Matrix Characterization of 4-Ary Algebraic Operations of Idempotent Algebras

Let (U; F) be an idempotent algebra. There is an r-ary essentially algebraic operation in F where there is not any (r + 3)-ary algebraic operation depending on at least r + 1 variables. In this paper, we prove that the set of all 4-ary algebraic operations of this algebras forms a finite De Morgan algebra, and then we characterize this De Morgan algebra.     

On the shape of solution sets of systems of (functional) equations

Solution sets of systems of linear equations over fields are characterized as being affine subspaces. But what can we say about the “shape” of the set of all solutions of other systems of equations? We study solution sets over arbitrary algebraic structures, and we give a necessary condition for a set of n-tuples to be the set of solutions of a system of equations in n unknowns over a given algebra. In the case of Boolean equations we obtain a complete characterization, and we also characterize solution sets of systems of Boolean functional equations.     

Fuzzy Boolean algebras

The purpose of this paper is to generalize the following situation: from the concrete structure $\text{B}$, we define the notion of Boolean algebras; the Stone representation theorem allows us to replace the algebraic study of Boolean algebras by a topological one. Let E be a non-empty set, and J a non-empty ordered set. Note $\text{B}$ the set of all fuzzy subsets of (E,J). We shall introduce the concept of fuzzy Boolean algebra and find a representation theorem. But it will be difficult to speak of the dual fuzzy topological space of a fuzzy Boolean algebra as we shall see further, except in certain particular cases.     

Algebraic aspects of generalized approximation spaces

The concept of approximation spaces is a key notion of rough set theory, which is an important tool for approximate reasoning about data. This paper concerns algebraic aspects of generalized approximation spaces. Concepts of R-open sets, R-closed sets and regular sets of a generalized approximation space (U,R) are introduced. Algebraic structures of various families of subsets of (U,R) under the set-inclusion order are investigated. Main results are: (1) The family of all R-open sets (respectively, R-closed sets, R-clopen sets) is both a completely distributive lattice and an algebraic lattice, and in addition a complete Boolean algebra if relation R is symmetric. (2) The family of definable sets is both an algebraic completely distributive lattice and a complete Boolean algebra if relation R is serial. (3) The collection of upper (respectively, lower) approximation sets is a completely distributive lattice if and only if the involved relation is regular. (4) The family of regular sets is a complete Boolean algebra if the involved relation is serial and transitive.     

SECOND-ORDER BOOLEAN ALGEBRAS

Opgedra aan Prof. Hennie Schutte by geleentheid van sy sestigste verjaarsdag.

Abstract

A Boolean algebra is the algebraic version of a field of sets. The complex algebra C(B) of a Boolean algebra B is defined over the power set of B; it is a field of sets with extra operations. The notion of a second-order Boolean algebra is intended to be the algebraic version of the complex algebra of a Boolean algebra. To this end a representation theorem is proved.     

An Abstract Characterization of De Morgan (2, 5)-Semigroup of 5-ary Functions

Let Ω be a set and T _Ω be the set of all 5-ary functions over Ω. In this paper, by considering five Mann’s compositions on T _Ω, we obtain a (2, 5)-semigroup and an abstract characterization of this algebra.     

On the number of essential arguments of homomorphisms between products of median algebras

In this paper we characterize classes of median-homomorphisms between products of median algebras, that depend on a given number of arguments, by means of necessary and sufficent conditions that rely on the underlying algebraic and on the underlying order structure of median algebras. In particular, we show that a median-homomorphism that take values in a median algebra that does not contain a subalgebra isomorphic to the m-dimensional Boolean algebra as a subalgebra cannot depend on more than \(m-1\) arguments. In view of this result, we also characterize the latter class of median algebras. We also discuss extensions of our framework on homomorphisms over median algebras to wider classes of algebras.     

Algebras and differential equations with additive symmetry

The set of all m-ary algebra structures on a given vector space affords, by the change of basis action, a representation of the general linear group. The invariants of a given subgroup are identified with those algebras admitting that subgroup as algebra automorphisms. Any finite dimensional representation of the additive group as automorphisms is obtained as the exponential of a nilpotent derivation. The latter can be embedded in the Lie algebra sl(2) so that the maximal vectors in an irreducible decomposition of the set of algebras as an sl(2) module are the invariants of the given action of the additive group. Dimension formulas and explicit bases are computed for the space of algebras with certain additive group actions. Employing the equivalence of the categories of m-ary algebras and systems of autonomous mth order homogeneous differential equations, the algebraic results are connected to the construction of first integrals and semi-invariants.     

The Schur group of an abelian number field

We characterize the maximum r-local index of a Schur algebra over an abelian number field K in terms of global information determined by the field K for an arbitrary rational prime, r. This completes and unifies previous results of Janusz in [G.J. Janusz, The Schur group of an algebraic number field, Ann. of Math. (2) 103 (1976) 253-281] and Pendergrass in [J.W. Pendergrass, The 2-part of the Schur group, J. Algebra 41 (1976) 422-438].     

Maximal number of Boolean functions realized by an initial Boolean automaton with two constant states

The problem of realization of Boolean functions by initial Boolean automata with two constant states and n inputs is considered. An initial Boolean automaton with two constant states and n inputs is an initial automaton with output such that in all states the output functions are n-ary constant Boolean functions 0 or 1. The maximum cardinality of set of n-ary Boolean functions, where n > 1, realized by an initial Boolean automaton with two constant states and n inputs is obtained.     

On a method of derivation of lower bounds for the nonlinearity of Boolean functions

The calculation of the exact value of the rth order nonlinearity of a Boolean function (the power of the distance between the function and the set of functions is at most r) or the derivation of a lower bound for it is a complicated problem (especially for r > 1). Lower bounds for nonlinearities of different orders in terms of the value of algebraic immunity were obtained in a number of papers. These estimates turn out to be sufficiently strong if the value of algebraic immunity is maximum or close to maximum. In the present paper, we prove a statement that allows us to obtain fairly strong lower bounds for nonlinearities of different orders and for many functions with low algebraic immunity.     

Implicative twist-structures

The twist-structure construction is used to represent algebras related to non-classical logics (e.g., Nelson algebras, bilattices) as a special kind of power of better-known algebraic structures (distributive lattices, Heyting algebras). We study a specific type of twist-structure (called implicative twist-structure) obtained as a power of a generalized Boolean algebra, focusing on the implication-negation fragment of the usual algebraic language of twist-structures. We prove that implicative twist-structures form a variety which is semisimple, congruence-distributive, finitely generated, and has equationally definable principal congruences. We characterize the congruences of each algebra in the variety in terms of the congruences of the associated generalized Boolean algebra. We classify and axiomatize the subvarieties of implicative twist-structures. We define a corresponding logic and prove that it is algebraizable with respect to our variety.     

On second-order nonlinearity and maximum algebraic immunity of some bent functions in \mathcal{PS}^{+}

The rth-order nonlinearity and algebraic immunity of Boolean function play a central role against several known attacks on stream and block ciphers. Since its maximum equals the covering radius of the rth-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the rth-order nonlinearity of a Boolean function is very complected/challenging problem, especially when r>1. In this article, we identify a subclass of \({\mathcal{D}}_{0}\) type bent functions constructed by modifying well known Dillon functions having sharper bound on their second-order nonlinearity. We further, identify a subclass of bent functions in \({\mathcal {PS}}^{+}\) class with maximum possible algebraic immunity. The result is proved by using the well known conjecture proposed by Tu and Deng (Des. Codes Cryptogr. 60(1):1–14, 2011). To obtain rth-order nonlinearity (r>2), that is, whole nonlinearity profile of the constructed bent functions is still an open problem.     

Algebraic Topological Closure Operators

For any closure operator c there is a T_o-closure operator whose lattice of closed subsets are isomorphic to that of c. A correspondence between algebraic topological (T_o) closure operators on a nonempty set X and pre-orderes (partial orders) on X is established. Equivalent conditions are obtained for a T_o-lattice to be a complete atomic Boolean algebra and for the lattice of closed subsets of an algebraic topological closure operator to be a complete atomic Boolean algebra. Further it is proved that a complete lattice is an algebraic T_o-lattice if and only if it is isomorphic to the lattice of closed subsets of some algebraic topological closure operator on a suitable set.AMS Subject Classification (1991): 06A23, 54D65.     

A combinatorial characterization of Boolean algebras

In this paper we prove that a groupoid is term equivalent to a Boolean algebra if and only if the number of n-ary term operations of the groupoid is equal to 2²ⁿ2^{2^n} for n = 0, 1, 2 and 3. This yields a partial solution of a problem posed by Berman in 1986.     

On generalized circulants over a Boolean algebra

For any given positive integer n, we give a necessary and sufficient condition for an n × nr-circulant (generalized circulant) over the Boolean algebra B = {0, 1} to be idempotent, and we present an algorith to obtain all n × n indempotent r-circulants over B for r = 0, 1,…,n ? 1.     

Boolean algebras R-generated by MV-effect algebras

We prove that every MV-effect algebra M is, as an effect algebra, a homomorphic image of its R-generated Boolean algebra. We characterize central elements of M in terms of the constructed homomorphism.     

Estimates for the number of Boolean functions realized by an initial Boolean automaton with three constant states

The problem of realization of Boolean functions by initial Boolean automata with constant states and n inputs is considered. Such automata are those whose output function coincides with one of n-ary constant Boolean functions 0 or 1 in all states. The exact value of the maximum number of n-ary Boolean functions, where n > 1, realized by an initial Boolean automaton with three constant states and n inputs is obtained.     

Polynomial parallel volume, convexity and contact distributions of random sets

We characterize convexity of a random compact set X in ℝ^d via polynomial expected parallel volume of X. The parallel volume of a compact set A is a function of r≥0 and is defined here in two steps. First we form the parallel set at distance r with respect to a one- or two-dimensional gauge body B. Then we integrate the volume of this (relative) parallel set with respect to all rotations of B. We apply our results to characterize convexity of the typical grain of a Boolean model via first contact distributions.     

On the number of cycles in local tournaments

A digraph without loops, multiple arcs and directed cycles of length two is called a local tournament if the set of in-neighbors as well as the set of out-neighbors of every vertex induces a tournament. A vertex of a strongly connected digraph is called a non-separating vertex if its removal preserves the strong connectivity of the digraph in question.In 1990, Bang-Jensen showed that a strongly connected local tournament does not have any non-separating vertices if and only if it is a directed cycle. Guo and Volkmann extended this result in 1994. They determined the strongly connected local tournament with exactly one non-separating vertex. In the first part of this paper we characterize the class of strongly connected local tournaments with exactly two non-separating vertices.In the second part of the paper we consider the following problem: Given a strongly connected local tournament D of order n with at least n+2 arcs and an integer 3≤r≤n. How many directed cycles of length r exist in D? For tournaments this problem was treated by Moon in 1966 and Las Vergnas in 1975. A reformulation of the results of the first part shows that we have characterized the class of strongly connected local tournaments with exactly two directed cycles of length n−1. Among other things we show that D has at least n−r+1 directed cycles of length r for 4≤r≤n−1 unless it has a special structure. Moreover, we characterize the class of local tournaments with exactly n−r+1 directed cycles of length r for 4≤r≤n−1 which generalizes a result of Las Vergnas.