3.1 ALGEBRA
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(resp., L(H)). For arbitrary X 1 ,-- . , Xk E L, pH(Xi n H,.. -, Xk n H) is a monotone function of H, that is, Hl C H2 implies
PH1(X1 n H,. -, Xk n Hl) c pH2 (X1 n H2, ... , Xk n H2). This follows easily from the monotonicity of u, n, and C, using induction on the length of the term p. We claim that for arbitrary X1,. .. , Xk E L and h E p(X1i ... , Xk) there is a finite H C En-1 such that h E H and h E pH(X1 n H,. - -, Xk n H). This statement will be proved by induction on the length of p. If p is just a variable, then we can choose H = {h}. Suppose that the statement is proved already for terms whose length is less than the length of p,,,,
and let p = p' A p". Then there are finite sets Hl and H2 such that h E p'H1(X1 nH1,...,Xk nH1), h E p'H2(X1 nH2,...,Xk nH2), and h E H1, h E H2. Let H = H1 U H2. Then we have
hEp'H1(X1nH1i...,XknHl)npH2(X1nH2i...,XknH2)
cpH(XinH,...,XknH)np"(X1nH,...,XknH) PH(Xl nH,...,XknH). Now let p = p' V p". Then h E p(X1, ... , Xk) C C (p(X1, ... , Xk) U P "(X1, ... , Xk))
By Caratheodory's theorem, there exist finitely many points h'1, ... , h' E
p'(Xl, . . . , Xk) and hl , . . . , by E P" (X1, ... , Xk) such that
hEC({h i,...,h',h',...,h';}) and u+v<n. (Actually, we could obtain u < 1, v < 1, but we don't need this now.) By the induction hypothesis there exist finite sets H,.... , Hu, G1, ... , G C_ Ere-1 such that h' E pH; (Xl n H2,.. . , Xk n Hi) and h' E p'' (X1 n .
j < v). Let H = {h} U Hl U
G;,...,Xk n G;) (1 < i < n,1
U
Now
h E C({hi,...,hu,hi,...,h';})nH c C(p'H(X1 nH,...,Xk nH) up" (X1 nH,...,Xk nH)) n H = pH (X1 n H,.. -, Xk n H). Now let p(x1, ... , xk) = 4(x1, ... , xk) be an identity of lattices that is satisfied by every n-distributive lattice, and let X1, ... , Xk E L be arbitrary. We have to prove p(X1i .... Xk) = q(X1,... , Xk). To prove the inclusion p(X1, ... , Xk) C q(X1, ... , Xk), take an arbitrary h E p(X1, ... , Xk). Then for a suitable finite H C Ere-1 we have h E pH (X1 nH, ... , Xk n H). Since L(H) is finite and n-distributive we have
h E gH(Xl nH,...,Xk nH)
gEn_1(Xl nE7L-1,...,Xk nEn-1) = q(Xl, ... 7 Xk) .
This establishes the inclusion p(X1, .... Xk) C q(Xi, ... , Xk), and the reverse inclusion can be proved in the same way.