1. Introduction. Boolean rings (or generalized Boolean algebras), defined by Stone, are rings in which every element is idempotent. Sets of postulates for these rings have been given by Stone(2) and by Stabler(8). I give in this paper a number of additional postulate-sets for such rings. The sets are all expressed in terms of ring addition and multiplication. Stone's postulates for Boolean ring...

and prove that the study of such structures is equivalent to the study of generalized Boolean algebras. We call such structures Boolean subtractive algebras since they are subtractive algebras in the sense of Crapo and Rota ([1], 3.7). Alternatively, such structures might be called generalized colonies since, as we later prove, every colony is a Boolean subtractive algebra. As an example of a B...

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 lo...

A <r-complete Boolean algebra is a Boolean algebra in which for every sequence of elements a$-, i = l, • • • , there is an element U?an, the countable union of the a», such that aiQU?an for every i, and such that if diQx for every i then U?anQx. The dual operation, countable intersection, can be introduced through complementation, and the distributive law afMJi*'a» = Uf {aC\an) and its dual can...

In this work we describe a novel approach to the classical problem of library binding. We de ne the concept of generalized matching, a Boolean constraint whose satisfaction allows us to nd all possible input assignments for which a library element matches a given Boolean relation. Completely and incompletely speci ed Boolean functions are treated as particular cases. Generalized matching suppor...

This paper presents a discrete device for neural network realized on field-programmable gate arrays (FPGA). A basic element of the implemented neural network is new type of neuron, called Boolean neuron that may be mapped directly to configurable logic blocks (CLB) or to look up table (LUT) of FPGAs. The structure and logic of the Boolean neuron allows a direct representation of the Boolean neu...

We show that Jacobs and Langen's domain for set-sharing analysis is isomorphic to the domain of positive Boolean functions, introduced by Marriott and SSndergaard for groundness dependency analysis. Viewing a set-sharing description as a minterm representation of a Boolean function leads to re-casting sharing analysis as an instantiation dependency analysis. The key idea is to view the sets of ...

We prove that the grades of simple modules indexed by boolean permutations, over incidence algebra symmetric group with respect to Bruhat order, are given Lusztig's $\mathbf{a}$-function. Our arguments combinatorial, and include a description intersection two principal order ideals when at least one permutation is boolean. An important object in our work reduced word written as minimally many r...

Given an infinite Boolean algebra B, we find a natural class of ∅-definable equivalence relations EB such that every imaginary element from B is interdefinable with an element from a sort determined by some equivalence relation from EB . It follows that B together with the family of sorts determined by EB admits elimination of imaginaries in a suitable multisorted language. The paper generalize...

We propose a new type of DNA chips with logical operations executable. In DNA computing researches, some methods have been developed to represent and evaluate Boolean functions on DNA strands. By employing the evaluation methods, we are able to deal with logical operations such as logical-and and logical-or for gene expressions. By combining with the DNA Coded Number method, we implement univer...