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{{Short description|Sifat matematis dari struktur aljabar}}{{about|aljabar abstrak|hukum fisika|Prinsip Archimedes}}
[[Berkas:Archimedean_property.png|jmpl|250x250px|Ilustrasi sifat Archimedes.]]
Dalam [[aljabar abstrak]] dan [[Analisis matematika|analisis]], '''Sifat Archimedes''', dinamai menurut ahli matematika Yunani kuno [[Archimedes]] dari [[Sirakusa]], adalah sifat yang dimiliki oleh beberapa [[struktur aljabar]], seperti [[Grup (matematika)|grup]], dan [[Medan (matematika)|medan]]. Secara kasar, ini adalah sifat yang tidak memiliki elemen '' jauh lebih besar '' atau '' jauh lebih kecil ''. Adalah [[Otto Stolz]] yang memberi nama pada aksioma Archimedes karena muncul sebagai [[Aksioma]] V dari Archimedes '' [[Pada Bola dan Tabung]] ''.<ref>G. Fisher (1994) in P. Ehrlich(ed.), Bilangan Riil, Generalisasi Realisasi, dan Teori Kontinua, 107-145, Kluwer Academic</ref>
Gagasan tersebut muncul dari teori [[Besaran (matematika)|besaran]] Yunani Kuno; itu masih memainkan peran penting dalam matematika modern seperti [[Aksioma Hilbert|aksioma]] [[David Hilbert]] untuk geometri, dan teori [[Grup terurut linear|grup terurut]], [[Bidang terurut|medan terurut]], dan [[medan lokal]].
Struktur aljabar di mana dua elemen bukan nol adalah ''sebanding '', dalam arti bahwa tidak satu pun dari mereka [[sangat kecil]] dibandingkan dengan yang lain, dikatakan '''Archimedes'''. Suatu struktur yang memiliki sepasang elemen bukan nol, yang salah satunya sangat kecil terhadap yang lain, dikatakan sebagai '''tak-Archimedes'''. Misalnya, [[grup terurut linear]] yang merupakan Archimedes adalah [[Grup Archimedean|grup Archimedes]].
Ini dapat dibuat tepat dalam berbagai konteks dengan rumusan yang sedikit berbeda.Misalnya, dalam konteks [[kolom terurut]], satu memiliki '''aksioma Archimedes''' yang merumuskan sifat ini, di mana medan [[bilangan riil]] adalah Archimedes, tetapi [[fungsi rasional]] dalam koefisien riil tidak.
In [[abstract algebra]] and [[Mathematical analysis|analysis]], the '''Archimedean property''', named after the ancient Greek mathematician [[Archimedes]] of [[Syracuse, Italy|Syracuse]], is a property held by some [[Algebraic structure|algebraic structures]], such as ordered or normed [[Group (algebra)|groups]], and [[Field (mathematics)|fields]]. The property, typically construed, states that given two positive numbers <math>x</math> and <math>y</math>, there is an integer <math>n</math> such that <math>nx > y</math>. It also means that the set of [[natural numbers]] is not bounded above.<ref>{{cite web|title=Math 2050C Lecture|url=https://www.math.cuhk.edu.hk/course_builder/2021/math2050c/MATH%202050C%20Lecture%204%20(Jan%2021).pdf|website=cuhk.edu.hk|access-date=3 September 2023}}</ref> Roughly speaking, it is the property of having no ''infinitely large'' or ''infinitely small'' elements. It was [[Otto Stolz]] who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ ''[[On the Sphere and Cylinder]]''.<ref>G. Fisher (1994) in P. Ehrlich(ed.), Real Numbers, Generalizations of the Reals, and Theories of continua, 107-145, Kluwer Academic</ref>
The notion arose from the theory of [[Magnitude (mathematics)|magnitudes]] of Ancient Greece; it still plays an important role in modern mathematics such as [[David Hilbert]]'s [[Hilbert's axioms|axioms for geometry]], and the theories of [[Linearly ordered group|ordered groups]], [[Ordered field|ordered fields]], and [[local fields]].
An algebraic structure in which any two non-zero elements are ''comparable'', in the sense that neither of them is [[infinitesimal]] with respect to the other, is said to be '''Archimedean'''. A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be '''non-Archimedean'''. For example, a [[linearly ordered group]] that is Archimedean is an [[Archimedean group]].
This can be made precise in various contexts with slightly different formulations. For example, in the context of [[Ordered field|ordered fields]], one has the '''axiom of Archimedes''' which formulates this property, where the field of [[Real number|real numbers]] is Archimedean, but that of [[rational functions]] in real coefficients is not.
== Sejarah dan asal nama sifat Archimedes ==
Konsep ini dinamai oleh [[Otto Stolz]] (pada tahun 1880-an) setelah ahli geografi dan fisikawan [[Yunani kuno]] [[Archimedes]] dari [[Sirakusa]].
Sifat Archimedes muncul di Buku V dari [[Elemen Euklides]] sebagai Definisi 4:
{{Quote|Besaran dikatakan memiliki rasio satu sama lain yang dapat, jika dikalikan, melebihi satu sama lain.}}
Karena Archimedes mengkreditkannya ke [[Eudoksos dari Knidos]] itu juga dikenal sebagai "Teorema Eudoxus" atau'' aksioma Eudoxus''<ref name="Knopp1951">{{cite book|last=Knopp|first=Konrad|year=1951|url=https://archive.org/details/theoryapplicatio00knop|title=Theory and Application of Infinite Series|location=London and Glasgow|publisher=Blackie & Son, Ltd.|isbn=0-486-66165-2|edition=English 2nd|page=[https://archive.org/details/theoryapplicatio00knop/page/7 7]|authorlink=Konrad Knopp|url-access=registration}}</ref>.
[[Archimedes menggunakan infinitesimal]] dalam argumen [[heuristik]], meskipun ia menyangkal bahwa argumen tersebut telah selesai [[bukti matematika]].
== Definisi untuk grup terurut linear ==
{{Main|Grup Archimedean}}
Misalkan''{{mvar|x}}'' dan''{{mvar|y}}'' menjadi [[Grup berurutan linear#Definisi|elemen positif]] dari [[grup terurut linier]] '' G ''.
Kemudian'''''{{mvar|x}}'' inifintesimal terhadap ''{{mvar|y}}''''' (atau ekuivalen,'''''{{mvar|y}}'' takhingga terhadap''{{mvar|x}}''''') jika, untuk setiap [[bilangan asli]] ''{{mvar|n}}'', kelipatan{{math|'' nx ''}} kurang dari ''{{mvar|y}}'', yaitu, pertidaksamaan berikut berlaku:<blockquote><math> \underset{n \text{ suku}}{\underbrace{x + \dots + x}} < y </math></blockquote>Definisi ini dapat diperluas ke seluruh kelompok dengan mengambil [[Nilai absolut|nilai mutlak]].
Grup ''{{mvar|G}}'' adalah '''Archimedes''' jika tidak ada pasangan {{math|(''x'', ''y'')}} sedemikian rupa sehingga ''{{mvar|x}}'' sangat kecil dibandingkan dengan ''{{mvar|y}}''.
Selain itu, jika ''{{mvar|K}}'' adalah [[struktur aljabar]] dengan satuan (1) misalnya, [[Gelanggang (matematika)|gelanggang]] ,definisi serupa berlaku untuk ''{{mvar|K}}''. Jika ''{{mvar|x}}'' sangat kecil dibandingkan dengan 1, maka ''{{mvar|x}}'' adalah '''elemen yang sangat kecil'''. Demikian juga, jika''{{mvar|y}}'' tak hingga 1, maka ''{{mvar|y}}'' adalah '''elemen takhingga'''. Struktur aljabar ''{{mvar|K}}'' adalah Archimedes jika tidak memiliki elemen takhingga dan tidak memiliki elemen takhingga.
== Medan terurut ==
[[Medan terurut]] memiliki beberapa sifat tambahan:
* [[Bilangan rasional]] adalah [[Pembenaman|terbenam]] di setiap medan terurut. Artinya, setiap kolom terurut memiliki [[Karakteristik (aljabar)|karakteristik]] nol.
* Jika''{{mvar|x}}'' Infinitesimal, maka {{math|1/''x''}} tidak terbatas, dan sebaliknya. Oleh karena itu, untuk memverifikasi bahwa medan adalah Archimedes, cukup dengan memeriksa hanya bahwa tidak ada elemen yang sangat kecil, atau untuk memeriksa bahwa tidak ada elemen yang tak terbatas.
* Jika''{{mvar|x}}'' sangat kecil dan<var> r </var>adalah bilangan rasional, maka {{math|'' rx ''}} juga sangat kecil. Akibatnya, diberi elemen umum ''{{mvar|c}}'', tiga bilangan {{math|''c''/2}}, ''{{mvar|c}}'', dan {{math|2''c''}} bisa jadi semua sangat kecil atau semua bukan sangat kecil.
Dalam setelan ini, medan terurut''{{mvar|K}}'' adalah Archimedes persis ketika pernyataan berikut, disebut '''aksioma Archimedes''', menyatakan:
: "Misalkan''{{mvar|x}}'' adalah elemen apa pun dari ''{{mvar|K}}''. Kemudian ada bilangan asli ''{{mvar|n}}'' sehingga {{math|''n'' > ''x''}}."
Sebagai alternatif, seseorang dapat menggunakan karakterisasi berikut:<blockquote><math>\forall\, \varepsilon \in K\big(\varepsilon > 0 \implies \exists\ n \in N : 1/n < \varepsilon\big).</math></blockquote>
== Definisi untuk medan ternorma ==
Kualifikasi "Archimedes" juga diformulasikan dalam teori [[Gelanggang penilaian|peringkat satu medan nilai]] dan ruang ternorma atas peringkat satu medan nilai sebagai berikut.
Misalkan ''{{mvar|F}}'' adalah medan yang diberkahi dengan fungsi nilai mutlak, yaitu fungsi yang mengaitkan bilangan real 0 dengan elemen medan 0 dan mengaitkan bilangan riil positif <math>|x|</math> dengan setiap bukan nol {{math|''x'' ∈ ''F''}} dan dirumuskan<blockquote><math> |xy|=|x| |y|</math> dan <math>|x+y| \le |x|+|y| </math>.</blockquote>Kemudian, ''{{mvar|F}}'' dikatakan '''Archimedes''' jika ada bukan nol {{math|''x'' ∈ ''F''}} ada [[bilangan asli]] ''{{mvar|n}}'' dirumuskan<blockquote><math> |\underbrace{x+\cdots+x}_{n\text{ terms}}| > 1. \, </math></blockquote>Demikian pula, ruang bernorma adalah Archimedes jika jumlah ''{{mvar|n}}'' suku, masing-masing sama dengan vektor bukan-nol ''{{mvar|x}}'', memiliki norma yang lebih besar dari satu untuk cukup besar ''{{mvar|n}}''.
medan dengan nilai mutlak atau ruang bernorma adalah Archimedes atau memenuhi ketentuan yang lebih kuat, yang disebut sebagai [[ultrametrik]] [[pertidaksamaan segitiga]]<math>|x+y| \le \max(|x|,|y|)</math>,
masing-masing. medan atau ruang bernorma yang memenuhi pertidaksamaan segitiga ultrametrik disebut '''tak-Archimedes'''.
Konsep ruang linier bernorma tak-Archimedes diperkenalkan oleh A. F. Monna.<ref name="monna1">Monna, A. F., Over een lineare P-adisches ruimte, Indag. Math., 46 (1943), 74–84.</ref>
== Lihat pula ==
* {{annotated link|0.999...#Infinitesimal|0.999...}}
* {{annotated link|Ruang vektor berurutan Archimedean}}
* {{annotated link|Konstruksi bilangan riil}}
== Catatan ==
<references responsive="1"></references>
== Referensi ==
{{refbegin}}
* {{Cite book|last=Schechter|first=Eric|year=1997|url=http://www.math.vanderbilt.edu/~schectex/ccc/|title=Handbook of Analysis and its Foundations|publisher=Academic Press|isbn=0-12-622760-8|ref=harv|postscript=.|authorlink=Eric Schechter|access-date=2009-01-30|archive-url=https://web.archive.org/web/20150307061351/http://www.math.vanderbilt.edu/%7Eschectex/ccc/|archive-date=2015-03-07|url-status=dead}}
{{refend}}
== History and origin of the name of the Archimedean property ==
The concept was named by [[Otto Stolz]] (in the 1880s) after the [[Ancient Greece|ancient Greek]] geometer and physicist [[Archimedes]] of [[Syracuse, Italy|Syracuse]].
The Archimedean property appears in Book V of [[Euclid's Elements|Euclid's ''Elements'']] as Definition 4:
{{Blockquote|Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.}}
Because Archimedes credited it to [[Eudoxus of Cnidus]] it is also known as the "Theorem of Eudoxus" or the ''Eudoxus axiom''.<ref name="Knopp19512">{{cite book|last=Knopp|first=Konrad|year=1951|url=https://archive.org/details/theoryapplicatio00knop|title=Theory and Application of Infinite Series|location=London and Glasgow|publisher=Blackie & Son, Ltd.|isbn=0-486-66165-2|edition=English 2nd|page=[https://archive.org/details/theoryapplicatio00knop/page/7 7]|author-link=Konrad Knopp|url-access=registration}}</ref>
[[Archimedes's use of infinitesimals|Archimedes used infinitesimals]] in [[heuristic]] arguments, although he denied that those were finished [[Mathematical proof|mathematical proofs]].
== Definition for linearly ordered groups ==
{{Main|Archimedean group}}
Let {{mvar|x}} and {{mvar|y}} be [[Linearly ordered group#Definitions|positive elements]] of a [[linearly ordered group]] ''G''. Then <math>x</math> '''is infinitesimal with respect to''' <math>y</math> (or equivalently, <math>y</math> '''is infinite with respect to''' <math>x</math>) if, for any [[natural number]] <math>n</math>, the multiple <math>nx</math> is less than <math>y</math>, that is, the following inequality holds:<math display="block"> \underbrace{x+\cdots+x}_{n\text{ terms}} < y. \, </math>This definition can be extended to the entire group by taking absolute values.
The group <math>G</math> is '''Archimedean''' if there is no pair <math>(x,y)</math> such that <math>x</math> is infinitesimal with respect to <math>y</math>.
Additionally, if <math>K</math> is an [[algebraic structure]] with a unit (1) — for example, a [[Ring (mathematics)|ring]] — a similar definition applies to <math>K</math>. If <math>x</math> is infinitesimal with respect to <math>1</math>, then <math>x</math> is an '''infinitesimal element'''. Likewise, if <math>y</math> is infinite with respect to <math>1</math>, then <math>y</math> is an '''infinite element'''. The algebraic structure <math>K</math> is Archimedean if it has no infinite elements and no infinitesimal elements.
=== Ordered fields ===
[[Ordered field|Ordered fields]] have some additional properties:
* The rational numbers are [[Embedding|embedded]] in any ordered field. That is, any ordered field has [[Characteristic (algebra)|characteristic]] zero.
* If <math>x</math> is infinitesimal, then <math>1/x</math> is infinite, and vice versa. Therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements.
* If <math>x</math> is infinitesimal and <math>r</math> is a rational number, then <math>rx</math> is also infinitesimal. As a result, given a general element <math>c</math>, the three numbers <math>c/2</math>, <math>c</math>, and <math>2c</math> are either all infinitesimal or all non-infinitesimal.
In this setting, an ordered field {{mvar|K}} is Archimedean precisely when the following statement, called the '''axiom of Archimedes''', holds:
: "Let <math>x</math> be any element of <math>K</math>. Then there exists a natural number <math>n</math> such that <math>n > x</math>."
Alternatively one can use the following characterization:<math display="block">\forall\, \varepsilon \in K\big(\varepsilon > 0 \implies \exists\ n \in N : 1/n < \varepsilon\big).</math>
== Definition for normed fields ==
The qualifier "Archimedean" is also formulated in the theory of [[Valuation ring|rank one valued fields]] and normed spaces over rank one valued fields as follows. Let <math>K</math> be a field endowed with an absolute value function, i.e., a function which associates the real number <math>0</math> with the field element 0 and associates a positive real number <math>|x|</math> with each non-zero <math>x \in K</math> and satisfies <math>|xy|=|x| |y|</math> and <math>|x+y| \le |x|+|y|</math>. Then, <math>K</math> is said to be '''Archimedean''' if for any non-zero <math>x \in K</math> there exists a [[natural number]] <math>n</math> such that<math display="block">|\underbrace{x+\cdots+x}_{n\text{ terms}}| > 1. </math>Similarly, a normed space is Archimedean if a sum of <math>n</math> terms, each equal to a non-zero vector <math>x</math>, has norm greater than one for sufficiently large <math>n</math>. A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as the [[ultrametric]] [[triangle inequality]],<math display="block">|x+y| \le \max(|x|,|y|) ,</math>respectively. A field or normed space satisfying the ultrametric triangle inequality is called '''non-Archimedean'''.
The concept of a non-Archimedean normed linear space was introduced by A. F. Monna.<ref name="monna1943">{{cite journal|last1=Monna|first1=A. F.|date=1943|title=Over een lineaire ''P''-adische ruimte|journal=Nederl. Akad. Wetensch. Verslag Afd. Natuurk.|issue=52|pages=74–84|mr=15678}}</ref>
== Examples and non-examples ==
=== Archimedean property of the real numbers ===
The field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function <math>|x|=1</math>, when <math>x \neq 0</math>, the more usual <math display="inline">|x| = \sqrt{x^2}</math>, and the <math>p</math>'''-adic absolute value''' functions. By [[Ostrowski's theorem]], every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some <math>p</math>-adic absolute value. The rational field is not complete with respect to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete. The completion with respect to the usual absolute value (from the order) is the field of real numbers. By this construction the field of real numbers is Archimedean both as an ordered field and as a normed field.<ref>[[Neal Koblitz]], "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Springer-Verlag,1977.</ref> On the other hand, the completions with respect to the other non-trivial absolute values give the fields of [[P-adic number|p-adic numbers]], where <math>p</math> is a prime integer number (see below); since the <math>p</math>-adic absolute values satisfy the [[ultrametric]] property, then the <math>p</math>-adic number fields are non-Archimedean as normed fields (they cannot be made into ordered fields).<!-- "by axiom" side -->In the [[axiomatic theory of real numbers]], the non-existence of nonzero infinitesimal real numbers is implied by the [[least upper bound property]] as follows. Denote by <math>Z</math> the set consisting of all positive infinitesimals. This set is bounded above by <math>1</math>. Now [[Proof by contradiction|assume for a contradiction]] that <math>Z</math> is nonempty. Then it has a [[least upper bound]] <math>c</math>, which is also positive, so <math>c/2 < c < 2c</math>. Since {{mvar|c}} is an [[upper bound]] of <math>Z</math> and <math>2c</math> is strictly larger than <math>c</math>, <math>2c</math> is not a positive infinitesimal. That is, there is some natural number <math>n</math> for which <math>1/n < 2c</math>. On the other hand, <math>c/2</math> is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal <math>x</math> between <math>c/2</math> and <math>c</math>, and if <math>1/k < c/2 \leq x</math> then <math>x</math> is not infinitesimal. But <math>1/(4n) < c/2</math>, so <math>c/2</math> is not infinitesimal, and this is a contradiction. This means that <math>Z</math> is empty after all: there are no positive, infinitesimal real numbers.
The Archimedean property of real numbers holds also in [[constructive analysis]], even though the least upper bound property may fail in that context.
=== Non-Archimedean ordered field ===
{{main article|Non-Archimedean ordered field}}
For an example of an [[ordered field]] that is not Archimedean, take the field of [[Rational function|rational functions]] with real coefficients. (A rational function is any function that can be expressed as one [[polynomial]] divided by another polynomial; we will assume in what follows that this has been done in such a way that the [[leading coefficient]] of the denominator is positive.) To make this an ordered field, one must assign an ordering compatible with the addition and multiplication operations. Now <math>f > g</math> if and only if <math>f - g > 0</math>, so we only have to say which rational functions are considered positive. Call the function positive if the leading coefficient of the numerator is positive. (One must check that this ordering is well defined and compatible with addition and multiplication.) By this definition, the rational function <math>1/x</math> is positive but less than the rational function <math>1</math>. In fact, if <math>n</math> is any natural number, then <math>n(1/x) = n/x</math> is positive but still less than <math>1</math>, no matter how big <math>n</math> is. Therefore, <math>1/x</math> is an infinitesimal in this field.
This example generalizes to other coefficients. Taking rational functions with rational instead of real coefficients produces a countable non-Archimedean ordered field. Taking the coefficients to be the rational functions in a different variable, say <math>y</math>, produces an example with a different [[order type]].
=== Non-Archimedean valued fields ===
The field of the rational numbers endowed with the p-adic metric and the [[p-adic number]] fields which are the completions, do not have the Archimedean property as fields with absolute values. <!-- Another example is the [[hyperreal numbers]] of [[nonstandard analysis]]. : (ed. I detest this, because the formal interpretation of the Axiom of Archimedes is indeed satisfied by hypernatural numbers in place of the "standard" natural numbers, which do not form a "hyperset" (or *-set, superset, whatever we call it) inside the system of the "hyperreal numbers".)--> All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value.<ref name="shell1">Shell, Niel, Topological Fields and Near Valuations, Dekker, New York, 1990. {{ISBN|0-8247-8412-X}}</ref>
=== Equivalent definitions of Archimedean ordered field ===
Every linearly ordered field <math>K</math> contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit <math>1</math> of <math>K</math>, which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an ordered [[monoid]]<!-- semigroup -->. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in <math>K</math>. The following are equivalent characterizations of Archimedean fields in terms of these substructures.<ref name="Schechter">{{harvnb|Schechter|1997|loc=§10.3}}</ref>
# The natural numbers are [[Cofinal (mathematics)|cofinal]] in <math>K</math>. That is, every element of <math>K</math> is less than some natural number. (This is not the case when there exist infinite elements.) Thus an Archimedean field is one whose natural numbers grow without bound.
# Zero is the [[infimum]] in <math>K</math> of the set <math>\{1/2, 1/3, 1/4, \dots\}</math>. (If <math>K</math> contained a positive infinitesimal it would be a lower bound for the set whence zero would not be the greatest lower bound.)
# The set of elements of <math>K</math> between the positive and negative rationals is non-open. This is because the set consists of all the infinitesimals, which is just the set <math>\{0\}</math> when there are no nonzero infinitesimals, and otherwise is open, there being neither a least nor greatest nonzero infinitesimal. Observe that in both cases, the set of infinitesimals is closed. In the latter case, (i) every infinitesimal is less than every positive rational, (ii) there is neither a greatest infinitesimal nor a least positive rational, and (iii) there is nothing else in between. Consequently, any non-Archimedean ordered field is both incomplete and disconnected.
# For any <math>x</math> in <math>K</math> the set of integers greater than <math>x</math> has a least element. (If <math>x</math> were a negative infinite quantity every integer would be greater than it.)
# Every nonempty open interval of <math>K</math> contains a rational. (If <math>x</math> is a positive infinitesimal, the open interval <math>(x,2x)</math> contains infinitely many infinitesimals but not a single rational.)
# The rationals are [[Dense set|dense]] in <math>K</math> with respect to both sup and inf. (That is, every element of <math>K</math> is the sup of some set of rationals, and the inf of some other set of rationals.) Thus an Archimedean field is any dense ordered extension of the rationals, in the sense of any ordered field that densely embeds its rational elements.
== See also ==
* {{annotated link|0.999...#Infinitesimals|0.999...}}
* {{annotated link|Archimedean ordered vector space}}
* {{annotated link|Construction of the real numbers}}
== Notes ==
{{reflist}}
== References ==
{{refbegin}}
* {{Cite book|last=Schechter|first=Eric|year=1997|url=http://www.math.vanderbilt.edu/~schectex/ccc/|title=Handbook of Analysis and its Foundations|publisher=Academic Press|isbn=0-12-622760-8|author-link=Eric Schechter|access-date=2009-01-30|archive-url=https://web.archive.org/web/20150307061351/http://www.math.vanderbilt.edu/%7Eschectex/ccc/|archive-date=2015-03-07|url-status=dead}}
{{refend}}
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----{{short description|Gambar invers nol di bawah homomorfisme}}
Dalam [[aljabar]], '''kernel''' dari [[homomorfisme]] (fungsi yang mempertahankan [[Struktur aljabar|struktur]]) umumnya [[gambar invers]] dari 0 (kecuali untuk [[Grup (matematika)|grup]] yang operasinya dilambangkan dengan multi, dimana kernel adalah kebalikan dari gambar 1). Kasus khusus yang penting adalah [[Kernel (aljabar linear)|kernel dari peta linear]]. [[Kernel (matriks)|kernel dari matriks]], juga disebut ''ruang nol'', adalah kernel dari peta linear yang ditentukan oleh matriks.
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