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By Stankey Burris, H. P. Sankappanavar

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The most objective of those lectures is first to in brief survey the elemental con­ nection among the illustration concept of the symmetric staff Sn and the speculation of symmetric features and moment to teach how combinatorial equipment that come up obviously within the idea of symmetric features result in effective algorithms to precise numerous prod­ ucts of representations of Sn by way of sums of irreducible representations.

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Now it is straightforward to verify that the diagram in Figure 6 gives the desired copy of N5 in L. 6 (Birkhoff). L is a nondistributive lattice iff M5 or N5 can be embedded into L. Proof. If either M5 or N5 can be embedded into L, then it is clear from earlier remarks that L cannot be distributive. For the converse, let us suppose that L is a nondistributive lattice and that L does not contain a copy of N5 as a sublattice. 5. §3. Distributive and Modular Lattices 15 Since the distributive laws do not hold in L, there must be elements a, b, c from L such that (a ∧ b) ∨ (a ∧ c) < a ∧ (b ∨ c).

Now, for any X ⊆ A define E(X) = X ∪ {f (a1 , . . , an ) : f is a fundamental n-ary operation on A and a1 , . . , an ∈ X}. Then define E n (X) for n ≥ 0 by E 0 (X) = X E n+1 (X) = E(E n (X)). As all the fundamental operations on A are finitary and X ⊆ E(X) ⊆ E 2 (X) ⊆ · · · one can show that (Exercise 1) Sg(X) = X ∪ E(X) ∪ E 2 (X) ∪ · · · , and from this it follows that if a ∈ Sg(X) then a ∈ E n (X) for some n < ω; hence for some finite Y ⊆ X, a ∈ E n (Y ). Thus a ∈ Sg(Y ). But this says Sg is an algebraic closure operator.

Since the inverse of an equivalence relation is just that equivalence relation, we have established (a). (b) ⇒ (c): Since θ2 ◦ θ1 ⊆ θ1 ∨ θ2 , from (b) we could deduce θ2 ◦ θ1 ⊆ θ1 ◦ θ2 , and then from the previous paragraph it would follow that θ2 ◦ θ1 = θ1 ◦ θ2 ; 2 hence (c) holds. 10 (Birkhoff). If A is congruence-permutable, then A is congruence-modular. Proof. Let θ1 , θ2 , θ3 ∈ Con A with θ1 ⊆ θ2 . We want to show that θ2 ∩ (θ1 ∨ θ3 ) ⊆ θ1 ∨ (θ2 ∩ θ3 ), so suppose a, b is in θ2 ∩ (θ1 ∨ θ3 ).

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