For I a proper, countably complete ideal on P(X) for some set X , can the quotient Boolean algebra P(X)/I be complete? This question was raised by Sikorski [Si] in 1949. By a simple projection argument as for measurable cardinals, it can be assumed that X is an uncountable cardinal κ, and that I is a κ-complete ideal on P(κ) containing all singletons. In this paper we provide consequences from ...

We prove that a balanced Boolean function on Sn whose Fourier transform is highly concentrated on the first two irreducible representations of Sn, is close in structure to a dictatorship, a function which is determined by the image or pre-image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on Sn generated by the tran...

Circuits of threshold elements ( Boolean input, Boolean output neurons ) have been shown to be surprisingly powerful. Useful functions such as XOR, ADD and MULTIPLY can be implemented by such circuits more efficiently than by traditional AND/OR circuits. In view of that, we have designed and built a programmable threshold element. The weights are stored on polysilicon floating gates, providing ...

1. Preliminaries. Let A be a dyadic relation defined on a finite set of n elements. It is convenient to regard A as an nXn matrix over the two-element Boolean algebra, so that a,, = l if and only if iAj. There is then an obvious correspondence of relational concepts [l; 2]2 with matrix notions, interpreting conjunction as "coordinatewise" multiplication, relative product as matrix product, etc....