For certain weak versions of the Axiom of Choice (most notably, the Boolean Prime Ideal theorem), we obtain equivalent formulations in terms of partial orders, and filter-like objects within them intersecting certain dense sets or antichains. This allows us to prove some consequences of the Boolean Prime Ideal theorem using arguments in the style of those that use Zorn’s Lemma, or Martin’s Axiom.

The filter generator consists of a linear feedback shift register (LFSR) and a Boolean filtering function that combines bits from the shift register to create a key stream. The nonlinear combiner generator employs several (LFSRs) and a Boolean function that combines bit from all the registers to generate the key stream. A new attack on the filter generator has recently been described by Rønjom ...

Let us say that a class of upward closed sets (upsets) distributive lattices is finitary filter if it under homomorphic preimages, intersections, and directed unions. We show the only classes upsets are formed by what we call n-filters. These related to finite Boolean lattice with n atoms in same way filters two-element lattice: n-filters precisely intersections prime pre-images n-filter non-ze...

in this paper the concept of an $omega$- almost boolean ring is introduced and illistrated how a sheaf of algebras can be constructed from an $omega$- almost boolean ring over a locally boolean space.

EQ-algebras were introduced by Nov?ak in [16] as an algebraic structure of truth values for fuzzy type theory (FTT). Nov?k and De Baets [18] various kinds such good, residuated, lattice ordered EQ-algebras. In any logical structures, using filters, one can construct other structures. With this inspirations, means fantastic filters we MV-algebras. Also, study prelinear introduce a new kind filte...

In order to research the logical system whose propositional value is given in a lattice from the semantic viewpoint, Xu [7] proposed the concept of lattice implication algebras, and discussed some of their properties. Xu and Qin [8] introduced the notion of implicative filters in a lattice implication algebra, and investigated some of their properties. Turunen [5] introduced the notion of Boole...

This paper presents a characteristic set method for solving Boolean equations, which is more efficient and has better properties than the general characteristic set method. In particular, the authors give a disjoint and monic zero decomposition algorithm for the zero set of a Boolean equation system and an explicit formula for the number of solutions of a Boolean equation system. The authors al...

We present a characteristic set method for solving Boolean equations, which is more efficient and has better properties than the general characteristic set method. In particular, we give a disjoint and monic zero decomposition algorithm for the zero set of a Boolean equation system and an explicit formula for the number of solutions of a Boolean equation system. We also prove that a characteris...

At Crypto 2008, Shamir introduced a new algebraic attack called the cube attack, which allows us to solve black-box polynomials if we are able to tweak the inputs by varying an initialization vector. In a stream cipher setting where the filter function is known, we can extend it to the cube attack with annihilators: By applying the cube attack to Boolean functions for which we can find low-degr...

Stack filters are a class of non-linear, digital filters whose behaviour is defined by Positive Boolean Functions rather than the multiply-accumulate, arithmetic operators that are widely used in conventional digital filters. The implementation of stack filters that can be adapted in real-time is feasible using dynamically reconfigurable FPGAs. This paper reports on efforts to map adaptive stac...