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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 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]

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition. [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]

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]

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]

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]

Let G be a graph with V(G) = {v1,...,vn} and E(G) = {e1,...,.em}. In this paper, only graphs with no multiple edges are observed. The n × n matrix A = [aij}, where aij = 1 if e = v1vj and aij = 0 otherwise, is the adjacency matrix of G and is denoted by A(G). The n × n matrix B = [bij], where bij = 0 if e = vivj ϵ E (G) and bij = 1 otherwise, is the antiadjacency matrix of G and is denoted by B(G). Boolean operations on two graphs have been examined by Harary and Wilcox. Hence, this paper will consider Boolean operations on two adjancency and antiadjancency matrices of two graphs G1 and G2 with V(G2) = V(G2). Boolean operations which are reviewed on this paper are OR (V), AND (Λ), XOR(⊕), and NXOR(⊕) However, the paper only focus on operation ⊕ and ⊕. The purposes of this paper are to introduce the operations on two adjacency and antiadjacency matrices of two graphs G1 and G2 with V(G1) = V(G2), to reveal the effect on the represented graph using operations ⊕ and ⊕ on both adjacency and antiadjacency matrices, and to compare the largest eigenvalues between the matrices generated by the Boolean operations. [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]

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]

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 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]

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]

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 deals with few spectral properties for elements from a fuzzy normed algebra. It is introduced the definition of the spectral radius of an element from the fuzzy normed linear space (X, N, Λ) and some basic properties of it are shown. More precisely, we will prove that almost all the properties of this spectrum from some particular case of locally convex spaces are also true for this setup. [ABSTRACT FROM AUTHOR]

Let C be an R-coalgebra. We defined a tidy clean C-comodule M if M is a clean module over the dual algebra of C. Since every coalgebra C is a comodule over itself, a clean R-coalgebra C is a clean comodule over C. The cleanness of coalgebra C depends on its dual. In this paper, we have some properties of clean coalgebras based on the relationship between the cleanness in R-coalgebra C with the cleanness of its opposite, a clean coproduct of the family of clean coalgebras and a clean coalgebra from a dual algebra. [ABSTRACT FROM AUTHOR]

In this paper the authors studied ordered algebraic structures (semimodules) which generalize rings, fields, modules and vector spaces as known from the theory of Algebra. Additionally these structures will be ordered and will satisfy monotonicity conditions similar to the case of ordered semigroups. In this paper we discuss the following results. (1) A linearly ordered integral domain R can be embedded in a linearly ordered field. (2) Let H be an ordered semimodule over R. (a) If H is a group then x ≤ y ⇒ x □ c ≤ y □ c for all x, y ∈ R and c ∈ H, implies x □ d ≥ y □ d for d ∈ H and d ≤ e (d ∈ H). (b) If R is a ring then a ≤ b ⇒ r □ a ≤ r □ b for all a, b and r ∈ R implies s □ a ≥ s □ b for all a, b ∈ H and s ∈R. (c) If R is the positive cone of a linearly ordered ring R and H is the positive cone of a linearly ordered group H then the external composition can be continued (extended) in a unique way on R x H such that H is a linearly ordered module over R. In fact result (1) is useful in the study of Algebraic path problems [2]. [ABSTRACT FROM AUTHOR]

Sets of n-place maps closed with respect to the Menger’s superposition are called Menger algebras of n-place functions. In this paper we give characterization of a wide class of topological spaces by Menger algebras of n-place strongly quasi-open maps. [ABSTRACT FROM AUTHOR]

In this paper, we obtain Sheffer stroke reduction for some algebraic structures such as BCK-algebras, MV-algebras, Wajsberg algebras and etc. by means of representing all operators in these structures with only Sheffer stroke basic algebras. We also examine some equalities and inequalities which are used in these constructions. In addition, we examine whether there is a transition from one of these structures to another by the help of this reduction. [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]

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 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, the simple fractional iterative differential equation will be the focus of study Dβv(s) = vn(s); v(s0) = a: where s0; v0 ∈ I = [0, b], and 0 < β < 1. One class of the iterative fractional differential equations is the simple iterative fractional differential equation. In this paper, the local existence and uniqueness results for the fractional order of degree n of simple iterative fractional differential equation are proven and the solutions found by using power series. [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]

Given commutative algebra R and R -module U, Hadi et al constructed the category of supermodules CU whose objects are all pairs of R -modules (N, M) where N is direct sum of U and M contains U. If U and W are R-modules then there were relations between the categories CU and CW under some conditions. For instance, if W is direct sum of U then there exists a covariant functor from CU to CW. Further, if U and W are isomorphic then such functor is a category equivalence. In this paper we investigate the relation between two categories of supermodules over different algebras. [ABSTRACT FROM AUTHOR]

In this paper, a notion of near-subtraction (NS) ordered semigroup has been studied. Some of their properties are discussed and some characterization theorems are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we extend some definitions and properties for the 2-dimensional matrices to the 3-dimensional matrices. We present some basic concepts of the 3-dimensional matrices. Moreover, we introduce the matrix inversion, determinant and condition number vectors for the 3-D matrices. Furthermore, we state and prove Cauchy-Schwarz inequality and some other inequalities related to the 3-dimensional matrix norms. We finally investigate the effects of the third dimension with the new definitions and inequalities by some examples. [ABSTRACT FROM AUTHOR]

In 2017 we introduced a new notion of algebra called IKN-algebra. Motivated by some result on derivations (rightleft)-derivation and (leftright)- derivation in ring. In this paper we introduce derivation in INK-Algebras and investigate some important result. [ABSTRACT FROM AUTHOR]

A multi-dimensional QTRU public-key cryptosystem improves NTRU by replacing the original ring in NTRU by quaternion algebra. In this paper, we introduce QMNTR an improvement of QTRU by using a new mathematical structure of two public keys and five private keys. This modification has enhanced the public key system to be more robust and secure. [ABSTRACT FROM AUTHOR]

In this paper we study on Monte Carlo method for solving systems of linear algebraic equations (SLAE) based on shared memory. Former research demostrated that GPU can effectively speed up the computations of this issue. Our purpose is to optimize Monte Carlo method simulation on GPU memory achritecture specifically. Random numbers are organized to store in shared memory, which aims to accelerate the parallel algorithm. Bank conflicts can be avoided by our Collaborative Thread Arrays(CTA) scheme. The results of experiments show that the shared memory based strategy can speed up the computaions over than 3X at most. [ABSTRACT FROM AUTHOR]

In this paper we present a new numerical method for solving fractional differential equations (FDEs) based on Genocchi polynomials operational matrix through collocation method. The operational matrix of fractional integration in Riemann-Liouville sense is derived. The upper bound for the error of the operational matrix of fractional integration is also shown. The properties of Genocchi polynomials are utilized to reduce the given problems to a system of algebraic equations. Illustrative examples are finally given to show the simplicity, accuracy and applicability of the method. [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]

As pointed out by Professor Berndt that the following two interesting summations due to Ramanujan [1] 1 + ( x/1!)2 x/3x + (x(x - 1)/2! )2 x(x - 1)/3x(3x - 1) + ... = г3(2x + 1)/ г3(x + 1)г(3x + 1) 1 + x/1! (x - 1)(x + 1) x /(4x - 1) + x(x - 1)/2! (x - 1)(x - 2)/(x + 1)(x + 2) x(x - 1) (4x - 1)(4x - 2) + . . . = 8 г3(3x + 1) г(x + 1) 9г3(2x + 1)г(4x + 1) can be obtained quite simply by employing the classical Saalschutz's summation theorem. Recently an interesting extension of the classical Saalschutzs summation theorem has been given by Rakha and Rathie. The aim of this paper is to provide entensions of the above summation formulae due to Ramanujan. The results obtained in this paper are interesting, easily established and useful. [ABSTRACT FROM AUTHOR]

Some results concerning the Yosida approximation are given in this paper in the framework of the algebra Bp(X) which is more general than the standard Banach algebra of operators. As Yosida approximation is useful for demonstrating generation theorem for semigroups, this more general setting allows for working with universally bounded linear operators defined on locally convex spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce signed symmetric group algebras as a symmetric algebras with each edge of basis elements of symmetric group algebra are indexed by the group elements of ℤ2. We establish the generators and its relations for signed symmetric group algebras. Applying, the definition of cellularity of [2], we prove that the signed symmetric group algebras are cellular over the field of characteristic zero. Finally, we find the dimension for the irreducible representations of signed symmetric group algebras over the field of characteristic zero. [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 this paper, a collocation method based on Taylor polynomials is developed for solving systems linear differentialdifference equations with variable coefficients defined in large intervals. By using Taylor polynomials and their properties in obtaining operational matrices, the solution of the differential-difference equation system with given conditions is reduced to the solution of a system of linear algebraic equations. We first divide the large interval into M equal subintervals and then Taylor polynomials solutions are obtained in each interval, separately. Some numerical examples are given and results are compared with analytical solutions and other techniques in the literature to demonstrate the validity and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of solving a generalized algebraic Riccati equation with an indefinite sign of its quadratic term. We extend the approach introduced by Lanzon, Feng, Anderson and Rotkowitz (2008) for solving similar Riccati equations. We numerically investigate two types of iterative methods for computing the stabilizing solution. The first type of iterative methods constructs two matrix sequences, where the sum of them converges to the stabilizing solution. The second type of methods defines one matrix sequence which converges to the stabilizing solution. Computer realizations of the presented methods are numerically tested and compared on the test of family examples. Based on the experiments some conclusions are derived. [ABSTRACT FROM AUTHOR]

In this paper, we construct the element of Tanabe algebras, using that, we prove Tanabe algebras are cellular and also the dimensions of its cell modules. [ABSTRACT FROM AUTHOR]

A family of four stages high algebraic order embedded explicit six-step methods, for the numerical solution of second order initial or boundary-value problems with periodical and/or oscillating solutions, are studied in this paper. The free parameters of the new proposed methods are calculated solving the linear system of equations which is produced by requesting the vanishing of the phase-lag of the methods and the vanishing of the phase-lag's derivatives of the schemes. For the new obtained methods we investigate: • Its local truncation error (LTE) of the methods. • The asymptotic form of the LTE obtained using as model problem the radial Schrödinger equation. • The comparison of the asymptotic forms of LTEs for several methods of the same family. This comparison leads to conclusions on the efficiency of each method of the family. • The stability and the interval of periodicity of the obtained methods of the new family of embedded finite difference pairs. • The applications of the new obtained family of embedded finite difference pairs to the numerical solution of several second order problems like the radial Schrödinger equation, astronomical problems etc. The above applications lead to conclusion on the efficiency of the methods of the new family of embedded finite difference pairs. [ABSTRACT FROM AUTHOR]

Due to the dependence on computers and the internet in many areas as a result of what it provides in terms of storing and transmitting information with high speed and accuracy, but there are many obstacles appeared when using a lot of computers, and the biggest of these obstacles and problems is the protection and securing of information that is stored in the computer and sent over the internet has become a necessity Urgency to the encryption system in order to maintain the security of the information and hide it so that even if it is reached, you ensure that they cannot understand it. NTRU is a good way to maintain information security characterized by its great resistance to attacks. In this paper we define a new multidimensional algebra called tripternion algebra and created a new algebraic structure to design analogue of NTRU public key cryptosystem which called NTRTRN with high security. [ABSTRACT FROM AUTHOR]

In this paper, we present TOTRU a new multi-dimensional public-key cryptosystem. TOTRU is based on the octonion algebra as an alternative to the NTRU-mathematical structure which non-commutative, non-associative. It has been differentiated from NTRU by creating two public keys. This structure increases the complexity and security of TOTRU. We also show that TOTRU is an improvement over NTRU in terms of security and efficiency. [ABSTRACT FROM AUTHOR]