We show that the principal order ideal below an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motz...

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...

A logic may be called local if each model may be used to deene the truth. In the proposition calculus each boolean algebra deene the truth by its greatest element. In the equational logic the equality between the algebra elements deene the truth. In the clause Horn logic the truth is given by the interpretation of the predicats in the model. We give a categorial framework for these logics and w...

We give exact formulas for the growth of the ideal Aλ for λ a quadratic element of the algebra of Boolean functions over the Galois field GF (2). That is, we calculate dimAkλ where Ak is the subspace of elements of degree less than or equal to k. These results clarify some of the assertions made in the article of Yang and Chen [YC] concerning the efficiency of the XL algorithm in cryptography.

The water crisis is harmful effects on human life. Also one of the most problems in the human society is water resources management. In this study, site selection of most suitable areas for artificial recharge using Boolean model. Criteria used to select the most suitable area for artificial recharge included: infiltration, slope, aquifer thickness, aquifer quality.  The layers were produced an...

A voting rule in a Boolean algebra B is an upward closed subset that contains, for each element x ∈ B, exactly one of x and ¬x. We study several aspects of voting rules, with special attention to their relationship with ultrafilters. In particular, we study the set-theoretic hypothesis that all voting rules in the Boolean algebra of subsets of the natural numbers modulo finite sets are nearly u...

We develop alternative semantics for à Lukasiewicz logic and for cancellative hoop logic according to the following idea. We formalize statements reflecting an inexact knowledge of certain (sharp) properties; we assume that all what can be known about a property is its expressive strength. To this end, we consider a Boolean algebra endowed with an automorphism group or, alternatively, with a me...

We construct and analyze an explicit basis for the homology of the boolean complex of a finite simple graph. This provides a combinatorial description of the spheres in the wedge sum representing the homotopy type of the complex. We assign a set of derangements to any finite simple graph. For each derangement, we construct a corresponding element in the homology of the complex, and the collecti...