equals the
2499:isomorphism
2493:Isomorphism
2385:, called a
867:operations.
136:isomorphism
114:linear maps
107:Felix Klein
9653:0232.18001
9563:References
9471:0962.16026
9462:0824704819
9425:0232.18001
9241:holohedric
8838:projection
8792:(that is,
8180:such that
8006:) implies
7917:such that
7630:isomorphic
7395:. It is a
6772:. Clearly
6694:for every
6635:such that
6378:such that
6272:semigroups
6039:surjective
5966:identities
5896:(see also
5734:, one has
5602:such that
4901:semigroups
4873:is simply
4818:such that
4638:. Given a
4431:for every
4264:such that
4161:semigroups
3766:induces a
3758:of finite
3411:, one has
3253:. One has
3218:such that
2998:, one has
2956:surjective
2603:such that
2235:amount to
2000:Note that
1177:surjective
1175:, but not
921:satisfies
835:linear map
775:semigroups
160:operations
150:Definition
9687:Morphisms
9589:262693179
9273:120022176
9224:121524108
9157:Citations
9137:). Hence
8902:Σ
8878:Σ
8856:∗
8852:Σ
8774:Σ
8770:∈
8673:Σ
8669:∈
8599:∗
8590:Σ
8567:∗
8558:Σ
8554:→
8549:∗
8540:Σ
8536::
8508:∗
8499:Σ
8478:ε
8475:≠
8452:ε
8449:≠
8414:ε
8374:ε
8352:∗
8343:Σ
8300:∗
8291:Σ
8260:Σ
8256:∈
8166:∗
8157:Σ
8153:→
8148:∗
8139:Σ
8135::
8106:Σ
8079:Σ
8068:morphisms
7797:∼
7649:∼
7536:∗
7510:∗
7489:∗
7439:∼
7359:∼
7292:∼
7249:∼
7223:→
7132:∘
7120:∘
7074:≠
7048:≠
7002:→
6996::
6926:→
6920::
6867:zero maps
6863:cokernels
6841:∘
6829:∘
6803:≠
6705:∈
6620:→
6614::
6510:→
6504::
6454:⟹
6444:⟹
6389:∘
6363:→
6357::
6331:→
6325::
6191:∘
6179:∘
6073:→
6054:morphisms
5757:∘
5745:∘
5383:→
5377::
5213:…
5194:…
5158:−
5147:…
5136:−
5125:…
5069:…
5050:…
4972:…
4953:…
4803:→
4351:∘
4339:∘
4275:∘
4249:→
4243::
4217:→
4211::
4134:varieties
4080:∘
4068:∘
3962:→
3926:injective
3821:
3760:dimension
3735:, form a
3646:∗
3582:∗
3539:∗
3478:∗
3432:∗
3339:∗
3294:∘
3283:
3264:∘
3134:→
3102:→
2814:injective
2737:→
2646:∘
2617:∘
2585:→
2547:→
2507:bijective
2475:morphisms
2406:→
2276:×
2194:×
2050:≠
1764:That is,
1173:injective
1047:
1029:
1008:
896:↦
636:…
602:μ
576:…
551:μ
458:μ
435:→
308:⋅
284:⋅
248:⋅
182:→
57:, or two
9681:Category
9623:(1971),
9487:, 2010.
9395:(1971).
9210:: 1–82.
9182:29857037
9141:= 0 and
8944:Morphism
8922:See also
8843:The set
8759:for all
8712:-uniform
8658:for all
8464:for all
8239:for all
6280:integers
6206:implies
4603:category
4369:one has
4095:implies
3939:that is
3937:morphism
3746:or of a
3733:category
3726:morphism
3722:codomain
2860:implies
2518:morphism
2501:between
1808:. Then
1193:matrices
1171:. It is
1078:Examples
264:), then
125:morphism
9340:0598630
9048:) with
7889:be two
7843:modules
7695:of the
6655:is the
6253:modules
6031:algebra
5940:of the
5893:variety
5298:modules
5255:is the
5111:is the
4992:monoids
4640:variety
4185:modules
4165:monoids
4057:, then
3866:over a
3750:form a
2772:, then
2522:inverse
1784:is the
1682:form a
1126:, ×, 1)
1116:, +, 0)
865:algebra
854:modules
805:inverse
786:monoids
96:ähnlich
32:algebra
9667:
9651:
9641:
9609:
9587:
9577:
9555:p. 287
9539:
9518:
9495:
9469:
9459:
9423:
9413:
9370:
9338:
9328:
9271:
9222:
9180:
8834:coding
7855:ideals
7811:kernel
7419:. The
7373:kernel
7196:Kernel
6985:. Let
6909:, and
6257:groups
5891:for a
5366:: let
5089:groups
4899:; for
4177:fields
4169:groups
4157:magmas
4130:fields
3894:fields
3748:module
3737:monoid
3718:domain
3042:, and
2934:, and
2381:has a
2113:monoid
1277:where
1187:are a
1087:Monoid
990:, the
801:groups
90:morphe
53:, two
51:groups
9603:(PDF)
9364:(PDF)
9269:S2CID
9220:S2CID
9194:See:
8955:Notes
8836:or a
8429:when
8314:. If
7944:)) =
7897:from
7670:group
6282:into
6276:rings
5672:. As
5562:from
5458:such
5233:rings
4511:. If
4328:: If
4173:rings
3902:roots
3868:field
3791:group
3764:basis
3663:arity
2816:, as
1684:group
816:rings
472:arity
142:, an
138:, an
84:μορφή
78:homos
55:rings
38:is a
9665:ISBN
9639:ISBN
9607:ISBN
9585:OCLC
9575:ISBN
9537:ISBN
9516:ISBN
9493:ISBN
9457:ISBN
9411:ISBN
9368:ISBN
9326:ISBN
9289:(63)
9283:(63)
9178:OCLC
9089:) =
9081:) =
8987:and
8366:and
8097:and
8021:),…,
7959:),…,
7853:and
7841:and
6818:and
6493:Let
6297:and
6274:and
6263:and
6241:sets
5637:and
5542:and
5418:and
5292:for
4491:and
4183:and
4153:sets
3752:ring
3618:and
2700:and
2387:norm
2300:and
1199:and
1189:ring
1183:The
824:ring
533:if
513:and
199:sets
102:ὁμός
72:ὁμός
34:, a
9649:Zbl
9467:Zbl
9421:Zbl
9261:doi
9231::
9212:doi
8581:on
8490:in
8282:on
8042:in
7997:,…,
7980:in
7935:,…,
7901:to
7873:In
7865:).
7813:of
7748:mod
7632:to
7588:in
7548:of
7399:on
7375:of
7281:by
7261:on
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