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Con A, ⊆ is a complete sublattice of Eq(A), ⊆ , the lattice of equivalence relations on A. Proof. To verify that Con A is closed under arbitrary intersection is straightforward. For arbitrary joins in Con A suppose θi ∈ Con A for i ∈ I. Then, if f is a fundamental n-ary operation of A and θi , a1 , b1 , . . 7 it follows that one can find i0 , . . , ik ∈ I such ai , bi ∈ θi0 ◦ θi1 ◦ · · · ◦ θik , 0 ≤ i ≤ n. An easy argument then suffices to show that f (a1 , . . , an ), f (b1 , . . , bn ) ∈ θi0 ◦ θi1 ◦ · · · ◦ θik ; hence i∈I θi is a congruence relation on A.

N − 1}, ∨, ∧, 0, 1 is a bounded chain with 0 < n − 1 < n − 2 < · · · < 2 < 1, and 1 = 2, 2 = 3, . . , (n − 2) = n − 1, (n − 1) = 0, and 0 = 1. See Figure 8, where the unary operation is depicted by arrows. In IV§7 we will give a structure theorem for all n-valued Post algebras, and in V§4 show that they can be defined by a finite set of equations. 1 .. 2 n-1 0 Figure 8 The Post algebra Pn (13) Cylindric Algebras of Dimension n. If we are given n ∈ ω, then an algebra A, ∨, ∧, , c0 , . . , cn−1 , 0, 1, d00, d01 , .

If K ⊆ Su(A), show that K is a closed set system for A iff K is closed under arbitrary intersections. Given a set A and a family K of subsets of A, K is said to be closed under unions of chains if whenever C ⊆ K and C is a chain (under ⊆) then C ∈ K; and K is said to be closed under unions of upward directed families of sets if whenever D ⊆ K is such that A1 , A2 ∈ D implies A1 ∪ A2 ⊆ A3 for some A3 ∈ D, then D ∈ K. A result of set theory says that K is closed under unions of chains iff K is closed under unions of upward directed families of sets.

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