On the dimension of the algebra generated by a boolean matrix

We investigate the subspace of the space of all n × n Boolean (0,1)-matrices, spanned by the powers of an arbitrary matrix. We estimate the maximum dimension of such spaces as a function of n and show that their bases consists of consecutive integer powers of the matrix, starting at I. We also determine the maximum dimension of the space spanned by the powers of as symmetric matrix and characterise the matrices achieving that maximum.     

On the variety generated by Murskii's algebra

It is shown that the variety generated by Murskii's algebra contains uncountably many subvarieties.Presented by K. A. Baker.     

On the global dimension of an algebra

Let algebra R = Λ/P, where Λ is a free algebra over a field w. gl. dim R: = {min n ¦? R-modules X, Y, Tor _n+1 ^R (X, Y)=0}. In order that w. gl. dim R≤2n (w. gl. dim R≤2n+1), it is necessary and sufficient that, for any two ideals of algebra Λ, a left ideal A and a right ideal B, containing ideal P, the following equation holds: $$AP^n \cap P^n B = AP^n B + P^{n + 1} (AP^n B \cap P^{n + 1} = AP^{n + 1} + P^{n + 1} B).$$      

On a function algebra which is not generated by its idempotents

Translated from Matematicheskie Zametki, Vol. 51, No. 1, pp. 3–7, January, 1992.     

Identification of boolean functions by methods of linear algebra

We prove that the problem of identification of a Boolean function by using methods of the theory of linear spaces over finite fields is solvable.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 260–268, February, 1995.     

The Banach algebra generated by a contraction

Let be a contraction on a Banach space and the Banach algebra generated by . Let be the unitary spectrum (i.e., the intersection of with the unit circle) of . We prove the following theorem of Katznelson-Tzafriri type: If is at most countable, then the Gelfand transform of vanishes on if and only if

     

On a superatomic Boolean algebra which is not generated by a well-founded sublattice

Let b denote the unboundedness number of ω^ω. That is, b is the smallest cardinality of a subset such that for everyg∈ω^ω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that , and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ₀, ℵ₁, ℵ₁, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ₀, ℵ₀, λ, 1〉, andB is not well-generated, then λ≥b.     

Krull dimension of the enveloping algebra of a semisimple Lie algebra

Let be a complex semisimple Lie algebra and be its enveloping algebra. We deduce from the work of R. Bezrukavnikov, A. Braverman and L. Positselskii that the Krull-Gabriel-Rentschler dimension of is equal to the dimension of a Borel subalgebra of .

     

On the matrix algebra of elliptic biquaternions

In this study, we introduce the concept of elliptic biquaternion matrices. Firstly, we obtain elliptic matrix representations of elliptic biquaternion matrices and establish a universal similarity factorization equality for elliptic biquaternion matrices. Afterwards, with the aid of these representations and this equality, we obtain various results on some basic topics such as generalized inverses, eigenvalues and eigenvectors, determinants, and similarity of elliptic biquaternion matrices. These valuable results may be useful for developing a perfect theory on matrix analysis over elliptic biquaternion algebra in the future.     

Transitive reduction of a rectangular boolean matrix

We consider a problem of reducing rectangular boolean matrices. Some operations of reduction are defined and some interesting properties are obtained. The results are useful for models of information retrieval systems. Especially they are useful for simplification of document-term matrices.     

Transitive reduction of a nilpotent boolean matrix

Boolean matrices are widely used in many fields, and the theory of boolean matrices is related to the algebra of relations, switching theory, and graph theory. First some basic properties of nilpotent matrices are shown in the paper. A nilpotent boolean matrix plays an important role in the theory of boolean matrices. The purpose is to present those properties of boolean matrices which are related to finding the transitive reduction of a nilpotent matrix, or an acyclic graph.     

On ideals generated by two generic quadratic forms in the exterior algebra

Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.     

The Banach algebra generated by a C0-semigroup

Let $\mathbf{T}={\{T(t)\}}_{t?0}$ be a bounded $C_0$-semigroup on a Banach space with generator A. We define $A_{\mathbf{T}}$ as the closure with respect to the operator-norm topology of the set $\{\stackrel{?}{f}(\mathbf{T}):f\in L^1({\mathbb{R}}_{+})\}$, where $\stackrel{?}{f}(\mathbf{T})={\int }_0^{\infty }f(t)T(t)\mathrm{d}t$ is the Laplace transform of $f\in L^1({\mathbb{R}}_{+})$ with respect to the semigroup T. Then $A_{\mathbf{T}}$ is a commutative Banach algebra. It is shown that if the unitary spectrum $\sigma (A)\cap \mathrm{i}\mathbb{R}$ of A is at most countable, then the Gelfand transform of $S\in A_{\mathbf{T}}$ vanishes on $\sigma (A)\cap \mathrm{i}\mathbb{R}$ if and only if, ${\lim }_{t\to \infty }6T(t)S6=0$. Some applications to the semisimplicity problem are given. To cite this article: H. Mustafayev, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Radicals of matrix rings generated by a companion matrix

The radicals of full matrix rings as well as structural matrix rings have been studied extensively and there is a well-developed theory for both. Here we initiate the radical theory for another class of matrix rings over a given ring; the matrix ring generated by the companion matrix of a polynomial over the ring.     

Model theoretic properties in the variety generated by a primal algebra

We show that the model theoretic and syntactic properties of the structures in the variety generated by a primal algebra are essentially the same as those in the variety generated by the two element Boolean algebra. The particular properties that we study here are the finite axiomatizability of their theories, the quantifier complexity of axioms for their theories, and various kinds of model completeness for these theories. It turns out that the proofs of these results can be extracted from the corresponding results for Boolean algebras if one takes the proper viewpoint.Presented by S. Burris.The author acknowledges the useful comments of the editor which led to an improved version of Proposition 1.17 as well as the formulation of some questions at the end of this paper.