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 , ..