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The notion of a rough set is an extension of set theory. The notion of set theory such as intersection and union can be applied in rough sets. The properties of the rough set in detail were introduced by Pawlak in 1982. The construction of notions in algebraic structures is motivated by set theory. This is also done in rough sets so that a notion called rough algebraic structures is obtained. Some concepts of rough algebraic structures that have been constructed include rough groups, rough rings, and rough modules. Furthermore, in multilinear algebra, we know the concept (R, S) - bimodules. Motivated by the relationship of the concept of the rough sets with algebraic structures, this paper aims to provide a definition (R, S) -bimodules in rough algebra, namely by replacing the R and S rings with the rough rings R1 and S1. Furthermore, we will give some examples of (R1, S1) -rough bimodules and it will also be shown that some of the properties (R1, S1) -bimodules still apply in (R1, S1) -rough bimodules. On the other hand, we will also give the definition of rough bisubmodules and the homomorphism of rough bimodules. [ABSTRACT FROM AUTHOR]

For a given Lie algebra L, the main problem about local 1 2 -derivations is to prove that they automatically become an 1 2 -derivation or to give examples of local 1 2 -derivation of L, which are not 1 2 -derivation. In this paper, we investigate local 1 2 -derivations on quasi-filiform Leibniz algebras. It is proved that quasi-filiform Leibniz algebras of type I, as a rule, admit a local 1 2 -derivation which is not 1 2 -derivation. [ABSTRACT FROM AUTHOR]

The focus of this paper is to examine 2-local automorphisms of the Arens algebra Lω (M,τ)) that is connected to a von Neumann algebra. Consider a von Neumann algebra M of type I accompanied by a trustworthy, regular, and semi-finite trace τ, we consider the noncommutative Arens algebra L ω (M , τ) = ∩ p ≥ 1 L p (M , τ) , where Lp (M,τ) is a Banach space defined Lp (M,τ)={x ∈ S(M,τ) :τ (| x |p)<∞} for p≥1. We will show that the surjective 2-local involutive automorphism Φ of the Arens algebra Lω (M,τ) identically acting on the center is an inner automorphism, i.e. there is a unitary element u of the Arens algebra Lω (M,τ) such that Φ(x)=uxu* for all x ∈ Lω (M,τ). It is required that the automorphism ϕx, y (x) for an arbitrary pair (x, y), satisfying the conditions ϕx, y (x)=Φ(x) and ϕx, y (y)=Φ(y), acts identically on the center of the Arens algebra. [ABSTRACT FROM AUTHOR]

We consider genetic and evolution algebras generated by non-ergodic Volterra operator. It is known that a zero-sum game generated by Volterra operator be a RPS game if and only if the operator is a non-ergodic transformation. We will call the genetic (evolution) algebra generated by non-ergodic Volterra operator RPS genetic (respectively RPS evolution) algebra. In this paper, we investigate the problem of isomorphism of two RPS genetic (evolutionary) algebras and establish necessary and sufficient conditions when two such algebras will be isomorphic. [ABSTRACT FROM AUTHOR]

In the paper local and 2-local *-automorphisms on real algebra B(X) of all bounded linear operators on a real Banach space are considered. In particular, 2-local *-automorphisms of real W*-algebra B(Hr) is described. Namely, it is proved that on real W*-algebra B(Hr) each *-automorphism is an inner and any 2-local *-automorphism is a *-automorphism. Moreover, it is proved that if X is a real Banach spaces and θ : B(X)→B(X) is a local automorphism, then θ is an automorphism. [ABSTRACT FROM AUTHOR]

In this paper, we study the holomorphic representation of Olshanetsky-Perelomov operators associated with the weyl group W of type Bn. Thus, we view the algebra ℋ (ℂn) of holomorphic functions as a module the ring 풟 of invariant differential operators under the group W. Then we study the holomorphic representation of 풟. Using the representation theory of the Weyl group W, we construct the irreducible components of ℋ (ℂn), by explicity providing their generators. [ABSTRACT FROM AUTHOR]

In this paper, we recall the notion of a quadratic stochastic operator (QSO) generated by a special distribution on infinite state space. Also, we consider the concept of genetic algebras generated by these QSOs. This paper aims to study the idempotency in the genetic algebras generated by QSO and defined by special distributions, including geometric, Poisson, mixture geometric, mixture Poisson and heterogeneous mixture geometric and Poisson distributions. [ABSTRACT FROM AUTHOR]

This paper is about the classification, up to isomorphism, of two-dimensional algebras over any basic field. It provides a full classification of such algebras in terms of their matrices of structure constants which is helpful in solving many other problems related to two-dimensional algebras. [ABSTRACT FROM AUTHOR]

Proof is important in mathematics both as a component of mathematics and as a mathematics learning tool. The ability to construct proof is one indicator of mathematical reasoning, which is a critical component of mathematics learning outcomes, particularly in Algebra. This research aims to describe the students' mathematical proving ability. This design research consists of three stages: preparing for the experiment, the design experiment, and the retrospective analysis. Focus discussion of this paper is only students' mathematical proving ability in the design experiment and the retrospective analysis. This research involved 9 students of mathematics education. The test (post-test) was conducted to collect data about students' mathematical proving ability. This research resulted in a Local Instruction Theory (LIT) of learning algebra proofs which contains 4 activities: reading proof, completing proof, examining proof, and constructing the proof. In the experiment activities, students experienced the common errors, such as assuming the conclusion to prove the conclusion, proving general statements using specific examples, not proving both conditions in a bi conditional statement, and misusing definitions. The common errors Student errors are reduced after all activities are completed. The result of this research showed that mathematical proving ability can be categorized into three score categories, low, middle, and high level. From 9 students involved in this research: 2 students in high level and 7 students in low level or 2 students in high score category, 5 students in middle score category, and 1 student in low score category. [ABSTRACT FROM AUTHOR]

In this paper, we recall the notion of a quadratic stochastic operator (QSO) generated by a special distribution on infinite state space. Also, we consider the concept of genetic algebras generated by these QSOs. This paper aims to study the idempotency in the genetic algebras generated by QSO and defined by special distributions, including geometric, Poisson, mixture geometric, mixture Poisson and heterogeneous mixture geometric and Poisson distributions. [ABSTRACT FROM AUTHOR]

This paper is about the classification, up to isomorphism, of two-dimensional algebras over any basic field. It provides a full classification of such algebras in terms of their matrices of structure constants which is helpful in solving many other problems related to two-dimensional algebras. [ABSTRACT FROM AUTHOR]

In this paper, we discuss and study the structure of inner ideals of the special linear Lie algebras of associative simple algebras. We prove that if A is an associative finite dimensional simple algebra over algebraically closed fields of positive characteristic, then every inner ideal of regular type of [A, A] is generated by a pair of idempotents. [ABSTRACT FROM AUTHOR]

In this paper the classification theorem on two dimensional real algebras is used to describe their automorphism groups and derivation algebras. [ABSTRACT FROM AUTHOR]

Fuzzy mathematics is an area of mathematics concerned with fuzzy set theory and fuzzy logic. The description provided by fuzzy sets is more accurate than the description provided by conventional sets. The paper highlights a bunch of new fuzzy mathematics approaches. The structure of Algebra G-fuzzy metric space (AGFM-space) was introduced, a new type for the concept of G-fuzzy metric space, gives a consistent extension of G-metric space, and leads to further investigations and applications. A condition was proposed for a G*-convergence sequence in the suggested area to be G*-Cauchy. Moreover, some enhanced versions of theoretical applications in AGFM-spaces are developed. [ABSTRACT FROM AUTHOR]

Hom-Lie superalgebras can be considered as the deformation of Lie superalgebras; which are ℤ2-graded generalization of Hom-Lie algebras. The motivation of this paper is to introduce the concept of isoclinism and factor set in regular Hom-Lie superalgebras. Finally, we obtain that two finite same dimensional regular Hom-Lie superalgebras are isoclinic if and only if they are isomorphic. [ABSTRACT FROM AUTHOR]

This paper is motivated by GABOR SZASZ's introduction of outer measure based on valuation for lattices. We introduce order-preserving valuation on a relative sub lattice of the lattice, a countable cover of an element in a relative sub lattice, outer measure induced by a valuation of a relative sub lattice, Ɲ* measurability and to establish certain elementary properties of induced outer measure via order preserving valuation on a relative sub lattice. To prove outer measure via order preserving valuation on a relative sub lattice is also valid the outer measure generated by measure, we define the definition of outer measure on relative sub lattices of a lattice L induced by a measure Ɲ on Q of relative sub lattices of L, measure on algebra of relative sub lattices and outer measure of relative sub lattices, we prove L(ℬ) is a sigma-algebra where L(ℬ) is the class of Ɲ* measurable relative sub lattices and deduce a corollary that Ɲ*(S) = Ɲ(S), also we prove that outer measure to any subset E of a relative sub lattice SEQσand outer measure to relative sub lattice BEQσδ are equal. Finally, by defining sigma-finite measure on Q and Ɲ* generated by Ɲ we prove a relative sub lattice E is measurable Ɲ* ⇔ E is the proper difference S ∼ B of a relative sub lattice A in Qσδ and a relative sub lattice B with Ɲ*(B) = 0. Further each relative sub lattice B with Ɲ*(B) = 0 is contained in a relative sub lattice C in Qσδ with Ɲ*(C) = 0. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present new results obtained in the study of spaces with a Δ-symmetric Banach Algebra, we defined the δ-characters functional and discuss the space of measurable function(Lo(µ)) is Δ- symmetric and is Banach algebra. Also, we define sℂ(Lo(µ)) by depended on Lo(µ) and proved the spaces is satisfies new properties in symmetric Banach algebra and prove any function in sℂ(Lo(µ)) satisfies all conditions of state functional and found corresponding between the space sℂ(Lo(µ)) and the space of all o- characters functional δch(Lo(µ)) on Lo(µ) as well as well defined. Also, we proved the functional ψ from Æ into sℂ(Lo(µ)) is continuous and state functional if every non-zero element has an inverse. We proved the quotient sℂ(Lo(µ))/Ι is also symmetric Δ –Banach algebra. [ABSTRACT FROM AUTHOR]

In this paper the correspondences between electromagnetic and fluid dynamic elements of many authors have been studied in detail, using the mathematical tool-geometric algebra. The physical as well as topological properties of the components are then analysed. Studying the different analogies indeed help to have a clear and compact visualisation of both fields and thus to understand the nature of their field mechanisms. The use of geometric algebra within mathematical physics, further envisages geometrical meaning as well as the physical interpretation of mathematical elements. [ABSTRACT FROM AUTHOR]

This paper aims to present new results obtained from the study of the set of all self-adjoint elements and symmetric ideals of Æ. Also, include defining symmetric-representation linear functional on Æ. We define p and prove every p is state functional. We discuss a new functional is a q-bounded linear functional in Æ and discover some results depended on it. We define symmetric almost sublinear functional and depended on it since that every symmetric– representation of Æ is uniformly continuous. We introduce some results that discuss connected between every position linear functional in Æ is q-bounded linear functional if it is continuous functional. And we connect between every a positive linear functional on Æ is representable if satisfying q-bounded functional condition. [ABSTRACT FROM AUTHOR]

An algorithm for solving Min-Goguen fuzzy linear systems of equations is presented in this paper. Solving linear systems of equations is subject of great scientific interest. The authors have developed fast an efficient algorithms over several algebras. Here we present, relatively simple, fast and efficient algorithm to solve fuzzy linear systems of equations for Min-Goguen algebra, logically backed by previously developed by the authors algorithms in max-min and min-max algebras. [ABSTRACT FROM AUTHOR]

The paper contains a description of generating elements of algebras of block-symmetric polynomials on finite and infinite ℓp-sums of spaces ℂs, s ∈ ℕ. In the case of a finite sum and s > 1 there are no algebraic bases in these algebras and we propose explicit formulas for algebraic dependencies between generating elements. [ABSTRACT FROM AUTHOR]

In the paper multiplicative polynomials and analytic functions on Fréchet algebras are investigated. Some connections with homomorphisms on symmetric tensor products of the Fréchet algebras are obtained. A method of constructing of multiplicative G-analytic mapping on Fréchet algebras is proposed. [ABSTRACT FROM AUTHOR]

Homomorphic Encryption, a magic wand in the field of cryptography that enables the user to perform secure computation over encrypted data, without decrypting the data. It allows cloud service providers to have access to encrypted data instead of original data, which also facilitates secure computation over encrypted data along with the secure transmission of it. NTRU, open-source public-key cryptography, based on lattices, is one of the standards of Homomorphic Encryption, can be used for secure outsourcing, the transmission of data, and performing computation on encrypted data. NTRU scheme is not vulnerable to quantum computer-based attacks, unlike existing traditional cryptographic techniques. In this paper, NTRU, the homomorphic encryption scheme is reviewed. The different variants of NTRU are also explainedthrough a detailed literature studybased on ring structure.The analysis also has been done that may help researchers to have a clear understanding of NTRU and its different variants and to distinguish them easily. The availability of different types of fields, rings based on different structures, different algebra, etc. for implementation of NTRU opens a new way for researchers to explore them to achieve a more efficient NTRU scheme for real-life applications. [ABSTRACT FROM AUTHOR]

This paper presents the developed algorithms for a tool environment for the design of complex information systems based on automata models using the algebra of high-dimensional vectors. The paper raises the problem of the little knowledge of methods of coding complex systems in the format of automata based on the algebra of hyperdimensional vectors. Methods of formalized description of systems in the form of hyperdimensional vectors and description of algorithms for transformation of formalized description using the algebra of hyperdimensional vectors are introduced. The features, advantages and disadvantages of modeling automata models using hyperdimensional vectors are listed. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the ideas of an AlA – ideal of BRK – algebra, closed AlA – ideal of BRK – algebra, and a fuzzy closed AlA – ideal of BRK – algebra. Some kinds of stuff of these ideals and relate it with others are given. And we constructed and show some propositions of it. [ABSTRACT FROM AUTHOR]

In this paper, the relationship between the algebra and graph theory are been studied especially the zero divisor graph in the ring ZN and constitute graph of these element Γ(Zn). The graph obtained by the above relation are isomorphic to some certain graphs as a complete and complete bipartite graph. The number of some vertices of Γ(Zn) are been introduced. Also, the domination of the zero divisor graph to some certain graphs are been determined. [ABSTRACT FROM AUTHOR]

In a two-dimensional case, it is shown that at n ≥ 4 every algebra with well-defined n-powers(n-powers-associative algebra) is a power-associative algebra. In connection with this result, two open questions are formulated as well. [ABSTRACT FROM AUTHOR]

In a two-dimensional case, it is shown that at n ≥ 4 every algebra with well-defined n-powers(n-powers-associative algebra) is a power-associative algebra. In connection with this result, two open questions are formulated as well. [ABSTRACT FROM AUTHOR]

The main aim of this work is to give some insights on various aspects of lower dimensional gravity via constructing the consistent cross-couplings that can be introduced in D = 2 spacetimes between a collection of BF models and a set of arbitrary matter fields. We work in a standard QFT ansatz related to space-time locality and other a priori requests on the interaction vertices and perform various cohomological computations based on the local BRST cohomology of the free model. These results are further translated at the level of the overall functional that governs the interacting BF-matter models, known as the deformed solution to the master equation, from which we deduce all the properties and features of the allowed couplings and their underlying gauge algebra. [ABSTRACT FROM AUTHOR]