By R. Chuaqui

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ZID. The f o l l o w i n g theorem, whose p r o o f i s l e f t t o t h e r e a d e r , g i v e s severI n t h i s book, R o R - l c I D a l equivalent possible d e f i n i t i o n s o f functions. i s used t o say t h a t R i s a f u n c t i o n . We a l s o assume i n t h i s Chapter t h a t R, S, T a r e r e l a t i o n s . 1 (i) R0R-l cID(ii) R O R - ~ 510- R O R - ~ (v) R OR-^ LIDc -D I (vi) RoR-'cID- ( ~ v Ro) n R = 0. WSWT(SnT)oR = (SoR) n ( T O R ) . ( i i i ) R o R - lcID(iv) ( R o R - 1) n D v = 0 .

C (iv) R* ( A n 8 ) c (R* A ) n (R* 8 ) (v) = U (R* % (R* 8 ) . 5 (R* 2 A) . 8). 8 . (R* A ) n 8 . (ix) B n D R - c ~ R* R - ~ 8*. (x) R* A (xi) R* % = R* ( A n R-'* R* A ) c - R* R-'* R-'* 2. R* A = R* A . R* A . 7. Since (R-')-' = R, it is enough to show, AnR* 8 # O + -1* +. f 0. So suppose q E A n R* 8 . e. xEBnR-l*A. Thus, 8 n R-'*A f 0 , PROOF OF (i). PROOF OF (ii). R*A) n (R*A) = OF (iii). By (ii), we have that 8 n(R-'* R*8) is assumed, A n (R-l*2. R*B) = 0. R*B) n R*A PROOF R*A o o But the right side is obviously true.

The i m p l i c a t i o n o f ( v ) t o ( i ) i s proved s i m i l a r l y . I n a s i m i l a r way, t h e f o l l o w i n g can be proved. xtlq F(x) u F ( q ) gF(xuq), (v) W x w q F ( x n q ) 5F(x) " F ( Y ) . We now begin t h e s t u d y o f f i x e d p o i n t s o f monotone o p e r a t i o n s . T h i s study i s based on T a r s k i 1955. The theorems proved here w i l l be useful i n several chapters o f t h e book. We say t h a t a c l a s s X i s a hixed p a i d of a unary o p e r a t i o n F, i f We have t h a t Y, Z-monotone o p e r a t i o n s always have f i x e d p o i n t s .

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