Boolean Product Research Articles

The Boolean prime ideal theorem and products of cofinite topologies

We show: The Boolean Prime Ideal theorem is equivalent to each one of the statements: “For every family of compact spaces, for every family of basic closed sets of the product with the fip there is a family of subbasic closed sets () with the fip such that for every ”. “For every compact Loeb space (the family of all non empty closed subsets of has a choice function) and for every set X the product is compact”. (: the axiom of choice restricted to families of finite sets) implies “every well ordered product of cofinite topologies is compact” and “every well ordered basic open cover of a product of cofinite topologies has a finite subcover”. (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”. (: every filter of extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space having a base of size and for every set X of size the product is compact”.

Characteristic matrix of covering and its application to Boolean matrix decomposition

Covering-based rough sets provide an efficient means of dealing with covering data, which occur widely in practical applications. Boolean matrix decomposition has frequently been applied to data mining and machine learning. In this paper, three types of existing covering approximation operators are represented by Boolean matrices, and then used in Boolean matrix decomposition. First, we define two characteristic matrices of a covering. Through these Boolean characteristic matrices, three types of existing covering approximation operator are concisely and equivalently represented. Second, these operator representations are applied to Boolean matrix decomposition, which has a close relationship with nonnegative matrix factorization, a popular and efficient technique for machine learning. We provide a sufficient and necessary condition for a square Boolean matrix to decompose into the Boolean product of another matrix and its transpose. We then develop an algorithm for this Boolean matrix decomposition. Finally, these three covering approximation operators are axiomatized using Boolean matrices. This work presents an interesting viewpoint from which to investigate covering-based rough set theory and its applications.

On Deducing the Clique Potential of Nanoscale Combinational Circuits

In the backdrop of probabilistic computing defining the landscape of nanoscale digital logic circuits, a combinational circuit is usually specified in terms of a set of fully connected variable nodes called the ‘clique’. The clique energy or clique potential is an indicator of correct circuit operation under the premise of non-deterministic nature of device parameters at nanoscale dimensions. Also, the clique potential forms an important parameter in the probabilistic analysis of digital circuits with respect to the Markov random field model as well as in computing the probability density function. The major problem inherent in determining the clique energy expression of an arbitrary logic function that features multiple inputs and outputs is the computational complexity. The number of primary inputs and outputs governs the order of the clique potential, and hence an explosion of input and output state spaces is imminent. In this context, this paper presents a novel scheme of deriving the clique expression by segregating the primary outputs and operating upon them in parallel, thus restraining the state space expansion significantly by O(2n-1) where n signifies the number of primary outputs of the combinational logic. The important result shown in this work is that the clique potential of an arbitrary combinational circuit is equal to the Boolean product of clique potentials of its primary outputs.

Equational type characterization for σ-complete MV-algebras

In the framework of algebras with infinitary operations, an equational base for the category of σ-complete MV-algebras is given. In this way, we study some particular objects as simple algebras, directly irreducible algebras, injectives, etc. A completeness theorem with respect to the standard MV-algebra, considered as σ-complete MV-algebra, is obtained. Finally, we apply this result to the study of σ-complete Boolean algebras and σ-complete product MV-algebras.

Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term

Let $${\mathbb{BRL}}$$ denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety $${\mathbb{V}}$$ of $${\mathbb{BRL}}$$ is a unary term t in the language of bounded residuated lattices such that for every $${{\bf A} \in \mathbb{V}, t^{A}}$$ , the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense that there is a variety $${\mathbb{V}_{t} \subsetneq \mathbb{BRL}}$$ such that a variety $${\mathbb{V} \subsetneq \mathbb{BRL}}$$ admits the unary term t as a Boolean retraction term if and only if $${\mathbb{V} \subseteq \mathbb{V}_{t}}$$ . Moreover, the equation s(x) = t(x) holds in $${\mathbb{V}_{s} \cap \mathbb{V}_{t}}$$ . The radical of $${{\bf A} \in \mathbb{BRL}}$$ , with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety $${\mathbb{V} \subseteq \mathbb{V}_{t}}$$ is associated a variety $${\mathbb{V}^{r}}$$ formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in $${\mathbb{V}}$$ . Each free algebra in such $${\mathbb{V}}$$ is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety $${\mathbb{V}^{r}}$$ , also with the same number of free generators. A hierarchy of subvarieties of $${\mathbb{BRL}}$$ admitting Boolean retraction terms is exhibited.

CONDITIONALLY MONOTONE INDEPENDENCE I: INDEPENDENCE, ADDITIVE CONVOLUTIONS AND RELATED CONVOLUTIONS

We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in noncommutative probability theory and unifies the monotone and Boolean products, and moreover, the orthogonal product. Then we define the associated cumulants and calculate the limit distributions in central limit theorem and Poisson's law of small numbers. We also prove a combinatorial moment-cumulant formula using monotone partitions. We investigate some other topics such as infinite divisibility for the additive convolution and deformations of the monotone convolution. We define cumulants for a general convolution to analyze the deformed convolutions.

A Fast Output-Sensitive Algorithm for Boolean Matrix Multiplication

We use randomness to exploit the potential sparsity of the Boolean matrix product in order to speed up the computation of the product. Our new fast output-sensitive algorithm for Boolean matrix product and its witnesses is randomized and provides the Boolean product and its witnesses almost certainly. Its worst-case time performance is expressed in terms of the input size and the number of non-zero entries of the product matrix. It runs in time $\widetilde{O}(n^{2}s^{\omega/2-1})$ , where the input matrices have size n×n, the number of non-zero entries in the product matrix is at most s, ? is the exponent of the fast matrix multiplication and $\widetilde{O}(f(n))$ denotes O(f(n)log? d n) for some constant d. By the currently best bound on ?, its running time can be also expressed as $\widetilde{O}(n^{2}s^{0.188})$ . Our algorithm is substantially faster than the output-sensitive column-row method for Boolean matrix product for s larger than n 1.232 and it is never slower than the fast $\widetilde{O}(n^{\omega})$ -time algorithm for this problem. By applying the fast rectangular matrix multiplication, we can refine our upper bound further to the form $\widetilde{O}(n^{\omega(\frac{1}{2}\log_{n}s,1,1)})$ , where ?(p,q,t) is the exponent of the fast multiplication of an n p ×n q matrix by an n q ×n t matrix. We also present a partial derandomization of our algorithm as well as its generalization to include the Boolean product of rectangular Boolean matrices. Finally, we show several applications of our output-sensitive algorithms.

On the Semigroup Algebra of Binary Relations

The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB n contains an ideal K n of dimension (2 n − 1)2, and we construct an explicit isomorphism of K n with the matrix algebra M 2 n −1(F).

Boolean products of R0‐algebras

AbstractIn this paper, the (weak) Boolean representation of R0‐algebras are investigated. In particular, we show that directly indecomposable R0‐algebras are equivalent to local R0‐algebras and any nontrivial R0‐algebra is representable as a weak Boolean product of local R0‐algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Generalizations of Boolean products for lattice-ordered algebras

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FL w -algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n -potent GBL-algebras are represented as Esakia products of simple n -potent MV-algebras.

Discovery of optimal factors in binary data via a novel method of matrix decomposition

We present a novel method of decomposition of an n × m binary matrix I into a Boolean product A ○ B of an n × k binary matrix A and a k × m binary matrix B with k as small as possible. Attempts to solve this problem are known from Boolean factor analysis where I is interpreted as an object–attribute matrix, A and B are interpreted as object–factor and factor–attribute matrices, and the aim is to find a decomposition with a small number k of factors. The method presented here is based on a theorem proved in this paper. It says that optimal decompositions, i.e. those with the least number of factors possible, are those where factors are formal concepts in the sense of formal concept analysis. Finding an optimal decomposition is an NP-hard problem. However, we present an approximation algorithm for finding optimal decompositions which is based on the insight provided by the theorem. The algorithm avoids the need to compute all formal concepts and significantly outperforms a greedy approximation algorithm for a set covering problem to which the problem of matrix decomposition is easily shown to be reducible. We present results of several experiments with various data sets including those from CIA World Factbook and UCI Machine Learning Repository. In addition, we present further geometric insight including description of transformations between the space of attributes and the space of factors.

Applying Universal Algebra to Lambda Calculus

The aim of this article is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories (equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the long-standing open questions concerning the representability of λ-theories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λ-abstraction algebras. In every combinatory and λ-abstraction algebra, there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λ-abstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e. algebras that cannot be decomposed as the Cartesian product of two other non-trivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e. the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λ-theories. In one of the main results of the article we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.

BM-CENTRAL LIMIT THEOREMS FOR POSITIVE DEFINITE REAL SYMMETRIC MATRICES

The bm-central limit theorem for positive-definite real symmetric matrices and for homogeneous rooted trees are studied. The combinatorial nature of it is described. In addition, the construction of bm-product of graphs is given, which generalizes comb and boolean products. General properties of bm-extension operators are shown.

Boolean representation of bounded BCK-algebras

We define the Boolean center and the Boolean skeleton of a bounded BCK-algebra, and we use the Boolean skeleton to obtain a representation of bounded BCK-algebras, called (weak) Pierce $$b\mathbb{BCK}$$ -representation, as (weak) Boolean products of bounded BCK-algebras. We analyze the cases in which the stalks in these representations are directly indecomposable, finitely subdirectly irreducible or simple algebras. We give some examples of algebras and relative subvarieties of bounded BCK-algebras to illustrate the results.

Free algebras in varieties of Stonean residuated lattices

Given a variety $$\mathbb{V}$$ of bounded residuated lattices satisfying the Stone identity $$\neg x \lor \neg\neg x = \top$$ , the free algebras in $$\mathbb{V}$$ over a set X of cardinality |X| are represented as weak Boolean products over the Cantor space 2|X| of a family of free algebras in an associated variety of (not necessarily bounded) residuated lattices with a bottom added.

Local algebras in the representation of MV-algebras

In this paper we present several results about local MV-algebras, extending existing results given for MV-chains. The role of local MV-algebras in sheaf representation and weak boolean product is stressed and the relationship of local MV-algebras with varieties of MV-algebras is analyzed.

Weak Boolean products of bounded dually residuated $l$-monoids

Weak Boolean products of bounded dually residuated $l$-monoids

Some model theory of Bezout difference rings- a survey

This survey article is based on a joint work with Ehud Hrushovski on some Bezout difference rings ([19]). We study in a model-theoretic point of view certain classes of difference rings, namely rings with a distinguished endomorphism and more particularly rings of sequences over a field with a shift. First we will recall some results on difference fields. A difference field is a difference ring which is a field (in this case, the endomorphism is necessarily injective); and one can show that any difference field can be embedded in an inversive difference field, namely a field with an automorphism ([5]). The model theory of difference fields started in the nineties; one motivation was to understand the model-theory of non principal ultraproducts of the algebraic closure Fp of the prime fields endowed with the Frobenius maps. The existence of a model-companion for the theory of difference fields was shown by L. van den Dries and A. Macintyre ([27]). Here, we will recall the geometric axiomatization ACFA of the class of existentially closed difference fields given by Z. Chatzidakis and E. Hrushovski ([7]); they identified the different completions of ACFA, and deduce its decidability. The proofs that the non principal ultraproducts of the Fp endowed with the Frobenius maps, are models of ACFA, are much more difficult (?). An easy consequence of the decidability of ACFA is the decidability of some von Neumann commutative regular difference rings. Indeed, these are Boolean products of fields and whenever the endomorphism σ fixes the prime spectrum of the ring, we may apply transfer results due to S. Burris and H. Werner in these products ([4]). More generally we will consider Bezout difference rings in the point of view

Anillos separables, de Baer y PP-anillos

Keimel introdujo en [5], la noción de proyectabilidad en la clase de anillos reticulados y f-anillos. Aquí introducimos una noción similar olvidando la estructura reticular y de orden, e interpretando la perpendicularidad mediante la estructura multiplicativa del anillo (en la clase de anillos reducidos). Esta noción resulta coincidir con la que define la clase de PP-anillos y se relaciona también con la compacidad del espacio de ideales primos minimales y los anillos de Baer débil.

Free algebras in varieties of BL-algebras generated by a BLn-chain

AbstractFree algebras with an arbitrary number of free generators in varieties of BL-algebras generated by one BL-chain that is an ordinal sum of a finite MV-chain Ln, and a generalized BL-chain B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln, and free algebras in the variety of basic hoops generated by B. The Boolean products are taken over the Stone spaces of the Boolean subalgebras of idempotents of free algebras in the variety of MV-algebras generated by Ln.2000 Mathematics subject classification: primary 03G25, 03B50, 03B52, 03D35, 03G25, 08B20.