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.     

AN AXIOMATIC APPROACH TO CATEGORIES OF TOPOLOGICAL ALGEBRAS

Abstract

Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the sense of Herrlich), and T and U are (regular epi, monosource)—topological and fibre small. Such a square is called a Topological Algebraic Situation (TAS) when the following two conditions are satisfied:

1. if h: UA → UB and g: VA → VB are morphisms with Gh = Tg, there exists a morphism f: A → B such that Uf = h and Vf = g;

2. V carries U-initial monosources into T-initial mono-sources.

The functor V has many nice properties which shed light on the blending of the “topology” and “algebra”; e.g., V is a topologically algebraic functor in the sense of Y.H. Hong. An ([Etilde],[Mtilde]) version of O. Wyler's “Taut Lift Theorem” is used to show that the existence of a left adjoint to V is related to Condition (ii). It is also shown that certain topological algebraic reflections arise as Topological Algebraic Situations from algebraic and topological surjective reflections.     

An Algebraic Study of Affine Real Ultrafilters

Abstract

The families of affine semi-algebraic sets over a real-closed field Kand semi-linear sets over an ordered field enjoy many closure properties with algebraic and geometric significance. This paper studies the natural closure properies of Minkowski sums and scalar dilation. It gives an extension of the underlying vector space structure that enables the study of an arithmetic on the abstract points of their associated spectra. This arithmetic satisfies certain cancellation principles that motivates an investigation into an algebraic object weaker than a group and culminates with a version of the Jordan-Hölder theorem. With the subsequent definition of dimension we show that the collection of affine real ultrafilters in K ⁿ is n-dimensional over the scalar ultrafilters.     

Characterization of binary polynomials of idempotent algebras

In this paper, we characterize the set of all binary algebraic (or polynomial) operations of an idempotent algebra that has at least one r-ary algebraic operation, (r ≥ 2), depending on every variable such that there is no an (r+2)-ary algebraic operation depending on at least (r+1) variables. We prove that this set forms a finite Boolean algebra, and then we characterize this Boolean 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.     

Boolean Metric Spaces and Boolean Algebraic Varieties

Abstract

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps A ⁿ ?→?A ^m in a class of commutative Von Neumann regular rings including p-rings, that we have called CFG-rings. In those rings, the study of the category of algebraic varieties (i.e., sets of solutions to a finite number of polynomial equations with polynomial maps as morphisms) is equivalent to the study of a class of Boolean metric spaces, that we call here CFG-spaces.     

Finitely Deep Matrices

In the algebraic study of deep matrices ? _X () on a finite set of indices over a field, Christopher Kennedy has recently shown that there is a unique proper ideal  whose quotient is a central simple algebra. He showed that this ideal, which doesn't appear for infinite index sets, is itself a central simple algebra. In this article we extend the result to deep matrices with a finite set of 2 or more indices over an arbitrary coordinate algebra A, showing that when the coordinates are simple there is again such a unique proper ideal, and in general that the lattice of ideals of ? _X (A)/ and  are isomorphic to the lattice of ideals of the coordinate algebra A.     

COMPONENTS OF POSITIVE ELEMENTS IN RIESZ SPACES

Abstract

We find a necessary and sufficient condition on a Riesz space E such that a sub-Boolean algebra of components of a positive element generates the Boolean algebra of all components of the element. This condition yields a dual characterization of principal A-modules in the sense of D. Vuza. When applied to l-algebras, one finds improvements of results of J. Synnatzschke.     

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.     

Intuitionistic choice and classical logic

Abstract. The effort in providing constructive and predicative meaning to non-constructive modes of reasoning has almost without exception been applied to theories with full classical logic [4]. In this paper we show how to combine unrestricted countable choice, induction on infinite well-founded trees and restricted classical logic in constructively given models. These models are sheaf models over a -complete Boolean algebra, whose topologies are generated by finite or countable covering relations. By a judicious choice of the Boolean algebra we can directly extract effective content from -statements true in the model. All the arguments of the present paper can be formalised in Martin-L?f's constructive type theory with generalised inductive definitions. Received: 20 March 1997 / Revised version: 20 February 1998     

On extended algebraic immunity

Algebraic immunity (AI) measures the resistance of a Boolean function f against algebraic attack. Extended algebraic immunity (EAI) extends the concept of algebraic immunity, whose point is that a Boolean function f may be replaced by another Boolean function f ^c called the algebraic complement of f. In this paper, we study the relation between different properties (such as weight, nonlinearity, etc.) of Boolean function f and its algebraic complement f ^c . For example, the relation between annihilator sets of f and f ^c provides a faster way to find their annihilators than previous report. Next, we present a necessary condition for Boolean functions to be of the maximum possible extended algebraic immunity. We also analyze some Boolean functions with maximum possible algebraic immunity constructed by known existing construction methods for their extended 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.     

Homological Perturbation Theory and Mirror Symmetry

We explain how deformation theories of geometric objects such as complex structures,Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson al-gebras.We use homological perturbation theory to construct A∞ algebra structures on the cohomology,and their canonically defined deformations.Such constructions are used to formulate a version of A∞ algebraic mirror symmetry.     

A characterization of free generating sets in MV-algebras

MV-algebras are a generalization of Boolean algebras. As is well known, a free generating set for a Boolean algebra is characterized by the following simple algebraic condition: whenever A and B are finite disjoint subsets of X then . Our aim in this note is to give a similar characterization of free generating sets in MV-algebras. Received January 30, 2005; accepted in final form March 13, 2007.     

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.     

Free hypermodules: a categorical approach

Abstract

In category theory, the existence of free objects is very important, especially free modules that play an important role in homological algebra. Although algebraic hyperstructures are a natural extension of algebraic structures, due to the major difference between them, study-free objects in algebraic hyperstructures become very difficult. In this article, we provide a categorical approach for the construction of free hypermodules. In fact, by considering appropriate morphisms between hypermodules, we characterize free hypermodules from three different perspectives.     

Bounded Algebraic Geometry over a Free Lie Algebra

Bounded algebraic sets over a free Lie algebra F over a field k are classified in three equivalent languages: (1) in terms of algebraic sets; (2) in terms of radicals of algebraic sets; (3) in terms of coordinate algebras of algebraic sets.Supported by RFBR grant No. 02-01-00192a.__________Translated from Algebra i Logika, Vol. 44, No. 3, pp. 269–304, May–June, 2005.     

ON AN ALGEBRAIZATION OF MEASURE THEORY. ABSTRACT SEMIRINGS

Abstract

The concept of a semiring is introduced as a semilattice analogue of a semiring of sets. A detailed study of its embeddability into r-complete Boolean algebras is carried out. Some of its properties which are important from the standpoint of Measure Theory are elucidated. A representation of a semiring by a Semiring of sets is derived. A relation to implicative BCK-algebras is also mentioned.     

Skew Boolean algebras derived from generalized Boolean algebras

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/ where is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras constructed from generalized Boolean algebras B by a twisted product construction for which . In particular we study the congruence lattice of with an eye to viewing as a minimal skew Boolean cover of B. This construction is the object part of a functor from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor . This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

Coherent Continuation on the Category of (𝔤, K)-Modules

Abstract

Let 𝔤 be a complex semisimple Lie algebra. Let K be an algebraic group acting on the flag variety of 𝔤 with finitely many orbits. We give a geometric interpretation of the coherent continuation on the category of finitely generated (𝔤, K )-modules in terms of the intertwining functors on the category of K-equivariant 𝒟-modules.