By Leopoldo Nachbin (Eds.)

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Since Ax Class and Ax E x t a r e common t o b o t h t h e o r i e s , i t i s enough t o show Ax Em and Ax Num as theorems o f B. A l l theorems i n t h i s s e c t i o n a r e theorems o f B. 3 depend o n l y on Ax Class and Ax Ext. 1 a l s o be used f o r B. 4 @' f o r formulas $ w r i t t e n i n t h e p r i m i t i v e n o t a t i o n . I s h a l l l a t e r extend t h i s r e l a t i v i z a t i o n t o d e f i n e d concepts b u t we s h a l l n o t need i t i n t h i s s e c t i o n . 10, W A ( @ + 3 u ( U u-C u A @ ~ [ A n u ] ,) Ax Ref where $ i s a f o r m u l a i n L which does n o t c o n t a i n u a n d c o n t a i n s a t most A free.

4 can be g i v e n by, F*A = { F ' x : x E A } , and F-l*A have e a s i l y , tl x ( x t h e image o f a f u n c t i o n F, = { x : F'xEA}. 7 THEOREM, R0R-l c CZD-I D A S0S-l E DR+ R ' x = S I X ) ) . I n t h e r e s t o f t h i s s e c t i o n , t h e l e t t e r s F, G, H, s t r i c t e d t o functions. 8 (i) (ii) = B n D F -1 F* F-'*B . F* F - ~ * B c -B . A n B = o -+ = (F*A) n B . (F-~*A) n (F-~*B) = 0 . (v) F-l*(AnB) = (F-'*A) n (F-l*B). (vi) F-l*(A%B) = (F-'*A) % (vii) (viii) (ix) 6, g, t--f DR = D S A and h.

F i s a b i u n i q u e funct i o n from A o n t o A. A i s t h e c l a s s o f a l l permutations o f A. 1 1. 2. 3. Characterize a l l r e l a t i o n s R t h a t s a t i s f y R = R o R - l o R 4. 3 = PDF-' A (F* I . P D F i s a biunique function). - 5. 8 6. F i n d a r e l a t i o n R which i s n o t a f u n c t i o n b u t s a t i s f i e s 7. Characterize t h e r e l a t i o n s (x) (xii). W B 3 A ( B n D R - ~= R*A). R that satisfy R* R'l* R*A = R*A . 2 57 MONOTONE O P E R A T I O N S , z A unary o p e r a t i o n F i s Y,Z-monutone i f i t s a t i s f i e s : Y LA c B C Y c F [ A ) C_ F ( B ) 5 2 .