By C.-G. Schmidt

ISBN-10: 3540138633

ISBN-13: 9783540138631

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**Extra info for Arithmetik Abelscher Varietaeten mit komplexer Multiplikation**

**Example text**

11): A[X] =< a[X] U {fa I a E A} > A(X) . 13) U {XjA I j E J} , which is included both in TE(G) and A(X) . Denote the elements of TE(G) by G WG 'Wi"" et c. 1. For every subalgebra S of A(X) , every wG E TE(G) and every (... 14) PROOF: (wG)s( ... ,gk, ... )(v) = (WG)A(' " ,gk(V), .. ) (Vv E AX) . 11) that (xf)s( ... )(v) = gk(V) = (Xf)A('" ,gk(V), .. s«.... )(v) = fi((wf)s(... ), .. s«,... ))(v) G G . = fi((wi )s( ... )(v), , (Wn(i»)S('" ,9k, . ))(v) = fi((wf)A('" , (W~(i»)A('" ,9k(V), . )) ,9k(V), ..

If f, 9 : An ---+ A are functions with the substitution property such that fiB = glB then 'ljJf = 'ljJg. PROOF: Let f', g' : B" ---+ B be the functions associated with f and s. 3. Then for every bl , . , bn E B, f' (bl, ... ,bn) = f' ('IjJ(bt) , ... ,1/J(bn)) = 'IjJ(J(b l, . ,bn )) = 1/J(g(b l, ... , bn)) = . = g'(bl, . . , bn) , that is, f' = s'. Hence for every Xl, ... ,X n E A, 1/J(J(Xl,"" x n)) = J'(1/J(Xt} , ... , 'IjJ(x n)) = g'('IjJ(Xl) , ... ,1/J(xn)) = 1/J(g(Xl, ... ,x n )) . 5. The set of polynomials (of algebraic functions) is closed under composition of functions.

Trivially a lattice is both a join semilattice and a meet semilattice. If E is an infinite set, then (Pi(E); ~), where Pi(E) stands for the family of all infinite subsets of E, is a join semilattice but not a lattice. As we have done for lattices, if (S ;:s;) is a join semilattice we can define xVy = sup{x, y} for every x, yES and obtain an algebra (S; V) of type (2) whose operation is commutative, associative and idempotent (cf. 13)). We arrive at the same result if we start from a meet semilattice and define x /\ y = inf {x, y}.