700:. The sentence is undoubtful correct if they are sets, modules or groups. If Birkhoff allows more general categories for them, this definition is clearly in contradiction with the present standard of mathematics. One must keep in mind that when he wrote his book, the language of categories was not yet popularized in all mathematics. Therefore, this book is not reliable for the modern terminology. Moreover, for some authors the term "homomorphism" is used only for algebraic structures, while, for others, this is a synonymous of "morphism". As we cannot know the background of the readers, we must have a formulation which is correct, whichever is this background. This is the reason for giving definitions that are correct in any case, and then explain that "in general" isomorphism = bijective, monomorphism = injective and epimorphism = surjective, with a short explanation of what "in general" means in each case. This is what I have tried to do with my edit. In any case, I am
750:, and that most definitions given in textbooks are about specific kinds of homomorphisms, commonly, homomorphisms of groups, vector spaces and modules. As all definitions are equivalent in these cases, these textbooks are not useful for sourcing a general definition. As far as I know, there exist three theories providing a general definition of isomorphism, epimorphism and monomorphism. The first one is Bourbaki's definition of algebraic structures. It has never been accepted by the mathematicians community, nor really used by other authors, and thus cannot be used as a source. The second one is the theory of
853:, and cannot be introduced in Knowledge. IMO, the problem with the present state of section "Types" is that the bulleted list implies a very short description, and that this description, although correct (in my opinion) it too short for being clearly understood by a non-expert reader. Therefore, I suggest to merge the subsection in the section and to split the section in subsections, one for each item of the bulleted list. This would allow to clarify everything, to provide as many examples as needed, and so on.
95:
85:
64:
31:
876:
surjective one. It is easier to present that information if we first split into subsections like e.g. "category theory approaches" and e.g. "set theory approaches". The suffix "approaches" is intended to reflect that the notions inspired by category theory are nowadays used beyond that field, in particular in abstract algebra (this was Taku's and your point above, if I understand you right); maybe you have better title suggestions to express this intention. -
332:
with this clarity in the article. I'll try to reword the definition to make it clearer that the structure that is preserved by a homomorphism is nominated (and only implied as a default), and not specified by the objects. In particular, it should be clear that a "monoid homomorphism" is a homomorphism that preserves all structure that is required by the definition of monoids. My attempt may break subtler aspects, please feel free to rescue the correction. —
505:"... However, the definitions in category theory are somewhat technical. In the important special case of module homomorphisms, and for some other classes of homomorphisms, there are much simpler descriptions, as follows: ... An epimorphism (sometimes called a cover) is a surjective homomorphism. ... For endomorphisms and automorphisms, the descriptions above coincide with the category theoretic definitions; the first three descriptions do not. ..."
176:
1335:, one use the term of variety, but the term is too restricted, as fields do not form a variety. Moreover most people interested in algebra do not know the terminology of universal algebra. Thus, for not being too technical, some caution is needed for using the terminology of universal algebra in this article. Because of this, I am not fully satisfied by some of my sentences, and some help would be welcome.
22:
982:", given in the linked article, is not recalled here. Thus it is unclear that, here, an algebraic structure has a underlying set, and only one, while, for many mathematicians, "algebraic structure" is either not clearly defined or is much larger concept. Thus, we have first to state clearly that in this article, an algebraic structure has one and only one underlying set.
1331:, an algebraic structure is a triple consisting of a set, the operations and the axioms satisfied by these operations. But it seems that there is no commonly accepted term for the class of algebraic structures sharing the same operations and the same axioms. Bourbaki uses "species of algebraic structure", but it seems that this terminology is generally ignored. In
2080:
also should alter the definition of a general homomorphism in this article; the general definition given here does not line up with the definition of ring homomorphism given in its corresponding article. Unless of course the definition given on that wikipedia page is really of a non-zero homomorphism, which should be clarified over there.
824:" We could extend the former image to include all kinds of morphisms discussed, by adding "inj" and "surj" and extending the areas for "mon" and "epi" (then meaning "monic" and "epic") appropriately. And of course we should warn that "injective" and "surjective" don't make sense in category theory (if I remember right - ?).
928:"epimorphism" and "monomorphism" in earlier days meant "surjective" and "injective" homomorphism ("set-theoretically inspired meaning"), but has got a second ("category-theoretically inspired") meaning with the raise of category theory, viz. "left" and "right cancellable" (homo)morphism, respectively. If I understand
754:. Although being the object of some active research, universal algebra has rarely been considered in the main stream of mathematics, and its terminology is not widely accepted; a witness of this is that the term "variety" has a completely different meaning in universal algebra and in the main stream of mathematics (
600:. The simplest way to see it to look them at some algebra textbooks. But more conceptually the definitions should not refer to the underlying set; this allows in particular a smooth transition from sets to some more structured objects; for instance, one can talk about a homomorphism between sheaves of modules. --
1078:(Sorry, by "Dummit-Foote, I meant their "abstract algebra" and by "Eisenbud", "communicative algebra view toward algebraic geometry". I have deliberately avoided old, foreign or other less popular textbooks. The fact they avoid the term "epimorphism" is a good indication that we should do the same in Knowledge.)
1977:. Indeed this works not only with rings or groups, but with any algebraic structure; you can add as many operations as you want, binary or otherwise, and the argument still works. Therefore all groups are homomorphic, as are all rings, and indeed all algebraic structures. This is why we have a notation
1093:
As for an isomorphism, before defining it, if we like, we can explain it along the line: if a homomorphism is a bijection between the underlying sets (note algebraic structure is not a set!!), then the inverse function is necessarily a homomorphism. And, as D.Lazard suggested, with a proof, since the
331:
section is confusing in this respect. It suggests on first reading, for example, that a group homomorphism only applies between two groups, and not between say two rings (though on further thinking, of course every ring is a group under addition). EmilJ and
Deltahedron make it clear here; we could do
2519:
have the same type, means that there is a one to one correspondence between their operations, and that the corresponding operations have the same arity and satisfy the same axioms. In most cases, the corresponding operations have the same name, but this is not required. An example is the exponential
2079:
Thank you for correcting my example; I was not aware that preservation of the multiplicative identity was required in the definition of ring homomorphism, and as it turns out neither was the lecturer for my rings course at univeristy, whom I have now informed. However does this not then mean that we
902:
textbook authors (Dummit-Foote, Eisenbud) is simply to avoid such problematic terms like an epimorphism and use more down-to-earth unambiguous terms like surjective homomorphisms or injective morphisms. They then discuss category theory and mentions, for instance, a subjective module homomorphism is
424:
I agree. I have tagged the disputed definitions. It seems, from the comments in the text that previously these definitions were explicitly restricted to modules and that the present version is due to a troll that believed that abstract algebra is reduced to module theory. I have not yet got the time
481:
Your edit assert implicitly that "monomorphism", "epimorphism", ... have two non-equivalent meanings in mathematics. This is this implicit assertion which is wrong. Saying, as you did that in abstract algebra an epimorphism is always surjective, implies that this is true for rings or that the study
383:
The statement that "in abstract algebra a homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism" needs qualification. The definition in terms of injective and surjective maps makes this true, but the categorical definition in terms of left and right cancellation
1364:
I have completely rewritten, and expanded the section "Types", now renamed "Special homomorphisms". In my opinion, this solves the questions discussed in the previous section. Nevertheless some further work would be useful, in particular for improving references and settling the points that I have
654:
In any case, I suggest to keep the remarks (and also the proofs) on relations between the category theory and the (i.e. Birkhoff's) abstract algebra notions; this is possible even if "iso", "epi", "mono" is used only for the former, and only "bijective", "onto", "one-one" for the latter. Moreover,
260:
In any variety (or a more general class) of algebras, homomorphisms are required to preserve all operations in the signature of the variety. The signature of monoids includes the constant for identity, hence monoid homomorphisms have to preserve identity elements. The signature of, say, semigroups
1493:
In
January 2017, I edited the article for clarifying the relationships monomorphism–injection and epimorphism–surjection. Reading my edits again, I remark that I have left open the question of (non-necessarily commutative) groups: is every epimorphism (categorical meaning) of groups a surjection?
941:
that support your point, please do! In particular, could you please give full references of the "Dummit-Foote, Eisenbud" textbook(s)? Since the set-theory meaning is still (widely, I believe, based on the current distribution of citations) in use, the article should mention it; from your point of
520:
You are right that we should supply a citation for the non-category wording. I can offer the following one: Garrett
Birkhoff, "Lattice Theory", Providence, Am. Math. Soc., Vol.25 of Colloquium Publications, 1967, Sect.VI.3, p.134 (although the overall issue of the book are lattices, Sect.VI is on
936:
Concerning your second question, please note that up to now the article only has citations that support the set-theoretic meanings (viz. Birkhoff.1967, which is admittedly historical, and Burris.Sankappanavar.2012, which is undoubtedly not), apart from the pure category-theory book
Saunders.1971
704:
reverting to previous version, as 1/ it may be confusing for people who do not care that the given definition do not apply for rings and non-algebraic structures. 2/ it contains the wrong assertions that every epimorphism is a split epimorphism and that every monomorphism is a split monomorphism
1426:
Ok. I suppose an implicit question I had was whether the category theoretic definitions might be better placed either under a separate subheading or under morphism as opposed to here. I would guess that at least some readers landing here might be confused by the introduction of the more general
1085:
of isomorphism, epimorphism, etc. for general algebraic structures. Since the latter is not well-defined, that's not possible and we shouldn't try that. An "epimorphism" has specific meaning in the category theory but some authors, especially in historical sources, use it also mean a surjective
974:
I agree with your suggestion of using "surjective", "injective", "left" and "right cancelable". However, we have also to discuss all the names, their variants, and their relationships. As this discussion cannot be done in a single line, this is the main reason for splitting the section into one
774:). It follows that the only terminology that make sense for this article is that of category theory. Any other terminology would be confusing for many readers. However, I agree that if some sources use a terminology which is not compatible, this must be added to the article in a sentence like "
875:
No principal objection, but I see a problem of presentation: in each subsection we'd repeat something like "some authors call it '...', and some call it '...'", and the information was lost that the same authors that call an injective homomorphism "monomorphism" also use "epimorphism" for a
927:
I think it is a good idea to use unambiguous names in definitions, and that "surjective", "injective" (from your citations), and "left" and "right cancellable" (from Mac Lane's "Categories for the
Working Mathematician") is a good choice. Based on these definitions, we could then say that
897:
to give definitions of epimorphisms and such for some hypothetical algebraic object? Since, as I want to stress, an algebraic object need not have an underlying set, the set-theoretic definition cannot be used for that purpose (this was a point I wanted to make). It seems a solution among
245:
Doesn't a homomorphism need to preserve identity? The definition of monoid homomorphisms in the article on monoids requires this; the same goes for the definition of functors (and it says in the article about functors that one can think of them as homomorphisms between categories).
903:
precisely an epimorphism in the category of modules. Put in another way, do people use terms like epimorphisms aside from the category-theoretic meaning? Yes, some do but that's limited to historical sources. (There is no neee to use historical terms in
Knowledge.) --
1407:
synonymous of injective and surjective (respectively). Moreover, this figure introduces a confusion between "function" and "homomorphism": looking at the figure, a beginner could think that the hyperbolic sine is an automorphism of the reals, which is blatantly wrong.
1280:, there were already three proofs of that kind. If they are re-inserted, they should get a more explicit header such that it becomes obvious what they are about without having to read the corresponding section in the article. (I guess, due to the latter problem,
1427:
definitions, and could be looking for the algebraic definitions (injective / surjective). Not sure I understand the comment about functions, the assumption was that the figure would be placed in context, which is clearly that of homomorphisms, not functions.
482:
of rings do not belong to abstract algebra; both are wrong. I remove the word of troll. Your edit is only original research that is highly confusing for the reader and has been inserted with a summary saying that the previous correct version was confusing.
758:). Thus universal algebra, and Birkhoff book may not be used as a source for the terminology of this article. The last general definition of homorphisms that I know is that of category theory. It is presently used in many fields of mathematics, from
517:, it gives the right-cancellation definition in the context of category theory, and notes a deviating definition (viz. being surjective) by "Many authors in abstract algebra and universal algebra". This is consistent with the result of my edit.
932:
and you right, it is your point that the latter meaning is more elegant and therefore gets more and more widespread in abstract algebra, even beyond pure category theory; this should be made clear in the article (provided there is evidence for
821:
In category theory and some abstract algebra books, "monomorphism" and "epimorphism" refers to a monic and epic morphism, respectively, while other abstract algebra books use these notions to refer to an injective and surjective homomorphism,
235:(Z,.) (Z: set of integers , . : multiplication) between monoids given by f(x)=0. This is a homomorphism since f(x.y) = f(x).f(y) but identity is not preserved since f(1) = 0. The statement that identity is preservd seems wrong to me. --
1326:
I have encountered a problem of terminology: firstly it is not clear, what exactly means "algebraic structure", a specific instance, as the additive group of the integers or the whole class, as "structure of group". In general, and in
2409:
1057:
It would be useful to find an explicit reference for this result (it is certainly not new). If we do not find such a reference, I suggest to include the proof in the article, because this would avoid the vague "in may cases"
2138:
I think it would be good if we could add a section on the history of the homomorphism concept. In particular: Who invented the concept? Which motivation did he have? For which specific purposes is the concept used today?
2531:
I agree that presently, the formulation is confusing, but introducing the concept of "operation version" does not solve anything. Instead, the beginning of the section shoud be rewritten in the line of what precedes.
509:
So the non-equivalent meanings of (e.g.) epimorphism were present already before my edit, as was the definition of epimorphism to be a surjective homomorphism (which happens to be the definition I am familar with).
261:
does not include identity, hence semigroup homomorphisms do not have to preserve identity (even if the two semigroups happen to be monoids). Shahab’s map is a semigroup homomorphism, but not a monoid homomorphism.—
1948:
2520:
function, which is an homomorphism from the additive group of the complex numbers to the multiplicative group of the nonzero complex numbers. So, in the definition, one must distinguish between an operation on
668:, and the module homomorphism example can easily be added to the same paragraph; a more explicit warning about deviating notions in abtract algebra and category theory then should be added there, too. -
1859:
1123:'s claim that category theory notions are commonly used in abstract algebra, so we badly need them. I've been searching in our library, but found only these books, all sticking to set-based notions:
457:
Sorry, but I don't understand your point. First, a definition can't be wrong, but unusual at worst. Second, the case of category theory is handled in an own subsection where the inclusion Z--: -->
1320:
are essentially equivalent. As I do not know a good reference for this equivalence, I have included a proof, collapsed, because it is too technical and not really interesting for most readers.
664:
If there isn't sufficient evidence that
Birkhoff's terminology isn't used any longer today, I suggest just to revert D.Lazard's edits. The ring homomorphism example was already present in the
151:
2576:
1030:, for semigroups, the semigroup of positive integers, for monoids, the monoid of nonnegative integers, for groups, the additive group of the integers, for rings, the polynomial ring in
2062:
998:), "monomorphism" is equivalent with "injective". I do not know of an explicit source, but the proof may proceed as follows. Given a variety of algebraic structure, a free object over
1634:
1582:
2463:
1403:
This figure is not convenient for this article, because it is wrong in this context. In fact, a large part of this article is devoted to showing that monomorphism and epimorphism
1298:
It is not because I am against proofs in WP that I have removed these three proofs. It because they were confusing, as the statements that were supposed to be proved were lacking.
2415:
This sentence says that "preserves ««an»» operation". «an» means "a single" or "untyped" or "unversioned". If we should use versions, the sentence should be modified and make it
1666:
988:
It is a (rather easy) theorem that, for these algebraic structures, bijective is equivalent with having an inverse. This must be presented as a theorem, and possibly proved.
1770:
2199:
2012:
Being homomorphic is not a relevant concept, as, for most structures, there are many homomorphisms between any two objects. Your example is not a ring homomorphism, as
2566:
1995:
1975:
1713:
2483:
2219:
281:
Indeed, this is quite a nice illustration of the necessity for care over the codomain as well as the domain of a map. The map x → 0 is a monoid homomorphism from
2581:
2496:
There is something wrong here: we should either change the sentence, or change the formula, or somehow clarify the meaning of subscripts in one or two sentences.
2266:
2246:
1733:
1686:
849:
I agree with your overall goal, but not with the suggested use of epic and monic. In fact, making a difference between "being an epimorphism" and "being epic" is
364:
page and have formal definitions as well as examples because definitions may be helpful, but understanding won't come with a plethora of unorganized information.
1269:
In order to ease discussion, I suggest to upload scans of the most relevant pages locally for a short time to this page - I guess this could be allowed by the
1384:
Some visitors might be looking for a quick understanding of these concepts in the common case of sets, groups, etc. For beginners especially, would including
615:
I agree. Moreover, the "equivalently" assertion for epimorphisms is wrong, even for modules. I'll try to correct the section, if you do not do it before me.
35:
2591:
1340:
I intend to replace section "Types" by my new version. Before that, it would be better to have a consensus that this replacement will improve the article.
827:
Our overall goal should be that a by reading the section under consideration, the confusion (that caused our repeated discussions here) can be overcome. -
141:
2274:
2561:
1385:
819:
Q"). We comment on different terminologies and warn about confusions, using the nouns ("monomorphism", "epimorphism") for that. We could e.g. write "
763:
1022:
implies, almost immediately that every monomorphism is injective, and thus that "monomorphism" is equivalent with "injective". The free object over
2571:
117:
818:. We discuss the differences and relations between them (such as "surjective implies epic, but nor vice versa: e.g. ring homomorphism Z--: -->
810:
We use adjectives to name the (homo)morphism properties, as far as possible, i.e. "monic", "epic" (this is what S. Mac Lane uses in the cited
746:
This is not a problem of language nor a problem of availability of textbooks. The point is that this article is about a general definition of
2586:
1467:
1225:
1494:
More precisely, do there exist a right-cancelable group homomorphism that is not surjective? Does anybody know an answer to this question?
655:
the article's current version uses terms like "generally", "often" etc. in places where definitions are expected, and we should avoid that.
1038:, the subset of the preceding consisting of the polynomials without constant terms. For vector spaces and modules, the free object over
1864:
645:
according to
Birkhoff's definitions. If they are unusual today, we need other citations supporting that - nobody gave some since 2014.
247:
1182:
1135:
1119:
Could you please give a full citation of your books? Apparently, they are the only references up to now that could support your and
108:
69:
1010:
of any algebraic structure of the variety, there exists an homomorphism from the free object to the algebraic structure that maps
596:
but, as a remainder, I just want to point out the definitions of an isomorphism and an epimorphisms given in the article are ...
2556:
1054:. This cannot be done if some operations are not everywhere defined, such as the multiplicative inverse in the case of field.
402:
1775:
2501:
1232:— doesn't handle abstract algebras (although he mentions them informally on p.56,59); defines particular algebras, like
44:
593:
458:
Q is even used as example. The text there appears to say exactly the same as
Deltahedron. So, where is the problem? -
219:
189:
1323:
For epimorphisms, I do not know a condition implying the equivalence, nor the details of how to write the subsection.
1289:
947:
881:
832:
737:
673:
578:
471:
463:
994:
It is also a theorem that, for algebraic structures for which all operations are everywhere defined (varieties of
893:
The presentation of this materials (epimorphisms, etc) is still not optimal. I have come to wonder: do we really
2094:
The definition that is given here must not be changed. However, it is useful to clarify that is applies also to
198:
2065:
2497:
2015:
1027:
251:
2117:
2085:
2002:
1587:
1535:
1432:
1393:
1099:
978:
IMO, most of this discussion results of an ambiguity in the first line of the article: the definition of "
908:
605:
442:. Should we restore the previous version of the section or rewrite the section to have a correct version?
2422:
2160:
2069:
1302:
1285:
969:
943:
877:
828:
733:
687:
669:
574:
467:
459:
414:
306:
50:
2113:
2081:
1998:
392:
is a ring monomorphism and epimorphism in the latter sense, but not an isomorphism. See, for example,
94:
2072:. Nevertheless a few words may be added to the article for saying that many algebraic structures have
633:
Being not a native
English speaker, the only relevant textbook I have at hand is Birkhoff.1967, cited
1428:
1389:
466:) 22:28, 11 July 2014 (UTC) Btw: It was my edit, and I don't think I deserve to be called a troll. -
21:
1328:
979:
771:
361:
116:
on Knowledge. If you would like to participate, please visit the project page, where you can join
2537:
2103:
1639:
1514:
1499:
1413:
1370:
1345:
1063:
858:
783:
710:
620:
487:
447:
430:
369:
204:
100:
2528:. The easiest way for that is to index the name of the operation with the name of the structure.
84:
63:
1050:
may be constructed by considering the formal application of the operations of the structure to
2144:
1738:
1464:
1332:
1222:
1179:
1132:
1114:
1095:
995:
922:
904:
755:
751:
729:
601:
399:
339:
2172:
410:
302:
200:
175:
1980:
1953:
1691:
265:
2468:
2204:
1127:
Wolfgang Wechsler (1992). Wilfried Brauer and Grzegorz Rozenberg and Arto Salomaa (ed.).
942:
view, readers should be warned of the unelegent meaning still found in the literature. -
641:
and I had almost the same diccsussion in 2014. As a result, I then tried to fix section
2095:
811:
2251:
2231:
1718:
1671:
634:
2550:
2533:
2404:{\displaystyle f(\mu _{A}(a_{1},\ldots ,a_{k}))=\mu _{B}(f(a_{1}),\ldots ,f(a_{k})),}
2099:
2073:
1510:
1495:
1409:
1388:
figure be helpful? Book here. Also might address the removed figure mentioned above.
1366:
1341:
1281:
1170:(p.246) — the latter may be a good source for algebras with more than one carrier set
1120:
1059:
1035:
929:
854:
802:
779:
759:
723:
706:
638:
616:
483:
443:
426:
365:
2140:
1081:
I think the problem of the confusing state of the sectio stems from our attempt to
850:
642:
594:
Talk:Homomorphism/Archive 1#Incorrect definitions of special types of homomorphisms
333:
937:(which moreover isn't much more contemporary than Birkhoff.1967). So, if you can
514:
236:
113:
1301:
By the way, I have started to write a new version for the section "Types" (see
1131:. EATCS Monographs on Theoretical Computer Science. Vol. 25. Berlin: Springer.
1316:
I have tried to avoid the above debate in stating that the two definitions of
262:
90:
807:
Ok, now I understand your point of view. What about the following suggestion:
1668:
be the trivial homomorphism, i.e. the homomorphism mapping every element of
728:
So I understand you havn't an Englush textbook available either? Possibly,
1002:
is an algebraic structure of the variety that has a distinguished element
767:
202:
2541:
2505:
2148:
2121:
2107:
2089:
2006:
1518:
1503:
1436:
1417:
1397:
1374:
1349:
1293:
1103:
1067:
951:
912:
885:
862:
836:
787:
741:
714:
677:
624:
609:
582:
491:
475:
451:
434:
418:
373:
343:
310:
268:
255:
239:
2134:
Section on the history and application of the homomorphism concept?
1276:
I agree to include some elementary proofs into the article; in the
814:, p.19), "injective", "surjective", as there is no confusion about
2486:
2222:
2098:. I have added a paragraph in the article for this clarification.
1943:{\displaystyle \phi (r\times _{R}r')=\phi (r)\times _{S}\phi (r')}
1458:
1219:
Proofs and Fundamentals —- A First Course in Abstract Mathematics
394:
Dascalescu, Sorin; Nastasescu, Constantin; Raianu, Serban (2000).
1509:
I have found a reference, and I'll edit the article accordingly.
1090:
on the terminology and not try to give a mathematical exposition.
705:(this is true for vector spaces, but not for groups an modules).
521:"Universal Algebra", and VI.3 is on "Morphisms"). Birkhoff says:
439:
In fact, the wrong definitions have been introduced by this edit
379:
Isomorphism if and only if both a monomorphism and an epimorphism
1197:(p.395, for algebras with operations of an arity up to 2 only),
2064:
There is no ring homomorphism between two fields of different
1313:
is sill lacking. Here are some comments about this rewriting.
1178:. Texts and Monographs in Computer Science. Berlin: Springer.
205:
169:
15:
732:(Taku) can help with a few citations of algebra textbooks? -
1026:, which is unique up to an isomorphism is, for sets, any
289:,×. Of course, {0},× is a monoid, and a subsemigroup of
1277:
665:
498:
440:
1854:{\displaystyle \phi (r+_{R}r')=\phi (r)+_{S}\phi (r')}
1307:
the end of the (collapsed) proof for monomorphism, and
776:
Some author use the following different definition ...
297:,× since the identity of {0},× is not the identity of
2471:
2425:
2277:
2254:
2234:
2207:
2175:
2018:
1983:
1956:
1867:
1778:
1741:
1721:
1694:
1674:
1642:
1590:
1538:
565:
is an isomorphism of an algebra with itself, and an
112:, a collaborative effort to improve the coverage of
2477:
2457:
2403:
2260:
2240:
2213:
2193:
2068:. Your example works only for structures having a
2056:
1989:
1969:
1942:
1853:
1764:
1727:
1707:
1680:
1660:
1628:
1576:
1735:. This is indeed a homomorphism; we have for all
1529:Should we include the following in the article?
1460:Mathematics for Physics: An Illustrated Handbook
692:In your above citation, it is not said what are
2577:Knowledge level-5 vital articles in Mathematics
2417:
2167:
1950:since both sides of both equations evaluate to
1273:rule. Or does somebody see a problem with that?
1997:for isomorphy but no notation for homomorphy.
1046:} as a base. In general, the free object over
1042:is the vector space or free module which has {
592:This has already been pointed out in 2008 at
285:,× to the monoid {0},× but not to the monoid
213:This page has archives. Sections older than
8:
2165:What is operation version? The sentence is:
1365:listed at the end of the preceding section.
360:This page should follow the example of the
1174:David Gries and Fred B. Schneider (1994).
1006:and has the property that for any element
384:does not. For example, the injection i :
58:
2470:
2449:
2430:
2424:
2386:
2358:
2339:
2320:
2301:
2288:
2276:
2253:
2233:
2206:
2174:
2112:Thank you that is a perfect explanation.
2045:
2029:
2017:
1982:
1961:
1955:
1917:
1881:
1866:
1828:
1792:
1777:
1740:
1720:
1699:
1693:
1673:
1641:
1617:
1604:
1589:
1565:
1552:
1537:
1129:Universal Algebra for Computer Scientists
569:is a morphism of an algebra into itself.
553:is defined to be a one-one morphism from
1525:All algebraic structures are homomorphic
1086:homomorphism. We should stick to such a
2567:Knowledge vital articles in Mathematics
2057:{\displaystyle \phi (1_{R})\neq 1_{S}.}
1449:
223:when more than 10 sections are present.
60:
19:
1018:. The existence of a free object over
820:
775:
764:Wiles's proof of Fermat's Last Theorem
2582:C-Class vital articles in Mathematics
2524:, and the corresponding operation on
1629:{\displaystyle (S,+_{S},\times _{S})}
1577:{\displaystyle (R,+_{R},\times _{R})}
7:
106:This article is within the scope of
2458:{\displaystyle \mu _{A}and\mu _{B}}
1360:New section "Special homomorphisms"
1176:A Logical Approach to Discrete Math
1034:with integer coefficients, and for
49:It is of interest to the following
2592:High-priority mathematics articles
1309:the subsection about epimorphisms
1193:(p.387, single carrier-set only),
499:version immediately before my edit
14:
537:; a one-one morphism is called a
217:may be automatically archived by
126:Knowledge:WikiProject Mathematics
2562:Knowledge level-5 vital articles
234:Consider the map f:(Z,.)---: -->
230:Identity is not always preserved
174:
129:Template:WikiProject Mathematics
93:
83:
62:
29:
20:
146:This article has been rated as
2572:C-Class level-5 vital articles
2395:
2392:
2379:
2364:
2351:
2345:
2329:
2326:
2294:
2281:
2185:
2035:
2022:
1937:
1926:
1910:
1904:
1895:
1871:
1848:
1837:
1821:
1815:
1806:
1782:
1652:
1623:
1591:
1571:
1539:
344:08:31, 24 September 2013 (UTC)
1:
1104:21:54, 12 December 2016 (UTC)
1068:14:10, 11 December 2016 (UTC)
952:09:56, 11 December 2016 (UTC)
913:03:50, 11 December 2016 (UTC)
610:20:28, 30 November 2016 (UTC)
396:Hopf Algebra: An Introduction
311:20:33, 12 February 2013 (UTC)
269:17:55, 12 February 2013 (UTC)
256:17:36, 12 February 2013 (UTC)
240:12:14, 1 September 2006 (UTC)
120:and see a list of open tasks.
2587:C-Class mathematics articles
2149:23:48, 7 November 2019 (UTC)
1661:{\displaystyle \phi :R\to S}
1437:04:00, 1 February 2018 (UTC)
1418:10:38, 27 January 2018 (UTC)
1398:23:53, 26 January 2018 (UTC)
886:20:12, 7 December 2016 (UTC)
863:15:16, 7 December 2016 (UTC)
837:13:45, 7 December 2016 (UTC)
788:10:55, 7 December 2016 (UTC)
742:18:59, 6 December 2016 (UTC)
715:15:32, 6 December 2016 (UTC)
678:13:43, 6 December 2016 (UTC)
625:09:20, 1 December 2016 (UTC)
1375:14:26, 6 January 2017 (UTC)
1350:15:32, 3 January 2017 (UTC)
1294:13:14, 3 January 2017 (UTC)
1264:function (p.161) in general
1213:function in general (p.282)
1094:proof is illuminating. ---
293:,×, but not a submonoid of
2608:
1284:removed them on 1 Dec.) -
497:Please have a look at the
1765:{\displaystyle r,r'\in R}
1688:to the additive identity
583:07:09, 12 July 2014 (UTC)
492:23:19, 11 July 2014 (UTC)
476:22:32, 11 July 2014 (UTC)
452:22:10, 11 July 2014 (UTC)
435:19:49, 11 July 2014 (UTC)
419:18:14, 11 July 2014 (UTC)
145:
78:
57:
2542:17:16, 31 May 2022 (UTC)
2506:16:47, 31 May 2022 (UTC)
2194:{\displaystyle f:A\to B}
2122:11:07, 8 July 2018 (UTC)
2108:07:26, 8 July 2018 (UTC)
2090:20:36, 7 July 2018 (UTC)
2007:20:01, 5 July 2018 (UTC)
1168:many-sorted homomorphism
1083:give general definitions
374:22:20, 23 May 2014 (UTC)
152:project's priority scale
2201:preserves an operation
1519:13:14, 7 May 2018 (UTC)
1504:10:31, 7 May 2018 (UTC)
1217:Ethan D. Bloch (2000).
109:WikiProject Mathematics
2557:C-Class vital articles
2494:
2479:
2459:
2413:
2405:
2262:
2242:
2215:
2195:
2058:
1991:
1990:{\displaystyle \cong }
1971:
1944:
1855:
1766:
1729:
1709:
1682:
1662:
1630:
1578:
1489:Epimorphisms of groups
1278:version of 26 Nov 2016
1221:. Boston: Birkhäuser.
992:About "monomorphisms":
975:subsection by concept.
588:Inaccuracy (remainder)
571:
507:
425:to check the history.
327:The wording under the
220:Lowercase sigmabot III
2480:
2460:
2419:preserves operations
2406:
2263:
2243:
2216:
2196:
2059:
1992:
1972:
1970:{\displaystyle 0_{S}}
1945:
1856:
1767:
1730:
1710:
1708:{\displaystyle 0_{S}}
1683:
1663:
1631:
1579:
1303:User:D.Lazard/sandbox
1156:unsorted homomorphism
1152:unsorted term algebra
986:About "isomorphisms":
523:
503:
398:. CRC Press. p. 363.
36:level-5 vital article
2478:{\displaystyle \mu }
2469:
2423:
2275:
2252:
2232:
2214:{\displaystyle \mu }
2205:
2173:
2074:zero (homo)morphisms
2016:
1981:
1954:
1865:
1776:
1739:
1719:
1692:
1672:
1640:
1588:
1536:
1463:. World Scientific.
1457:Marsh, Adam (2017).
132:mathematics articles
1329:Algebraic structure
1305:). Presently, only
1164:many-sorted algebra
980:algebraic structure
772:homological algebra
666:15-Jul-2016 version
2498:Hooman Mallahzadeh
2475:
2455:
2401:
2258:
2238:
2228:, defined on both
2211:
2191:
2154:Operation version?
2054:
1987:
1967:
1940:
1851:
1762:
1725:
1705:
1678:
1658:
1626:
1574:
1238:group homomorphism
643:Homomorphism#Types
101:Mathematics portal
45:content assessment
2261:{\displaystyle B}
2241:{\displaystyle A}
1728:{\displaystyle S}
1681:{\displaystyle R}
1636:be rings and let
1469:978-981-3233-91-1
1333:universal algebra
1240:(p.267); defines
1227:978-0-8176-4111-5
996:universal algebra
967:
939:provide citations
756:algebraic variety
752:universal algebra
685:
227:
226:
166:
165:
162:
161:
158:
157:
2599:
2527:
2523:
2518:
2514:
2484:
2482:
2481:
2476:
2464:
2462:
2461:
2456:
2454:
2453:
2435:
2434:
2410:
2408:
2407:
2402:
2391:
2390:
2363:
2362:
2344:
2343:
2325:
2324:
2306:
2305:
2293:
2292:
2267:
2265:
2264:
2259:
2247:
2245:
2244:
2239:
2220:
2218:
2217:
2212:
2200:
2198:
2197:
2192:
2169:Formally, a map
2164:
2161:Jochen Burghardt
2096:0-ary operations
2063:
2061:
2060:
2055:
2050:
2049:
2034:
2033:
1996:
1994:
1993:
1988:
1976:
1974:
1973:
1968:
1966:
1965:
1949:
1947:
1946:
1941:
1936:
1922:
1921:
1894:
1886:
1885:
1860:
1858:
1857:
1852:
1847:
1833:
1832:
1805:
1797:
1796:
1771:
1769:
1768:
1763:
1755:
1734:
1732:
1731:
1726:
1714:
1712:
1711:
1706:
1704:
1703:
1687:
1685:
1684:
1679:
1667:
1665:
1664:
1659:
1635:
1633:
1632:
1627:
1622:
1621:
1609:
1608:
1583:
1581:
1580:
1575:
1570:
1569:
1557:
1556:
1481:
1480:
1478:
1476:
1454:
1286:Jochen Burghardt
1231:
1188:
1141:
1118:
973:
970:Jochen Burghardt
965:
944:Jochen Burghardt
926:
878:Jochen Burghardt
829:Jochen Burghardt
806:
734:Jochen Burghardt
727:
702:strongly against
691:
688:Jochen Burghardt
683:
670:Jochen Burghardt
575:Jochen Burghardt
468:Jochen Burghardt
460:Jochen Burghardt
408:
356:Poorly Explained
222:
206:
178:
170:
134:
133:
130:
127:
124:
103:
98:
97:
87:
80:
79:
74:
66:
59:
42:
33:
32:
25:
24:
16:
2607:
2606:
2602:
2601:
2600:
2598:
2597:
2596:
2547:
2546:
2525:
2521:
2516:
2512:
2467:
2466:
2445:
2426:
2421:
2420:
2382:
2354:
2335:
2316:
2297:
2284:
2273:
2272:
2250:
2249:
2230:
2229:
2203:
2202:
2171:
2170:
2158:
2156:
2136:
2041:
2025:
2014:
2013:
1979:
1978:
1957:
1952:
1951:
1929:
1913:
1887:
1877:
1863:
1862:
1840:
1824:
1798:
1788:
1774:
1773:
1748:
1737:
1736:
1717:
1716:
1695:
1690:
1689:
1670:
1669:
1638:
1637:
1613:
1600:
1586:
1585:
1561:
1548:
1534:
1533:
1527:
1491:
1486:
1485:
1484:
1474:
1472:
1470:
1456:
1455:
1451:
1382:
1362:
1228:
1216:
1185:
1173:
1138:
1126:
1112:
964:
920:
800:
721:
682:
590:
513:If you look at
405:
393:
381:
358:
232:
218:
207:
201:
183:
131:
128:
125:
122:
121:
99:
92:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
2605:
2603:
2595:
2594:
2589:
2584:
2579:
2574:
2569:
2564:
2559:
2549:
2548:
2545:
2544:
2529:
2511:The fact that
2474:
2452:
2448:
2444:
2441:
2438:
2433:
2429:
2412:
2411:
2400:
2397:
2394:
2389:
2385:
2381:
2378:
2375:
2372:
2369:
2366:
2361:
2357:
2353:
2350:
2347:
2342:
2338:
2334:
2331:
2328:
2323:
2319:
2315:
2312:
2309:
2304:
2300:
2296:
2291:
2287:
2283:
2280:
2257:
2237:
2210:
2190:
2187:
2184:
2181:
2178:
2155:
2152:
2135:
2132:
2131:
2130:
2129:
2128:
2127:
2126:
2125:
2124:
2066:characteristic
2053:
2048:
2044:
2040:
2037:
2032:
2028:
2024:
2021:
1986:
1964:
1960:
1939:
1935:
1932:
1928:
1925:
1920:
1916:
1912:
1909:
1906:
1903:
1900:
1897:
1893:
1890:
1884:
1880:
1876:
1873:
1870:
1850:
1846:
1843:
1839:
1836:
1831:
1827:
1823:
1820:
1817:
1814:
1811:
1808:
1804:
1801:
1795:
1791:
1787:
1784:
1781:
1761:
1758:
1754:
1751:
1747:
1744:
1724:
1702:
1698:
1677:
1657:
1654:
1651:
1648:
1645:
1625:
1620:
1616:
1612:
1607:
1603:
1599:
1596:
1593:
1573:
1568:
1564:
1560:
1555:
1551:
1547:
1544:
1541:
1526:
1523:
1522:
1521:
1490:
1487:
1483:
1482:
1468:
1448:
1447:
1443:
1442:
1441:
1440:
1439:
1421:
1420:
1381:
1378:
1361:
1358:
1357:
1356:
1355:
1354:
1353:
1352:
1338:
1337:
1336:
1324:
1321:
1299:
1274:
1267:
1266:
1265:
1226:
1214:
1183:
1171:
1136:
1107:
1106:
1091:
1079:
1075:
1074:
1073:
1072:
1071:
1070:
1055:
989:
983:
976:
957:
956:
955:
954:
934:
891:
890:
889:
888:
870:
869:
868:
867:
866:
865:
842:
841:
840:
839:
825:
808:
795:
794:
793:
792:
791:
790:
719:
718:
717:
659:
658:
657:
656:
649:
648:
647:
646:
628:
627:
589:
586:
525:A morphism of
495:
494:
455:
454:
437:
403:
380:
377:
357:
354:
353:
352:
351:
350:
349:
348:
347:
346:
318:
317:
316:
315:
314:
313:
274:
273:
272:
271:
231:
228:
225:
224:
212:
209:
208:
203:
199:
197:
194:
193:
185:
184:
179:
173:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
2604:
2593:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2554:
2552:
2543:
2539:
2535:
2530:
2510:
2509:
2508:
2507:
2503:
2499:
2493:
2491:
2488:
2472:
2465:(versions of
2450:
2446:
2442:
2439:
2436:
2431:
2427:
2416:
2398:
2387:
2383:
2376:
2373:
2370:
2367:
2359:
2355:
2348:
2340:
2336:
2332:
2321:
2317:
2313:
2310:
2307:
2302:
2298:
2289:
2285:
2278:
2271:
2270:
2269:
2255:
2235:
2227:
2224:
2208:
2188:
2182:
2179:
2176:
2166:
2162:
2153:
2151:
2150:
2146:
2142:
2133:
2123:
2119:
2115:
2111:
2110:
2109:
2105:
2101:
2097:
2093:
2092:
2091:
2087:
2083:
2078:
2077:
2075:
2071:
2067:
2051:
2046:
2042:
2038:
2030:
2026:
2019:
2011:
2010:
2009:
2008:
2004:
2000:
1984:
1962:
1958:
1933:
1930:
1923:
1918:
1914:
1907:
1901:
1898:
1891:
1888:
1882:
1878:
1874:
1868:
1844:
1841:
1834:
1829:
1825:
1818:
1812:
1809:
1802:
1799:
1793:
1789:
1785:
1779:
1759:
1756:
1752:
1749:
1745:
1742:
1722:
1700:
1696:
1675:
1655:
1649:
1646:
1643:
1618:
1614:
1610:
1605:
1601:
1597:
1594:
1566:
1562:
1558:
1553:
1549:
1545:
1542:
1530:
1524:
1520:
1516:
1512:
1508:
1507:
1506:
1505:
1501:
1497:
1488:
1471:
1466:
1462:
1461:
1453:
1450:
1446:
1438:
1434:
1430:
1425:
1424:
1423:
1422:
1419:
1415:
1411:
1406:
1402:
1401:
1400:
1399:
1395:
1391:
1387:
1379:
1377:
1376:
1372:
1368:
1359:
1351:
1347:
1343:
1339:
1334:
1330:
1325:
1322:
1319:
1315:
1314:
1312:
1308:
1304:
1300:
1297:
1296:
1295:
1291:
1287:
1283:
1279:
1275:
1272:
1268:
1263:
1259:
1255:
1252:function and
1251:
1247:
1243:
1239:
1235:
1229:
1224:
1220:
1215:
1212:
1208:
1205:function and
1204:
1200:
1196:
1192:
1186:
1184:3-540-94115-0
1181:
1177:
1172:
1169:
1165:
1161:
1157:
1153:
1149:
1145:
1139:
1137:3-540-54280-9
1134:
1130:
1125:
1124:
1122:
1116:
1111:
1110:
1109:
1108:
1105:
1101:
1097:
1092:
1089:
1084:
1080:
1077:
1076:
1069:
1065:
1061:
1056:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
990:
987:
984:
981:
977:
971:
963:
962:
961:
960:
959:
958:
953:
949:
945:
940:
935:
931:
924:
919:
918:
917:
916:
915:
914:
910:
906:
901:
896:
887:
883:
879:
874:
873:
872:
871:
864:
860:
856:
852:
848:
847:
846:
845:
844:
843:
838:
834:
830:
826:
823:
822:respectively.
817:
813:
809:
804:
799:
798:
797:
796:
789:
785:
781:
777:
773:
769:
765:
761:
760:number theory
757:
753:
749:
745:
744:
743:
739:
735:
731:
725:
720:
716:
712:
708:
703:
699:
695:
689:
681:
680:
679:
675:
671:
667:
663:
662:
661:
660:
653:
652:
651:
650:
644:
640:
636:
632:
631:
630:
629:
626:
622:
618:
614:
613:
612:
611:
607:
603:
599:
595:
587:
585:
584:
580:
576:
570:
568:
564:
560:
556:
552:
548:
544:
540:
536:
533:is called an
532:
528:
522:
518:
516:
511:
506:
502:
500:
493:
489:
485:
480:
479:
478:
477:
473:
469:
465:
461:
453:
449:
445:
441:
438:
436:
432:
428:
423:
422:
421:
420:
416:
412:
406:
401:
397:
391:
387:
378:
376:
375:
371:
367:
363:
355:
345:
341:
337:
336:
330:
326:
325:
324:
323:
322:
321:
320:
319:
312:
308:
304:
300:
296:
292:
288:
284:
280:
279:
278:
277:
276:
275:
270:
267:
264:
259:
258:
257:
253:
249:
244:
243:
242:
241:
238:
229:
221:
216:
211:
210:
196:
195:
192:
191:
187:
186:
182:
177:
172:
171:
168:
153:
149:
148:High-priority
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
73:High‑priority
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
2495:
2489:
2418:
2414:
2225:
2168:
2157:
2137:
2114:Joel Brennan
2082:Joel Brennan
1999:Joel Brennan
1531:
1528:
1492:
1473:. Retrieved
1459:
1452:
1444:
1404:
1383:
1363:
1318:monomorphism
1317:
1310:
1306:
1270:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1236:(p.258) and
1233:
1218:
1210:
1206:
1202:
1198:
1195:homomorphsim
1194:
1190:
1175:
1167:
1163:
1160:endomorphism
1159:
1155:
1151:
1147:
1143:
1128:
1115:TakuyaMurata
1087:
1082:
1051:
1047:
1043:
1039:
1031:
1023:
1019:
1015:
1011:
1007:
1003:
999:
991:
985:
938:
923:TakuyaMurata
900:contemporary
899:
894:
892:
815:
748:homomorphism
747:
730:TakuyaMurata
701:
697:
693:
597:
591:
572:
567:endomorphism
566:
563:automorphism
562:
558:
554:
550:
546:
542:
539:monomorphism
538:
534:
530:
526:
524:
519:
512:
508:
504:
501:. It says:
496:
456:
395:
389:
385:
382:
359:
334:
328:
298:
294:
290:
286:
282:
248:95.88.237.20
233:
214:
188:
180:
167:
147:
107:
51:WikiProjects
34:
2070:zero object
543:isomorphism
535:epimorphism
515:Epimorphism
411:Deltahedron
303:Deltahedron
123:Mathematics
114:mathematics
70:Mathematics
2551:Categories
1475:26 January
1445:References
1429:Adam Marsh
1390:Adam Marsh
1254:surjective
1246:one-to-one
1211:one-to-one
1199:surjective
1189:— defines
1148:surjection
1142:— defines
1088:commentary
404:0824704819
329:Definition
1242:injective
1207:injective
1144:injection
1028:singleton
39:is rated
2534:D.Lazard
2100:D.Lazard
1511:D.Lazard
1496:D.Lazard
1410:D.Lazard
1367:D.Lazard
1342:D.Lazard
1282:D.Lazard
1271:fair use
1162:(p.17);
1121:D.Lazard
1060:D.Lazard
930:D.Lazard
855:D.Lazard
803:D.Lazard
780:D.Lazard
768:topology
724:D.Lazard
707:D.Lazard
639:D.Lazard
637:, where
617:D.Lazard
484:D.Lazard
444:D.Lazard
427:D.Lazard
366:Therazzz
215:365 days
181:Archives
2141:Zaunlen
1405:are not
1191:algebra
1150:(p.3);
968:editor
686:editor
335:Quondum
150:on the
41:C-class
1380:Figure
237:Shahab
47:scale.
2487:arity
2485:) of
2223:arity
1250:monic
1234:group
851:WP:OR
770:(via
766:) to
635:above
598:wrong
561:; an
557:onto
549:with
541:. An
529:onto
362:Group
301:,×.
28:This
2538:talk
2515:and
2502:talk
2268:if
2248:and
2145:talk
2118:talk
2104:talk
2086:talk
2003:talk
1861:and
1584:and
1532:Let
1515:talk
1500:talk
1477:2018
1465:ISBN
1433:talk
1414:talk
1394:talk
1386:this
1371:talk
1346:talk
1290:talk
1262:epic
1258:onto
1223:ISBN
1203:onto
1180:ISBN
1133:ISBN
1100:talk
1096:Taku
1064:talk
1036:rngs
948:talk
933:it).
909:talk
905:Taku
895:need
882:talk
859:talk
833:talk
816:them
812:book
784:talk
738:talk
711:talk
696:and
674:talk
621:talk
606:talk
602:Taku
579:talk
488:talk
472:talk
464:talk
448:talk
431:talk
415:talk
400:ISBN
370:talk
340:talk
307:talk
263:Emil
252:talk
142:High
2221:of
1715:of
1311:are
1014:to
778:".
545:of
409:.
2553::
2540:)
2504:)
2473:μ
2447:μ
2428:μ
2371:…
2337:μ
2311:…
2286:μ
2209:μ
2186:→
2147:)
2120:)
2106:)
2088:)
2076:.
2039:≠
2020:ϕ
2005:)
1985:≅
1924:ϕ
1915:×
1902:ϕ
1879:×
1869:ϕ
1835:ϕ
1813:ϕ
1780:ϕ
1772:,
1757:∈
1653:→
1644:ϕ
1615:×
1563:×
1517:)
1502:)
1435:)
1416:)
1396:)
1373:)
1348:)
1292:)
1166:,
1158:,
1154:,
1146:,
1102:)
1066:)
966:To
950:)
911:)
884:)
861:)
835:)
786:)
740:)
713:)
684:To
676:)
623:)
608:)
581:)
573:-
490:)
474:)
450:)
433:)
417:)
388:→
372:)
342:)
309:)
266:J.
254:)
2536:(
2526:B
2522:A
2517:B
2513:A
2500:(
2492:,
2490:k
2451:B
2443:d
2440:n
2437:a
2432:A
2399:,
2396:)
2393:)
2388:k
2384:a
2380:(
2377:f
2374:,
2368:,
2365:)
2360:1
2356:a
2352:(
2349:f
2346:(
2341:B
2333:=
2330:)
2327:)
2322:k
2318:a
2314:,
2308:,
2303:1
2299:a
2295:(
2290:A
2282:(
2279:f
2256:B
2236:A
2226:k
2189:B
2183:A
2180::
2177:f
2163::
2159:@
2143:(
2116:(
2102:(
2084:(
2052:.
2047:S
2043:1
2036:)
2031:R
2027:1
2023:(
2001:(
1963:S
1959:0
1938:)
1934:′
1931:r
1927:(
1919:S
1911:)
1908:r
1905:(
1899:=
1896:)
1892:′
1889:r
1883:R
1875:r
1872:(
1849:)
1845:′
1842:r
1838:(
1830:S
1826:+
1822:)
1819:r
1816:(
1810:=
1807:)
1803:′
1800:r
1794:R
1790:+
1786:r
1783:(
1760:R
1753:′
1750:r
1746:,
1743:r
1723:S
1701:S
1697:0
1676:R
1656:S
1650:R
1647::
1624:)
1619:S
1611:,
1606:S
1602:+
1598:,
1595:S
1592:(
1572:)
1567:R
1559:,
1554:R
1550:+
1546:,
1543:R
1540:(
1513:(
1498:(
1479:.
1431:(
1412:(
1392:(
1369:(
1344:(
1288:(
1260:=
1256:=
1248:=
1244:=
1230:.
1209:=
1201:=
1187:.
1140:.
1117::
1113:@
1098:(
1062:(
1052:x
1048:x
1044:x
1040:x
1032:x
1024:x
1020:x
1016:a
1012:x
1008:a
1004:x
1000:x
972::
946:(
925::
921:@
907:(
880:(
857:(
831:(
805::
801:@
782:(
762:(
736:(
726::
722:@
709:(
698:B
694:A
690::
672:(
619:(
604:(
577:(
559:B
555:A
551:B
547:A
531:B
527:A
486:(
470:(
462:(
446:(
429:(
413:(
407:.
390:Q
386:Z
368:(
338:(
305:(
299:Z
295:Z
291:Z
287:Z
283:Z
250:(
190:1
154:.
53::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
