Introduction to lattice theory by G Szasz

By G Szasz

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It will often be necessary to dualize lattice theoretical propositions involving, in addition to the lattice operations, the order­ ing of the lattice as well. Therefore it will be useful to find out before­ hand the dual statement of “ a < 6” . By the definition ( 1 ) a < b means that a r\b = a. The dual of the latter statement is the state­ ment “ a w 6 = a” which by L4 and (2 ) means that a 6. Hence, in lattice-theoretical duality the dual of the statement “ a b” is the statement “ a ^ 6” . Hence, we have at once the special case (which is, however, directly apparent from the definition) that the dual o f the ordering of the lattice L is the ordering of the lattice %{L).

The element a w b is found by duality considerations. For example, the diagram o f Fig. 11 represents a lat­ tice in which the operations are defined as follows: For any element x o f the lattice, o r\ x = o , o \j x = = x i r\ x = x, i w x = i, and a b c d e r\ | a b c d e a b c d e a 0 a a 0 0 b 0 b b a 0 c a 0 a b a d b 0 b 0 b e a b c d e a d c d i d b i d e c i c i i d d i d i i e i i e 16. Sublattices. Ideals According to the definition o f “ subalgebras” o f an algebra, we call every non-empty subset R o f L such that (1 ) a ,b £ R = » a r\b, a w 6 6 R a sublattice o f the lattice L.

A„ o /£ . n n (4) 0 ai < «1» • • •. «n < U ay ;= i ;= i => u s ’ u ; > av . . n = (5) 7=1 n U a7 7= 1 The two statements comprised by (4) being mutually dual, and (6) being the dual o f (5), it is sufficient to prove (5) and the first half o f (4). The latter is obtained as follows: For any ak (k = 1 , . . n a j)r \ a k = al r\ . . r\ ak^x r\ (ak j=l n r 'v a k+1 r\ . . r\ an = f l « ; j=i ak) r\ by the subsequent application o f L3 and L7. As for (5), by the conditions regarding u we have u r\ a k = v, (k = 1 , ..

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