Some Applications of Matrix—and Computer Algebra in Relation Theory

We give necessary and sufficient conditions for cofactor expansibility of determinants along a row or column for Boolean square matrices over an arbitrary Boolean algebra. First of all we define a natural decomposition of an arbitrary Boolean matrix by interior, exterior, and determinate parts. The introduced notions allow us to establish the main result of this paper. It is shown that the formulas of the cofactor expansion along a row (column) of determinants of an arbitrary square Boolean matrix hold if and only if the formulas of the cofactor expansion along the corresponding row (column) hold for determinants of its interior part.

Binary relations, matrices and inference developments

Background: Using Boolean algebra for differential diagnosis of neurological disorders involves treating neurologic findings and disease entities such as Boolean variables and combining them via OR (∨) or other logical operations. Aim: To demonstrate how Boolean algebra (mathematical logic) and Boolean matrices can be applied to support differential diagnosis in neurology. Results: Each neurologic finding was encoded as a Boolean variable (1 = present, 0 = absent). Boolean variables and Boolean matrices reflect mutual interdependencies of neurologic findings across diseases, enabling reverse inference (i.e., inferring possible diseases from a pattern of findings). Conclusion: Boolean algebra offers a formal, rigorous framework to represent and manipulate differential diagnostic relationships in neurology; it may complement clinical reasoning and computational decision support.

Enumeration of Rosenberg-type hypercompositional structures defined by binary relations

Let n be a positive integer, and C n (r) the set of all n × n r-circulant matrices over the Boolean algebra B = {0, 1}, $$G_n = \bigcup\limits_{r = 0}^{n - 1} {C_n (r)} $$ . For any fixed r-circulant matrix C (C ≠ 0) in G n , we define an operation “*” in G n as follows: A * B = ACB for any A, B in G n , where ACB is the usual product of Boolean matrices. Then (G n , *) is a semigroup. We denote this semigroup by G n (C) and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix C. Let F be an idempotent element in G n (C) and M(F) the maximal subgroup in G n (C) containing the idempotent element F. In this paper, the elements in M(F) are characterized and an algorithm to determine all the elements in M(F) is given.

Factorizing Boolean matrices using formal concepts and iterative usage of essential entries

The k-undercover Boolean matrix factorization problem aims to approximate a m×n Boolean matrix X as the Boolean product of an m×k and a k×n matrices A◦B such that X is a cover of A◦B, i.e., no representation error is allowed on the 0’s entries of the matrix X. To infer an optimal and “block-optimal” k-undercover, we propose two exact methods based on MaxSAT encodings. From a theoretical standpoint, we prove that our method of inferring “block-optimal” k-undercover is a (1 - 1/e) ≈ 0.632 approximation for the optimal k-undercover problem. From a practical standpoint, experimental results indicate that our “block-optimal” k-undercover algorithm outperforms the state-of-the-art even when compared with algorithms for the more general k-undercover Boolean Matrix Factorization problem for which only minimizing reconstruction error is required.

In this paper we wish to point out the possibility of using a complex scalar field in a constant roll inflationary model, as needed for observational viability. We extend the idea of real field inflaton with constant rate of roll to a complex field, showing the feasibility of solving Einstein Klein-Gordon equations constrained by an appropriate form of constant roll definition. As compared to the well known (two-parametric class of) real field models, there is one more degree of flexibility in constant roll inflationary solutions which is represented by an arbitrary function of time, γ(t). We work with an arbitrary but constant function γ (where γ = 0 refers to the corresponding real field model) and find new inflationary class of potentials. In this class of models, the behavior of real and complex field models are similar in some aspects, for example the solutions with large constant roll parameter are not stable and should be considered as early time transients. These field solutions relax at late time on a dual attractor trajectory. However, complex fields phase space trajectories reach this stable regime after real fields. We performed the stability analysis on γ function space solutions and found that dynamically stable trajectories in phase space are stable under γ variations. We extended this study by considering multifield models of constant roll inflation with non-canonical kinetic terms. By enlarging the size of field space, we showed that a multifield constant roll model is dynamically a single field effective theory. If field space is parametrized by N non-canonical fields, there will be N free parameters in the potential that can be attributed to the interaction between the fields.

We consider nonlinear equations in linear spaces and algebras which can be solved by a “separation of variables” obtained due to Algebraic Analysis. It is shown that the structures of linear spaces and commutative algebras (even if they are Le

Let F = [0,1] be the fuzzy algebra with operations (+,·) and the standard order ≤ : a+b=max{a+b} and a·b = min {a· b}. The subalgebra B = {0,1} of F is called the Boolean algebra, and the set F n of all n ×n matrices over F (fuzzy matrices) and the set B n of all n × n matrices over B (Boolean matrices) form partially ordered multiplicative matrix semigroups under these fuzzy operations and order. For A and B in F n , if the (i, j)-entry A ij of A is less than or equal to Bij for each i and j, then we say B dominates A and is denoted by A≤B. If A dominates a permutation matrix, then A is called a Hall matrix and the set H n of all n × n Hall matrices is a subsemigroup of F n . A ∈ F n is a regular fuzzy matrix of F n if there exists X ∈ F n such that AX A = A. Note that every invertible matrix is also a regular matrix.

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.

This advanced textbook on linear algebra and geometry covers a wide range of classical and modern topics. Differing from existing textbooks in approach, the work illustrates the many-sided applications and connections of linear algebra with functional analysis, quantum mechanics and algebraic and differential geometry. The subjects covered in some

The motivation for this paper comes from the following sources. First, one can observe that the two major concepts underlying the methods of reasoning with incomplete information are the concept of degree of truth of a piece of information and the concept of approximation of a set of information items. We shall refer to the theories employing the concept of degree of truth as to theories of fuzziness and to the theories employing the concept of approximation as to theories of roughness (see [6] for a survey). The algebraic structures relevant to these theories are residuated lattices ([7], [12], [13], [16], [17], [18], [22], [23]) and Boolean algebras with operators ([19], [21], [10], [11]), respectively. Residuated lattices provide an arithmetic of degrees of truth and Boolean algebras equipped with the appropriate operators provide a method of reasoning with approximately determined information. Both classes of algebras have a lattice structure as a basis. Second, both theories of fuzziness and theories of roughness develop generalizations of relation algebras to algebras of fuzzy relations [20] and algebras of rough relations ([4], [5], [9]), respectively. In both classes a lattice structure is a basis. Third, not necessarily distributive lattices with modal operators, which can be viewed as most elementary approximation operators, are recently developed in [24] (distributive lattices with operators are considered in [14] and [25]). With this background, our aim is to begin a systematic study of the classes of algebras that have the structure of a (not necessarily distributive) lattice and, moreover, in each class there are some operators added to the lattice which are relevant for binary relations. Our main interest is in developing relational representation theorems for the classes of lattices with operators under consideration. More precisely, we wish to guarantee that each algebra of our classes is isomorphic to an algebra of binary relations on a set. We prove the theorems of that form by suitably extending the Urquhart representation theorem for lattices ([26]) and the representation theorems presented in [1]. The classes defined in the paper are the parts which put together lead to what might be called lattice-based relation algebras. Our view is that these algebras would be the weakest structures relevant for binary relations. All the other algebras of binary relations considered in the literature would then be their signature and/or axiomatic extensions.Throughout the paper we use the same symbol for denoting an algebra or a relational system and their universes.KeywordsBinary RelationDistributive LatticeRepresentation TheoremResiduated LatticeRelation AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

There is an isomorphism between the matrices over the Boolean algebra of subsets of a k-element set and the k-tuples of Boolean binary (i.e. zero-one) matrices. This isomorphism allows many problems concerning nonbinary Boolean matrices to the referred to the binary ease. However, there are some features of the general (i.e. nonbinary) case that have not been mentioned, although they differ somewhat from the binary case. We exhibit characterizations of the linear operators that preserve several invariants of matrices over finite Boolean algebras to illustrate the differences and similarities of the general vs. the binary cases. We employ a canonical form that is useful in applying the isomorphism.

We introduce a notion of dimension of an algebraic lattice and, treating such a lattice as the congruence lattice of an algebra, we introduce the dimension of an algebra, too. We define a star-product as a special kind of subdirect product. We obtain the star-decomposition of algebras into one-dimensional factors, which generalizes the known decomposition theorems e.g. for Abelian groups, linear spaces, Boolean algebras.