New PDF release: Proceedings of Symposium on Coastal Oceanography and

By Navy Committee, Ocean Studies Board, National Research Council.

ISBN-10: 0309584884

ISBN-13: 9780309584883

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Read or Download Proceedings of Symposium on Coastal Oceanography and Littoral Warfare (Unclassified Summary) Fleet Combat Training Center, Tactical Training Group, Pacific, San Diego, CA, August 2-5, 1993 PDF

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Additional resources for Proceedings of Symposium on Coastal Oceanography and Littoral Warfare (Unclassified Summary) Fleet Combat Training Center, Tactical Training Group, Pacific, San Diego, CA, August 2-5, 1993

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4. Let L be a lattice and let H be a nonempty subset of L. Then a ∈ sub(H) (the sublattice generated by H) iff a = p(h1 , . . , hn ), for some integer n ≥ 1, for some n-ary polynomial p, and for some h1 , . . , hn ∈ H. , p ≤ q), where p and q are polynomials. , p ≤ q) holds in the lattice L iff p(a1 , . . , an ) = q(a1 , . . , p(a1 , . . , an ) ≤ q(a1 , . . , an )) holds, for all a1 , . . , an ∈ L. The identity p = q is equivalent to the two inequalities p ≤ q and q ≤ p; the inequality p ≤ q is equivalent to the identity p ∨ q = q.

Such computations are often facilitated by the following lemma (Gr¨ atzer and E. T. 1. A reflexive binary relation Θ on a lattice L is a congruence relation iff the following three properties are satisfied, for x, y, z, t ∈ L: (i) x ≡ y (Θ) iff x ∧ y ≡ x ∨ y (Θ). (ii) x ≤ y ≤ z, x ≡ y (Θ), and y ≡ z (Θ) imply that x ≡ z (Θ). (iii) x ≤ y and x ≡ y (Θ) imply that x ∧ t ≡ y ∧ t (Θ) and x ∨ t ≡ y ∨ t (Θ). Let Con L denote the set of all congruence relations on L ordered by set inclusion (remember that we can view Θ ∈ Con L as a subset of L2 ).

The set Id L of all ideals of L is an order under set inclusion, and as an order it is a lattice. In fact, for I, J ∈ Id L, the lattice operations in Id L are I ∨ J = id(I ∪ J) and I ∧ J = I ∩ J. So we obtain the formula for the ideal join: x ∈ I ∨ J iff x ≤ i ∨ j, for some i ∈ I, j ∈ J. We call Id L the ideal lattice of L. Now observe the formulas: id(a) ∨ id(b) = id(a ∨ b), id(a) ∧ id(b) = id(a ∧ b). Since a = b implies that id(a) = id(b), these yield: The map a → id(a) embeds L into Id L. Since the definition of an ideal uses only ∨ and ≤, it applies to any joinsemilattice S.

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Proceedings of Symposium on Coastal Oceanography and Littoral Warfare (Unclassified Summary) Fleet Combat Training Center, Tactical Training Group, Pacific, San Diego, CA, August 2-5, 1993 by Navy Committee, Ocean Studies Board, National Research Council.


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