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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. 4. The congruence lattice of A, denoted by Con A, is the lattice whose universe is Con A, and meets and joins are calculated the same as when working with equivalence relations (see I§4). The following theorem suggests the abstract characterization of congruence lattices of algebras.

Bn ) ∈ θi0 ◦ θi1 ◦ · · · ◦ θik ; hence i∈I θi is a congruence relation on A. 4. The congruence lattice of A, denoted by Con A, is the lattice whose universe is Con A, and meets and joins are calculated the same as when working with equivalence relations (see I§4). The following theorem suggests the abstract characterization of congruence lattices of algebras. 5. For A an algebra, there is an algebraic closure operator Θ on A × A such that the closed subsets of A × A are precisely the congruences on A.

N − 1}, ∨, ∧, , 0, 1 where {0, 1, . . , 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 , .

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2^(x o) varieties of Heyting algebras not generated by their finite members by Blok W.J.


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