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Versiunea de la 2 februarie 2023 16:46

Nu confundați cu holomorfism sau homeomorfism.

În algebră, printr-un homomorfism este o „hartă” matematică care păstrează structura între două structuri algebrice de același tip (așa cum ar fi două grupuri, două inele sau două spații vectoriale). Cuvântul homomorfism provine din limba greacă veche - ὁμός (homos), care înseamnă „același” și - μορφή (morfon) care înseamnă „formă”. Cu toate acestea, cuvântul a fost aparent introdus în matematică din cauza unei (greșite) traduceri a cuvântului german germană ähnlich meaning similar comparativ cu ὁμός, care semnifică același.^[1]

Termenul folosit astăzi, homomorfism, a fost prima dată menționat în 1892, fiind atribuit matematicianului german Felix Klein (1849–1925).^[2]

Homomorfismele spațiilor vectoriale sunt numite și hărți liniare, iar studiul lor este subiectul algebrei liniare.

Conceptul de homomorfism a fost generalizat, sub denumirea de morfism, la multe alte structuri, care fie nu au o mulțime subiacentă, fie nu sunt algebrice. Această generalizare este punctul de plecare al teoriei categoriilor.

Un homomorfism poate fi, de asemenea, un izomorfism, un endomorfism, un automorfism, șamd, ... (vezi mai jos). Fiecare dintre acestea poate fi definit într-un mod care poate fi generalizat la orice clasă de morfisme.

Definiție

Un homomorfism este o hartă matematică între două structuri algebrice de același tip, care conservă operația matematică a structurilor.

Ceea ce se poate scrie formal $f:A\to B$ între două mulțimi $A$ , $B$ , care au aceeasși structură astfel încât, dacă $\cdot$ este operația structurii (presupusă aici, pentru simplitate, a fi o operație binară), atunci

f(x\cdot y)=f(x)\cdot f(y)

pentru orice pereche $x$ , $y$ de elemente ale mulțimii $A$ .<ref group="note">

Așa cum este adesea cazul, dar nu întotdeauna, același simbol pentru funcționarea ambelor $A$ și $B$ a fost folosit aici. One says often that $f$ preserves the operation or is compatible with the operation.

Note, referințe

^ Fricke, Robert (1897). Vorlesungen über die Theorie der automorphen Functionen. B.G. Teubner. OCLC 29857037.
^
A se vedea:
- Ritter, Ernst (1892). „Die eindeutigen automorphen Formen vom Geschlecht Null, eine Revision und Erweiterung der Poincaré'schen Sätze” [The unique automorphic forms of genus zero, a revision and extension of Poincaré's theorem]. Mathematische Annalen (în germană). 41: 1–82. doi:10.1007/BF01443449. From footnote on p. 22: "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 holoedrischen Isomorphismus zweier Gruppen einschränken, sonst aber von "Homomorphismus" sprechen, … " (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 holohedric isomorphism of two groups; otherwise, however, [I will] speak of a "homomorphism", … )
- Fricke, Robert (1892). „Ueber den arithmetischen Charakter der zu den Verzweigungen (2,3,7) und (2,4,7) gehörenden Dreiecksfunctionen” [On the arithmetic character of the triangle functions belonging to the branch points (2,3,7) and (2,4,7)]. Mathematische Annalen (în germană). 41: 443–468. doi:10.1007/BF01443421. From p. 466: "Hierdurch ist, wie man sofort überblickt, eine homomorphe*) Beziehung der Gruppe Γ₍₆₃₎ auf die Gruppe der mod. n incongruenten Substitutionen mit rationalen ganzen Coefficienten der Determinante 1 begründet." (Thus, as one immediately sees, a homomorphic relation of the group Γ₍₆₃₎ is based on the group of modulo n incongruent substitutions with rational whole coefficients of the determinant 1.) From footnote on p. 466: "*) 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"." (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".)

Bibliografie

Krieger, Dalia (2006). "On critical exponents in fixed points of non-erasing morphisms". Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, CA, USA, June 26–29, 2006. Oscar H. Ibarra, Zhe Dang. Springer-Verlag. pp. 280–291. ISBN 3-540-35428-X.
Stanley N. Burris; H.P. Sankappanavar (2012). A Course in Universal Algebra (PDF). ISBN 978-0-9880552-0-9.
Mac Lane, Saunders (1971), Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, Springer-Verlag, ISBN 0-387-90036-5, Zbl 0232.18001
Fraleigh, John B.; Katz, Victor J. (2003), A First Course in Abstract Algebra, Addison-Wesley, ISBN 978-1-292-02496-7

Legături externe

en Homomorphism - Homomorfism pe Britannica
en Homomorphism - Homomorfism ] pe web site-ul Universității Columbia

[1] Fricke, Robert (1897). Vorlesungen über die Theorie der automorphen Functionen. B.G. Teubner. OCLC 29857037.

[2] A se vedea:
Ritter, Ernst (1892). „Die eindeutigen automorphen Formen vom Geschlecht Null, eine Revision und Erweiterung der Poincaré'schen Sätze” [The unique automorphic forms of genus zero, a revision and extension of Poincaré's theorem]. Mathematische Annalen (în germană). 41: 1–82. doi:10.1007/BF01443449. From footnote on p. 22: "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 holoedrischen Isomorphismus zweier Gruppen einschränken, sonst aber von "Homomorphismus" sprechen, … " (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 holohedric isomorphism of two groups; otherwise, however, [I will] speak of a "homomorphism", … )

Fricke, Robert (1892). „Ueber den arithmetischen Charakter der zu den Verzweigungen (2,3,7) und (2,4,7) gehörenden Dreiecksfunctionen” [On the arithmetic character of the triangle functions belonging to the branch points (2,3,7) and (2,4,7)]. Mathematische Annalen (în germană). 41: 443–468. doi:10.1007/BF01443421. From p. 466: "Hierdurch ist, wie man sofort überblickt, eine homomorphe*) Beziehung der Gruppe Γ₍₆₃₎ auf die Gruppe der mod. n incongruenten Substitutionen mit rationalen ganzen Coefficienten der Determinante 1 begründet." (Thus, as one immediately sees, a homomorphic relation of the group Γ₍₆₃₎ is based on the group of modulo n incongruent substitutions with rational whole coefficients of the determinant 1.) From footnote on p. 466: "*) 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"." (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".)

[3] Ritter, Ernst (1892). „Die eindeutigen automorphen Formen vom Geschlecht Null, eine Revision und Erweiterung der Poincaré'schen Sätze” [The unique automorphic forms of genus zero, a revision and extension of Poincaré's theorem]. Mathematische Annalen (în germană). 41: 1–82. doi:10.1007/BF01443449. From footnote on p. 22: "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 holoedrischen Isomorphismus zweier Gruppen einschränken, sonst aber von "Homomorphismus" sprechen, … " (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 holohedric isomorphism of two groups; otherwise, however, [I will] speak of a "homomorphism", … )

[4] Fricke, Robert (1892). „Ueber den arithmetischen Charakter der zu den Verzweigungen (2,3,7) und (2,4,7) gehörenden Dreiecksfunctionen” [On the arithmetic character of the triangle functions belonging to the branch points (2,3,7) and (2,4,7)]. Mathematische Annalen (în germană). 41: 443–468. doi:10.1007/BF01443421. From p. 466: "Hierdurch ist, wie man sofort überblickt, eine homomorphe*) Beziehung der Gruppe Γ₍₆₃₎ auf die Gruppe der mod. n incongruenten Substitutionen mit rationalen ganzen Coefficienten der Determinante 1 begründet." (Thus, as one immediately sees, a homomorphic relation of the group Γ₍₆₃₎ is based on the group of modulo n incongruent substitutions with rational whole coefficients of the determinant 1.) From footnote on p. 466: "*) 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"." (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".)

[1]

[2]

@@ Linia 27: / Linia 27: @@
 == Bibliografie ==
+* Krieger, Dalia (2006). "On critical exponents in fixed points of non-erasing morphisms". ''Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, CA, USA, June 26–29, 2006''. Oscar H. Ibarra, Zhe Dang. [[Springer-Verlag]]. pp.&nbsp;280–291. [[ISBN (identifier)|ISBN]]&nbsp;[[Special:BookSources/3-540-35428-X|<bdi>3-540-35428-X</bdi>]].
+* {{cite book | url=http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf | isbn=978-0-9880552-0-9 | author1=Stanley N. Burris | author2=H.P. Sankappanavar | title=A Course in Universal Algebra | year=2012 }}
+* {{citation | first1 = Saunders | last1 = Mac Lane| author-link=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | volume=5 | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | isbn=0-387-90036-5 | year=1971 | zbl=0232.18001 }}
+* {{citation | first1 = John B. | last1 = Fraleigh | first2 = Victor J. | last2 = Katz | year = 2003 | title = A First Course in Abstract Algebra | publisher = Addison-Wesley | isbn= 978-1-292-02496-7}}
+== Legături externe ==
+* {{ien}}[https://www.britannica.com/science/homomorphism  Homomorphism - Homomorfism] pe [https://www.britannica.com/science/ Britannica]
+* {{ien}} [https://www.math.columbia.edu/~rf/homomorphisms.pdf  Homomorphism - Homomorfism] ] pe web site-ul [https://www.math.columbia.edu/ Universității Columbia]
+{{Informații bibliografice}}
+[[Categorie:Morfisme]]