Algebra, WS 2009 - download pdf or read online

By Martin Goldstern

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Dann ist auch j∈J Aj in V. ) • Sei (J, ≤) eine lineare Ordnung. Sei (Aj )j∈J eine Familie von Algebren in V, die i ≤ j ⇒ Ai ≤ Aj erf¨ ullt. 1) in V. 4 Oft schreibt man die Allquantoren nicht explizit an; statt ∀x ∀y (xy = yx) oder ∀x (x ∗ x−1 = e) schreibt man also k¨ urzer xy = yx bzw x ∗ x−1 = e, auch wenn man das allgemeine Gesetz und keine konkrete Instanz meint. 5 englisch: equationally defined class 52 Beweis. Man rechnet leicht nach, dass alle Gesetze, die in A gelten, erst recht“ in Unter” algebren und in homomorphen Bildern gelten.

Dann ist die Abbildung n Ea : AA → A, f → f (a) ein Homomorphismus. 6 Lemma (Einsetzungshomomorphismus mit variablem Parameter). Sei A Algebra, n AA die n-stellige Funktionenalgebra. n Dann gibt es einen (eindeutig bestimmten) Homomorphismus ϕ : T(x1 , . . , xn ) → AA , der jeder Variable xk die k-te Projektion πkn zuordnet. ϕ / AAn www www Ea w ϕa www ww&  T(x1 , . . 7 Anmerkung. h. eine Funktion von An nach A. Sei a ∈ An . Dann k¨onnen wir ϕ(t) an der Stelle a = (a1 , . . , an ) auswerten und erhalten ϕ(t)(a) = ϕa (t).

Un . i) Die oben definierte Abbildung ϕ : U1 × · · · × Un → G, (u1 , . . , un ) → u1 · · · un , ist bijektiv und b) ii) f¨ ur 1 ≤ i < j ≤ n, x ∈ Ui , y ∈ Uj gilt stets xy = yx. Beweis. a) ⇒ b): i) gilt, da ϕ Isomorphismus. Zu ii): F¨ ur x ∈ Ui , y ∈ Uj , 1 ≤ i < j ≤ n gilt xy = ϕ(e, . . , e, x i−te Stelle = ϕ((e, . . , e, , e, . . , e) · ϕ(e, . . , e, x , e, . . , e) · (e, . . , e, y j−te Stelle , e, . . , e) · (e, . . , e, y y , e, . . , e)) = , e, . . , e) = j−te Stelle = ϕ(e, .

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Algebra, WS 2009 by Martin Goldstern


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