# 2-affine complete algebras need not be affine complete by Aichinger E.

By Aichinger E.

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Extra info for 2-affine complete algebras need not be affine complete

Example text

E^° Por el Teorema 3, un vector ot está en el subespacio W de F5 generado por al, az, az si, y solo si, existen escalares cz, cz, cz en F, tales que 0 = C101 + C202 + Cada- Así, W consta de todos los vectores de la forma a = (cl) 2619 02; 361 + 462; donde cz, cz, cz son escalares arbitrarios de F. En forma alternativa, W puede ser escrito como el conjunto de todos los 5-tuples Of = (\$1, 252, 253, 254, 235) con x, en F, tal que 21: = 2271 174 = 331 + 427:- Así (-3, -6, 1, -5, 2) está en W, mientras que (2, 4, 6, 7, 8) no.

Iii-ai ciertos escalares cz, _ _ _ , ch. ,'y,,} es independiente, cada uno de los escalares z, = 0. Así, E Ita; + E Z/¡ﬁi = 0 y como {a¡,. ,a¡,, Bb. } es también un conjunto independiente, cada x, = 0 y cada y, = 0. _,a¡,, B¡,. ,B,,,, 71,. } es una base para W, + Wz_ Finalmente, dimW,+dimW2= (k+m)+(k+ﬂ) =k+(m+k+ﬂ) Cerremos esta sección con una observación referente a la independencia y dependencia lineal. Se han deﬁnido estos conceptos para conjuntos de vectores. Es útil haberlos deﬁnido para sucesiones ﬁnitas (n-tuples ordenados) de vectores: oil, _ _ _, oi,,_ Se dirá que los vectores az, _ _ _ , oi, son linealmente ili-pendientes si existen escalares ﬁjos cz, _ _ _, c,,, no todos nulos, tales que i-jor, + -- - + c,,oi,, = 0.