By Don Pigozzi

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Bn ) ∈ B. Let B α be the unique subalgebra of A with universe B α. Then B ⊆ B α ⊆ A. 25 (Third Isomorphism Theorem). Let A be a Σ-algebra, B ⊆ A, and α ∈ Co(A). Then B α (α ∩ (B α)2 ∼ = B (α ∩ B 2 ). Proof. Define h: B → B α β = α ∩ (B α)2 . α ∩ (B α)2 by h(b) = b α ∩ (B α)2 . h is surjective. Let h σ B (b1 , . . , bn ) = σ B (b1 , . . , bn )/β = σ B α (b1 , . . , bn )/β = σ B α/β (b1 /β, . . , bn /β) = σ B α/β h(b1 ), . . , h(bn ) . So h: B B α/β. For all b, b ∈ B, h(b) = h(b ) iff b/β = b /β iff b β b iff b α ∩ (B α)2 b iff b (α ∩ B 2 ) b since b, b ∈ B.

Then for each i ≤ n there is a ci ∈ A such that ai R ci S bi . Thus σ A(a1 , . . , an ) R σ A(c1 , . . , cn ) S σ A(b1 , . . , bn ). Hence σ A(a1 , . . , an ) R; S σ A(b1 , . . , bn ). 28. Let A be a Σ-algebra and let R be a directed set of binary relations on A. If each R ∈ R has the substitution property, then so does R. The proof is straightforward and is left as an exercise. 29. Let A be a Σ-algebra. , for every K ⊆ Co(A), Co(A) week 7 K= Eq(A) K. 36 Proof. The inclusion from right to left holds because Co(A) K is an equivalence relation that includes each congruence in K.

Suppose αi : i ∈ I ∈ Co(A)I and, for each i ∈ I, αi = ∆A . Then, for every i ∈ I, µ ⊆ αi . Hence ∆A ⊂ µ ⊆ i∈I αi . Using the Correspondence Theorem we can relativize this result to obtain a useful characterization of the quotients of an algebra that are SDEI. 70. Let A be a Σ-algebra and let α ∈ Co(A). , α ⊂ µα and, for every β ∈ Co(A) such that α ⊂ β we have µα ⊆ β. A graphical representation of the principal filter of Co(A) generated by α is given in the left-hand side of Figure 20. Proof.