Four Functions Theorem E. Roventa Computer Science Dept. Glendon College, York University 2275 Bayview Ave. Toronto (Ont.), Canada M4N 3M6 [email protected] T. Spircu Dept. of Medical Informatics "C. Davila" Univ. of Medicine and Pharmacy 8 Eroii Sanitari Blvd. Bucharest, Romania [email protected] Abstract – Isotone real functions defined over a finite Boolean algebra are studied. Weak maxima and weak minima are defined. A four functions theorem is proved. Index terms: Boolean algebra, isotonic function. 1. WEAK MAXIMA AND MINIMA It is well known [1] that any finite Boolean algebra = (X, 0, 1, ¯, , ) is isomorphic with the Boolean algebra (2N, , N, , , ) of the subsets of N = {e1, e2, …, en}, the set of atoms of . The inclusion between subsets of N transforms 2N into a partial ordered set. In what follows we will consider real functions defined on 2N that are compatible with this partial order and we will explore this structure. Definition. A real function : 2 N R is called isotonic if it satisfies the following condition: (ISO) if M P N , then ( M ) ( P) . Denote by Iso(N) the set of isotonic functions : 2 N R . Of course, the algebraic structure of R (additive semigroup, multiplicative group) is transferred on Iso(N) in an obvious manner. In what follows we will explore the transfer of the natural order structure of R , which seems slightly more interesting. Given , Iso( N ) , means ( M ) ( M ) for any M N . Of course, we could define max( , ) and min( , ) by ( M ) max( ( M ), ( M )) ( M ) min( ( M ), ( M )) for M N . But, because our aim is to extend the Ahlswede-Daykin theorem (i.e. the four functions theorem), the weak-max and the weak-min operations on Iso(N) will not be defined in the obvious manner. Definition. Given , : 2 N R , the function : 2 N R is defined recursively as follows: (1) ()() max{ (), ()} , (2) for M N , M { J M ()( M ) max ()( J ) max {( K ) ( L)}} . K , L N K L J K LM ( J ) ( J ) In a dual manner, the function : 2 N R is defined recursively as follows: (1’) ()() min{ (), ()} , (2’) for M N , M { J M ( )( M ) max ()( J ) min {(M ) (M )}}. K , L N K L J K LM ( K ) ( L) In particular, for N = {1} we have ( N ) ( N ) ()( N ) max {(), ()} max{ , }, () () ( N ) ( N ) ()( N ) min {(), ()} min { , }, () () Proposition 1. If , Iso( N ) , then , Iso( N ) . Proof. Let us consider M P N . Taking into account the recursive definition of , we have ( K ) ( L) ()( P) ()( M ) max { }. K , L N ( M ) ( M ) K LM K L P Because M K , L P , we have ( M ) ( K ) and ( M ) ( L) , thus the factor max is 1 . Hence ()( M ) ()( P) . In a dual manner, taking into account the recursive definition of , we have ( P) ( P) ( )( P) ()( M ) min { }. K , L N ( K ) ( L) K LM K L P This time the factor min is 1 , because ( P ) ( K ) and ( P ) ( L) , hence ( )( M ) ( )( P) . Proposition 2. If , Iso( N ) , then: (1) for any M , P N we have ( M ) ( P) ()( M P) ()( M P) and in particular (2) for any M N we have ( M ) ( M ) ()( M ) ( )( M ) . Proof. Let us begin with (2). For M we have, by definition, ()() ()() () () . Suppose M . Then there exists J * M and K *, L * such that K * L* J * , K * L* M and ( M ) ( M ) . ()( M ) ()( J *) ( K *) ( L*) By induction we know that ()( J *) ()( J *) ( J *) ( J *) . From here, taking into account the recursive definition of , we have ()( M ) ( )( M ) = ( M ) ( M ) ( K ) ( L) ()( J *) max ()( J ) max { } ( K , L ) ( J ) ( J ) ( K *) ( L*) J M ( M ) ( M ) ( K *) ( L*) ()( J *) ()( J *) ( M ) ( M ) . ( K *) ( L*) ( J *) ( J *) { } Let us prove now (1). Consider M , P N . If M P , then obviously we have the equality ( M ) ( P) ()( M P) ()( M P) because M N M P . If M P , then M P M P and we have ()( M P) ()( M P) ()( M P) max ( K ) ( L) } ( M P) ( M P) { K , L N K LM P K LM P ( M ) ( P) . ( M P) ( M P) Now we could exploit (recursively) the inequality ()( M P) ()( M P) ( M P) ( M P) . and the proposition is proved. The operations weak-max and the weak-min are not genuine max/min operations. For example, in general , all what can be established is that , . But the properties of these operations are well fitted for an extension of the AhlswedeDaykin theorem. 2. THE FOUR FUNCTIONS THEOREM Restate the classical 4-functions theorem as follows. Theorem (Ahlswede-Daykin). If = (L, , ) is a finite distributive lattice and if , , , : L [0, ) are such that ( x) ( y ) ( x y ) ( x y ) for every x, y L , then ( X ) (Y ) ( X Y ) ( X Y ) for every X , Y L . Here the extension of a function : L [0, ) is defined by (Z ) ( z) Z ( z) for Z L zL where Z is the characteristic function of Z (as a subset of L). Moreover, X Y and X Y are defined as follows: a) if X or Y , then X Y X Y ; b) if X , Y , then X Y {x y | x X , y Y } şi X Y {x y | x X , y Y } . The proof can be found in [2] or [3]. Consider now the Boolean algebra = (2N, , N, , , ) and suppose : 2 N [0, ) is ~ an ordinary function. Denote : Iso( N ) [0, ) the extension of to Iso(N), defined by ~ ( ) ( M ) ( M ) . M N Proposition 3. Suppose , , , : 2 N [0, ) are four ordinary functions satisfying the condition ( M ) ( P) ( M P) ( M P) for all M , P N . Then ~ ~ ~() () ~() () for any , Iso ( N ) where is the weak minimum and is the weak maximum of isotonic functions. Proof. Suppose , , , satisfy the condition ( M ) ( P) ( M P) ( M P) for M , P N . and , are isotonic functions defined on 2 N . Then, from Proposition 2 (1) above one has ( M ) ( P) ()( M P) ()( M P) If we denote ' ( X ) ( X ) ( X ) , ' ( X ) ( X ) ( X ) , ' ( X ) ( X ) ()( X ) , ' ( X ) ( X ) ()( X ) for X N , then it is obvious that ' ( M ) ' ( P) ' ( M P) ' ( M P) for M , P N . By applying Theorem 1 above (where L is the canonical Boolean algebra over 2 N . one obtains ' (2 N ) ' (2 N ) ' ( 2 N ) ' (2 N ) . Now, ' (2 N ) ' ( z) ' (M ) (M ) (M ) ~() , z2 N M N M N ~ ~ similarly ' (2 N ) () , ' (2 N ) ~() , ' (2 N ) () , and Proposition 3 is proved. REFERENCES [1] [2] [3] [4] G. Birkhoff, Lattice Theory. Amer. Math. Colloq. Publ. Vol. 25. New York, 1948. B. Bollobás, Combinatorics. Cambridge Univ. Press, Cambridge UK, 1986. P. C. Fishburn, Correlations in p. o. sets. Discrete Appl. Math. 39(1992), 173-191. P. C. Fishburn and L. Shepp, The Ahlswede-Daykin theorem. In: I. Althöfer, N. Cai, G. Dueck, L. Khachatrian, M. S. Pinsker, A. Sárközi, I. Wegener, Z. Zhang (Editors), Numbers, Information and Complexity. Kluwer Acad. Publ., Boston-Dordrecht-London, 2000.
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