45 zyxwvutsrqponmlk Notre Dame Journal of Formal Logic Volume 22, Number 1, January 1981 Consequences, Consistency, and Independence in Boolean Algebras FRANK MARKHAM BROWN and SERGIURUDEANU zyxwvutsrqponmlkjihgfedc Introduction In this paper we work within an arbitrary but fixed Boolean algebra (J5, + , , ; ,0,1)and with vectorsx =(x u . . ., x n ) eB n ,y = (y u . . ., y m )e B m , where n and m are two arbitrary but fixed positive integers.* A Boolean function f:B n +Bis characterized by the Boole expansion theorem [1], [2] 1 (1) /(*!,.. .,*„)= X) f(<Xi,...,<x n )x* 1 ...x% n , zyxwvutsrqponmlkjihgfe (α 1 ,...,α M )ezyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH \Ό ,l\ n where LJ denotes iterated sum (disjunction) when the vector (a u . . ., a n ) runs over {0, 1!" and x° =x', x 1 =x. If each (a u . . ., a n )e {0, l\ n is interpreted asa number i e ί θ , . . ., 2 n li written in basis 2 andthe corresponding minterm x" 1 . .. Xn n is denoted by m.(x), formula (1) becomes (2) /(Λ :)=Σ ! o " 1 /(Om / (^) In particular a Boolean function r: B n+m >B admits the expansions (3) r{x,y)=Σ J f = Q ι r{i, y )m i {x) = λ jj =0 r(x, J)mj(y) =Σ ;: o " 1 Σ ;Γ o" i κ u )m / wm / ( J ,) *The work of F.M. Brown was supported by the National Science Foundation under Grant MCS 77 01429. Received April 9, 1979; revised May 2, 1980