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Essential Arity Gap of Boolean Functions

Shtrakov, Slavcho (2008)

Serdica Journal of Computing

In this paper we investigate the Boolean functions with maximum essential arity gap. Additionally we propose a simpler proof of an important theorem proved by M. Couceiro and E. Lehtonen in [3]. They use Zhegalkin’s polynomials as normal forms for Boolean functions and describe the functions with essential arity gap equals 2. We use to instead Full Conjunctive Normal Forms of these polynomials which allows us to simplify the proofs and to obtain several combinatorial results concerning the Boolean functions...

Existence of Periodic Solutions for Nonlinear Neutral Dynamic Equations with Functional Delay on a Time Scale

Abdelouaheb Ardjouni, Ahcène Djoudi (2013)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Let 𝕋 be a periodic time scale. The purpose of this paper is to use a modification of Krasnoselskii’s fixed point theorem due to Burton to prove the existence of periodic solutions on time scale of the nonlinear dynamic equation with variable delay x ▵ t = - a t h x σ t + c ( t ) x ▵ ˜ t - r t + G t , x t , x t - r t , t ∈ 𝕋 , where f ▵ is the ▵ -derivative on 𝕋 and f ▵ ˜ is the ▵ -derivative on ( i d - r ) ( 𝕋 ) . We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a compact map. We show...

Flocks in universal and Boolean algebras

Gabriele Ricci (2010)

Discussiones Mathematicae - General Algebra and Applications

We propose the notion of flocks, which formerly were introduced only in based algebras, for any universal algebra. This generalization keeps the main properties we know from vector spaces, e.g. a closure system that extends the subalgebra one. It comes from the idempotent elementary functions, we call "interpolators", that in case of vector spaces merely are linear functions with normalized coefficients. The main example, we consider outside vector spaces, concerns Boolean algebras,...

Forcing for hL and hd

Andrzej Rosłanowski, Saharon Shelah (2001)

Colloquium Mathematicae

The present paper addresses the problem of attainment of the supremums in various equivalent definitions of the hereditary density hd and hereditary Lindelöf degree hL of Boolean algebras. We partially answer two problems of J. Donald Monk [13, Problems 50, 54], showing consistency of different attainment behaviour and proving that (for the variants considered) this is the best result we can expect.

Four-part semigroups - semigroups of Boolean operations

Prakit Jampachon, Yeni Susanti, Klaus Denecke (2012)

Discussiones Mathematicae - General Algebra and Applications

Four-part semigroups form a new class of semigroups which became important when sets of Boolean operations which are closed under the binary superposition operation f + g := f(g,...,g), were studied. In this paper we describe the lattice of all subsemigroups of an arbitrary four-part semigroup, determine regular and idempotent elements, regular and idempotent subsemigroups, homomorphic images, Green's relations, and prove a representation theorem for four-part semigroups.

Frankl’s conjecture for large semimodular and planar semimodular lattices

Gábor Czédli, E. Tamás Schmidt (2008)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ x . Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let m denote the number of nonzero join-irreducible elements of L . It is well-known that L consists of at most 2 m elements....

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