By H. Lausch, W. Nobauer
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Let n a 2, e E P, ( A ) , and y E F, ( A ) such that . * > gn), for (g1, y k , , * * * , g n#e(g,, ) - - - > g , )for , (gp v(g1, * * - 9 8,) = dg,, - . - > gn) f (~1,. 0 . -:,u,), for some (ul, . ,u,) E A", then y E Pn(A). proof. Suppose that y(u,, . ,u,) = c, and let a f b E A . 21. Then a,€P,,(A),thus a,(g,, . ,9,) = v,(aj, g,, . , gn), for some word wl. Let x E Fz ( A ) be defined by v, for u z b x(u,v> = c, for u = b. 23. We complete the proof of the theorem for case a). By definition of P,(A), every element of A is in P,(A).
G , - A g,), for all (gl, . ,g,) E A". Since y E F,-, ( A ) , by induction, we have y E P,-l ( A ) , thus tp(gl, . ,gn-l) = w,(a,, g,, . , g,-l), for some word wl. Also 3: E P 2 ( A ) by hypothesis, thus ~ ( uv), = w2(bj,u, v), for some word w2. Hence q(gl, . , g,) = w2(bj,wl(ai,g17 . e'p = w2(bj,wl(ui, El, . , &-J,En) is in P,(A). This completes the proof of the theorem. -. - 7 * ? 3. Proposition. Let A be I -poljwomially complete, then A is simple. Proof. By way of contradiction, suppose that A is not simple.
Then the results of 5 11 show that the following three cases are possible: a) A is n-polynomially complete for all n. b) A is n-polynomially complete for n = 1, but for no IZ =- 1. c) A is n-polynomially complete for no n. In case a) we say A is polynomially complete, in case b) A is polynomially semicomplete, and in case c) A is polynomially incomplete. For some important varieties, we are going to investigate now in what way the algebras of these varieties distribute over these three cases. We will consider just algebras A such that I A 1 # 1, for j A j = 1 implies that A is polynomially complete for all varieties.
Algebra of Polynomials by H. Lausch, W. Nobauer