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Lee, Jeong-Gon, Ravikumar Bandaru, Kul Hur, and Young Bae Jun. "Interior GE-Algebras." Journal of Mathematics 2021 (February 5, 2021): 1–10. http://dx.doi.org/10.1155/2021/6646091.
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The concepts of (commutative, transitive, left exchangeable, belligerent, antisymmetric) interior GE-algebras and bordered interior GE-algebras are introduced, and their relations and properties are investigated. Many examples are given to support these concepts. A semigroup is formed using the set of interior GE-algebras. An example is given that the set of interior GE-algebras is not a GE-algebra. It is clear that if X is a transitive (resp., commutative, belligerent, and left exchangeable) GE-algebra, then the interior GE-algebra X , f is transitive (resp., commutative, belligerent, and left exchangeable), but examples are given to show that the converse is not true in general. An interior GE-algebra is constructed using a bordered interior GE-algebra with certain conditions, and an example is given to explain this.
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Li, Guo, and Zheng Shanghua. "Left counital Hopf algebra structures on free commutative Nijenhuis algebras." SCIENTIA SINICA Mathematica 50, no. 6 (2019): 829. http://dx.doi.org/10.1360/scm-2017-0662.
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Gorshkov, Ilya, Ivan Kaygorodov, and Yury Popov. "Degenerations of Jordan Algebras and “Marginal” Algebras." Algebra Colloquium 28, no. 02 (2021): 281–94. http://dx.doi.org/10.1142/s1005386721000225.
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We describe all degenerations of the variety [Formula: see text] of Jordan algebras of dimension three over [Formula: see text]. In particular, we describe all irreducible components in [Formula: see text]. For every [Formula: see text] we define an [Formula: see text]-dimensional rigid “marginal” Jordan algebra of level one. Moreover, we discuss marginal algebras in associative, alternative, left alternative, non-commutative Jordan, Leibniz and anticommutative cases.
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Spiegel, Eugene. "Essential ideals of incidence algebras." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 68, no. 2 (2000): 252–60. http://dx.doi.org/10.1017/s1446788700001981.
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AbstractIt is determined when there exists a minimal essential ideal, or minimal essential left ideal, in the incidence algebra of a locally finite partially ordered set defined over a commutative ring. When such an ideal exists, it is described.
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Runde, Volker. "A functorial approach to weak amenability for commutative Banach algebras." Glasgow Mathematical Journal 34, no. 2 (1992): 241–51. http://dx.doi.org/10.1017/s0017089500008788.
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Let A be a commutative algebra, and let M be a bimodule over A. A derivation from A into M is a linear mapping D: A→M that satisfiesIf M is only a left A-module, by a derivation from A into M we mean a linear mapping D: A→M such thatEach A-bimodule M is trivially a left module. However, unless it is commutative, i.e.the two classes of linear operators from A into M characterized by (1) and (2), respectively, need not coincide.
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Dvurečenskij, Anatolij. "Central elements and Cantor-Bernstein's theorem for pseudo-effect algebras." Journal of the Australian Mathematical Society 74, no. 1 (2003): 121–44. http://dx.doi.org/10.1017/s1446788700003177.
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AbstractPseudo-effect algebras are partial algebras (E; +, 0, 1) with a partially defined addition + which is not necessary commutative and with two complements, left and right ones. We define central elements of a pseudo-effect algebra and the centre, which in the case of MV-algebras coincides with the set of Boolean elements and in the case of effect algebras with the Riesz decomposition property central elements are only characteristic elements. If E satisfies general comparability, then E is a pseudo MV-algebra. Finally, we apply central elements to obtain a variation of the Cantor-Bernstein theorem for pseudo-effect algebras.
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LYKOVA, ZINAIDA A. "Relations between the homologies of C*-algebras and their commutative C*-subalgebras." Mathematical Proceedings of the Cambridge Philosophical Society 132, no. 1 (2002): 155–68. http://dx.doi.org/10.1017/s0305004101005497.
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The paper concerns the identification of projective closed ideals of C*-algebras. We prove that, if a C*-algebra has the property that every closed left ideal is projective, then the same is true for all its commutative C*-subalgebras. Further, we say a Banach algebra A is hereditarily projective if every closed left ideal of A is projective. As a corollary of the stated result we show that no infinite-dimensional AW*-algebra is hereditarily projective. We also prove that, for a commutative C*-algebra A contained in [Bscr ](H), where H is a separable Hilbert space, the following conditions are equivalent: (i) A is separable; and (ii) the C*-tensor product A [otimes ]minA is hereditarily projective. Howerever, there is a non-separable, hereditarily projective, commutative C*-algebra A contained in [Bscr ](H), where H is a separable Hilbert space.
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KOLOGANI, MONA AALY, MOHAMMAD MOHSENI TAKALLO, RAJAB ALI BORZOOEI, and YOUNG BAE JUN. "Right and Left Mappings in Equality Algebras." Kragujevac Journal of Mathematics 46, no. 5 (2022): 815. http://dx.doi.org/10.46793/kgjmat2205.815k.
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The notion of (right) left mapping on equality algebras is introduced, and related properties are investigated. In order for the kernel of (right) left mapping to be filter, we investigate what conditions are required. Relations between left mapping and →-endomorphism are investigated. Using left mapping and →-endomorphism, a characterization of positive implicative equality algebra is established. By using the notion of left mapping, we define →-endomorphism and prove that the set of all →-endomorphisms on equality algebra is a commutative semigroup with zero element. Also, we show that the set of all right mappings on positive implicative equality algebra makes a dual BCK-algebra.
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Altayeva, A. B., and B. Sh Kulpeshov. "Algebras of Binary Formulas for Weakly Circularly Minimal Theories: Monotonic-to-left Case." Bulletin of Irkutsk State University. Series Mathematics 52 (2025): 120–36. https://doi.org/10.26516/1997-7670.2025.52.120.
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This article concerns the notion of weak circular minimality being a variant of o-minimality for circularly ordered structures. Algebras of binary isolating formulas are studied for ℵ0-categorical 1-transitive non-primitive weakly circularly minimal theories of convexity rank greater than 1 with a trivial definable closure having a non-trivial monotonic-to-left function acting on the universe of a structure. On the basis of the study, the authors present a description of these algebras. It is shown that for this case there exist only non-commutative algebras. A strict m-deterministicity of such algebras for some natural number m is also established.
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Singh, Surjeet, and Fawzi Al-Thukair. "Some chain conditions on weak incidence algebras." International Journal of Mathematics and Mathematical Sciences 2005, no. 15 (2005): 2389–97. http://dx.doi.org/10.1155/ijmms.2005.2389.
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LetXbe any partially ordered set,Rany commutative ring, andT=I∗(X,R)the weak incidence algebra ofXoverR. LetZbe a finite nonempty subset ofX,L(Z)={x∈X:x≤z for some z∈Z}, andM=Tez. Various chain conditions onMare investigated. The results so proved are used to construct some classes of right perfect rings that are not left perfect.
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Wardati, K. "The socle of Leavitt path algebras over a semiprime ring." Algebra and Discrete Mathematics 34, no. 1 (2022): 152–68. http://dx.doi.org/10.12958/adm1850.
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The Reduction Theorem in Leavitt path algebra over a commutative unital ring is very important to prove that the Leavitt path algebra is semiprime if and only if the ring is also semiprime. Any minimal ideal in the semiprime ring and line point will construct a left minimal ideal in the Leavitt path algebra. Vice versa, any left minimal ideal in the semiprime Leavitt path algebra can be found both minimal ideal in the semiprime ring and line point that generate it. The socle of semiprime Leavitt path algebra is constructed by minimal ideals of the semiprime ring and the set of all line points.
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JANSSEN, K., and J. VERCRUYSSE. "MULTIPLIER BI- AND HOPF ALGEBRAS." Journal of Algebra and Its Applications 09, no. 02 (2010): 275–303. http://dx.doi.org/10.1142/s0219498810003926.
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We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.
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Romano, Daniel A. "Pseudo Quasi-Ordered Residuated Systems, An Introduction." Pan-American Journal of Mathematics 1 (August 23, 2022): 12. http://dx.doi.org/10.28919/cpr-pajm/1-12.
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The concept of quasi-ordered residuated systems was introduced in 2018 by S. Bonzio and I. Chajda as a generalization of both hoop-algebras and commutative residuated lattices ordered by quasi-orders. The substructures of ideals and filters in such algebraic structures were considered by the author. This paper introduces and analyzes the concept of pseudo quasi-ordered residuated systems as a non-commutative generalization of quasi-ordered residuated systems with left and right residuum operations. Also, this paper discusses the concepts of ideals and filters in pseudo quasi-ordered residuated systems.
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ASCHIERI, PAOLO. "TWISTING ALL THE WAY: FROM ALGEBRAS TO MORPHISMS AND CONNECTIONS." International Journal of Modern Physics: Conference Series 13 (January 2012): 1–19. http://dx.doi.org/10.1142/s201019451200668x.
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Given a Hopf algebra H and an algebra A that is an H-module algebra we consider the category of left H-modules and A-bimodules [Formula: see text], where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist [Formula: see text] of H we then quantize (deform) H to [Formula: see text], A to A⋆ and correspondingly the category [Formula: see text] to [Formula: see text]. If we consider a quasitriangular Hopf algebra H, a quasi-commutative algebra A and quasi-commutative A-bimodules, we can further construct and study tensor products over A of modules and of morphisms, and their twist quantization. This study leads to the definition of arbitrary (i.e., not necessarily H-equivariant) connections on quasi-commutative A-bimodules, to extend these connections to tensor product modules and to quantize them to A⋆-bimodule connections. Their curvatures and those on tensor product modules are also determined.
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Moldovyan, Alexandr, and Nikolay Moldovyan. "New Forms of Defining the Hidden Discrete Logarithm Problem." SPIIRAS Proceedings 18, no. 2 (2019): 504–29. http://dx.doi.org/10.15622/sp.18.2.504-529.
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There are introduced novel variants of defining the discrete logarithm problem in a hidden group, which represents interest for constructing post-quantum cryptographic protocols and algorithms. This problem is formulated over finite associative algebras with non-commutative multiplication operation. In the known variant this problem, called congruent logarithm, is formulated as superposition of exponentiation operation and automorphic mapping of the algebra that is a finite non-commutative ring. Earlier it has been shown that congruent logarithm problem defined in the finite quaternion algebra can be reduced to discrete logarithm in the finite field that is an extension of the field over which the quaternion algebra is defined. Therefore further investigations of the congruent logarithm problem as primitive of the post-quantum cryptoschemes should be carried out in direction of finding new its carriers. The present paper introduces novel associative algebras possessing significantly different properties than quaternion algebra, in particular they contain no global unit. This difference had demanded a new definition of the discrete logarithm problem in a hidden group, which is different from the congruent logarithm. There are proposed several variants of such definition, in which it is used the notion of the local unite. There are considered right, left, and bi-side local unites. Two general methods for constructing the finite associative algebras with non-commutative multiplication operation are proposed. The first method relates to defining the algebras having dimension value equal to a natural number m > 1, and the second one relates to defining the algebras having arbitrary even dimensions. For the first time the digital signature algorithms based on computational difficulty of the discrete logarithm problem in a hidden group have been proposed.
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Saorín, Manuel. "Monoid gradings on algebras and the cartan determinant conjecture." Proceedings of the Edinburgh Mathematical Society 41, no. 3 (1998): 539–51. http://dx.doi.org/10.1017/s0013091500019878.
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In this work we tackle the Cartan determinant conjecture for finite-dimensional algebras through monoid gradings. Given an adequate ∑-grading on the left Artinian ring A, where ∑ is a monoid, we construct a generalized Cartan matrix with entries in ℤ∑, which is right invertitale whenever gl.dim A < ∞. That gives a positive answer to the conjecture when A admits a strongly adequate grading by an aperiodic commutative monoid. We then show that, even though this does not give a definite answer to the conjecture, it strictly widens the class of known graded algebras for which it is true.
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Schrempf, Konrad. "A standard form in (some) free fields: How to construct minimal linear representations." Open Mathematics 18, no. 1 (2020): 1365–86. http://dx.doi.org/10.1515/math-2020-0076.
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Abstract We describe a standard form for the elements in the universal field of fractions of free associative algebras (over a commutative field). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in a standard form). This completes “minimal” arithmetic in free fields since “minimal” constructions for the inverse are already known. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.
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Kadioglu, Hulya. "A computational approach for the classifications of all possible derivations of nilsolitons in dimension 9." Thermal Science 26, Spec. issue 2 (2022): 759–83. http://dx.doi.org/10.2298/tsci22s2759k.
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In mathematics and engineering, a manifold is a topological space that locally resembles Euclidean space near each point. Defining the best metric for these manifolds have several engineering and science implications from controls to optimization for generalized inner product applications of Gram Matrices that appear in these applications. These smooth geometric manifold applications can be formalized by Lie Groups and their Lie Algebras on its infinitesimal elements. Nilpotent matrices that are matrices with zero power with left-invariant metric on Lie group with non-commutative properties namely non-abelian nilsoliton metric Lie algebras will be the focus of this article. In this study, we present an algorithm to classify eigenvalues of nilsoliton derivations for 9-D non-abelian nilsoliton metric Lie algebras with non-singular Gram matrices.
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LIU, CHENG-KAI. "AN ENGEL CONDITION WITH AUTOMORPHISMS FOR LEFT IDEALS." Journal of Algebra and Its Applications 13, no. 02 (2013): 1350092. http://dx.doi.org/10.1142/s0219498813500928.
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Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[…[[σ(xn0), xn1], xn2], …], xnk] = 0 for all x ∈ L, where n0, n1, n2, …, nk are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.
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Sanghare, Mamadou. "Subrings of I-rings and S-rings." International Journal of Mathematics and Mathematical Sciences 20, no. 4 (1997): 825–27. http://dx.doi.org/10.1155/s0161171297001130.
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LetRbe a non-commutative associative ring with unity1≠0, a leftR-module is said to satisfy property (I) (resp. (S)) if every injective (resp. surjective) endomorphism ofMis an automorphism ofM. It is well known that every Artinian (resp. Noetherian) module satisfies property (I) (resp. (S)) and that the converse is not true. A ringRis called a left I-ring (resp. S-ring) if every leftR-module with property (I) (resp. (S)) is Artinian (resp. Noetherian). It is known that a subringBof a left I-ring (resp. S-ring)Ris not in general a left I-ring (resp. S-ring) even ifRis a finitely generatedB-module, for example the ringM3(K)of3×3matrices over a fieldKis a left I-ring (resp. S-ring), whereas its subringB={[α00βα0γ0α]/α,β,γ∈K}which is a commutative ring with a non-principal Jacobson radicalJ=K.[000100000]+K.[000000100]is not an I-ring (resp. S-ring) (see [4], theorem 8). We recall that commutative I-rings (resp S-tings) are characterized as those whose modules are a direct sum of cyclic modules, these tings are exactly commutative, Artinian, principal ideal rings (see [1]). Some classes of non-commutative I-rings and S-tings have been studied in [2] and [3]. A ringRis of finite representation type if it is left and right Artinian and has (up to isomorphism) only a finite number of finitely generated indecomposable left modules. In the case of commutative rings or finite-dimensional algebras over an algebraically closed field, the classes of left I-rings, left S-rings and rings of finite representation type are identical (see [1] and [4]). A ringRis said to be a ring with polynomial identity (P. I-ring) if there exists a polynomialf(X1,X2,…,Xn),n≥2, in the non-commuting indeterminatesX1,X2,…,Xn, over the centerZofRsuch that one of the monomials offof highest total degree has coefficient1, andf(a1,a2,…,an)=0for alla1,a2,…,aninR. Throughout this paper all rings considered are associative rings with unity, and by a moduleMover a ringRwe always understand a unitary leftR-module. We useMRto emphasize thatMis a unitary rightR-module.
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BRZEZIŃSKI, TOMASZ. "DIVERGENCES ON PROJECTIVE MODULES AND NON-COMMUTATIVE INTEGRALS." International Journal of Geometric Methods in Modern Physics 08, no. 04 (2011): 885–96. http://dx.doi.org/10.1142/s0219887811005440.
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A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first-order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a non-commutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras.
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Ashraf, Mohammad, Aisha Jabeen та Nazia Parveen. "On Jordan triple (σ,τ)-higher derivation of triangular algebra". Special Matrices 6, № 1 (2018): 383–93. http://dx.doi.org/10.1515/spma-2018-0032.
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Abstract Let R be a commutative ring with unity, A = Tri(A,M,B) be a triangular algebra consisting of unital algebras A,B and (A,B)-bimodule M which is faithful as a left A-module and also as a right B-module. In this article,we study Jordan triple (σ,τ)-higher derivation onAand prove that every Jordan triple (σ,τ)-higher derivation on A is a (σ,τ)-higher derivation on A.
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Krishna, Amalendu, and Husney Parvez Sarwar. "-THEORY OF MONOID ALGEBRAS AND A QUESTION OF GUBELADZE." Journal of the Institute of Mathematics of Jussieu 18, no. 5 (2017): 1051–85. http://dx.doi.org/10.1017/s1474748017000317.
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We show that for any commutative Noetherian regular ring $R$ containing $\mathbb{Q}$, the map $K_{1}(R)\rightarrow K_{1}\left(\frac{R[x_{1},\ldots ,x_{4}]}{(x_{1}x_{2}-x_{3}x_{4})}\right)$ is an isomorphism. This answers a question of Gubeladze. We also compute the higher $K$-theory of this monoid algebra. In particular, we show that the above isomorphism does not extend to all higher $K$-groups. We give applications to a question of Lindel on the Serre dimension of monoid algebras.
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Kleinfeld, Erwin, and Harry F. Smith. "Right alternative algebras with commutators in a nucleus." Bulletin of the Australian Mathematical Society 46, no. 1 (1992): 81–90. http://dx.doi.org/10.1017/s0004972700011692.
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Let A be a right alternative algebra, and [A, A] be the linear span of all commutators in A. If [A, A] is contained in the left nucleus of A, then left nilpotence implies nilpotence. If [A, A] is contained in the right nucleus, then over a commutative-associative ring with 1/2, right nilpotence implies nilpotence. If [A, A] is contained in the alternative nucleus, then the following structure results hold: (1) If A is prime with characteristic ≠ 2, then A is either alternative or strongly (–1, 1). (2) If A is a finite-dimensional nil algebra, over a field of characteristic ≠ 2, then A is nilpotent. (3) Let the algebra A be finite-dimensional over a field of characteristic ≠ 2, 3. If A/K is separable, where K is the nil radical of A, then A has a Wedderburn decomposition
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Wu, Xiaoying, and Xiaohong Zhang. "The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal." Symmetry 10, no. 10 (2018): 465. http://dx.doi.org/10.3390/sym10100465.
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For mathematical fuzzy logic systems, the study of corresponding algebraic structures plays an important role. Pseudo-BCI algebra is a class of non-classical logic algebras, which is closely related to various non-commutative fuzzy logic systems. The aim of this paper is focus on the structure of a special class of pseudo-BCI algebras in which every element is quasi-maximal (call it QM-pseudo-BCI algebras in this paper). First, the new notions of quasi-maximal element and quasi-left unit element in pseudo-BCK algebras and pseudo-BCI algebras are proposed and some properties are discussed. Second, the following structure theorem of QM-pseudo-BCI algebra is proved: every QM-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an anti-group pseudo-BCI algebra. Third, the new notion of weak associative pseudo-BCI algebra (WA-pseudo-BCI algebra) is introduced and the following result is proved: every WA-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an Abel group.
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Zhuchok, A. V. "The least dimonoid congruences on relatively free trioids." Matematychni Studii 57, no. 1 (2022): 23–31. http://dx.doi.org/10.30970/ms.57.1.23-31.
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When Loday and Ronco studied ternary planar trees, they introduced types of algebras,called trioids and trialgebras. A trioid is a nonempty set equipped with three binary associativeoperations satisfying additional eight axioms relating these operations, while a trialgebra is justa linear analog of a trioid. If all operations of a trioid (trialgebra) coincide, we obtain the notionof a semigroup (associative algebra), and if two concrete operations of a trioid (trialgebra)coincide, we obtain the notion of a dimonoid (dialgebra) and so, trioids (trialgebras) are ageneralization of semigroups (associative algebras) and dimonoids (dialgebras). Trioids andtrialgebras have close relationships with the Hopf algebras, the Leibniz 3-algebras, the Rota-Baxter operators, and the post-Jordan algebras. Originally, these structures arose in algebraictopology. One of the most useful concepts in algebra is the free object. Every variety containsfree algebras and free objects in any variety of algebras are important in the study of thatvariety. Loday and Ronco constructed the free trioid of rank 1 and the free trialgebra. Recently,the free trioid of an arbitrary rank, the free commutative trioid, the free n-nilpotent trioid, thefree rectangular triband, the free left n-trinilpotent trioid and the free abelian trioid wereconstructed and the least dimonoid congruences as well as the least semigroup congruence onthe first four free algebras were characterized. However, just mentioned congruences on freeleft (right) n-trinilpotent trioids and free abelian trioids were not considered. In this paper, wecharacterize the least dimonoid congruences and the least semigroup congruence on free left(right) n-trinilpotent trioids and free abelian trioids.
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Bagio, Dirceu, Daiana Flôres, and Alveri Sant’ana. "Inner actions of weak Hopf algebras." Journal of Algebra and Its Applications 16, no. 06 (2017): 1750118. http://dx.doi.org/10.1142/s0219498817501183.
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Let [Formula: see text] be an associative ring and [Formula: see text] idempotent elements of [Formula: see text]. In this paper we introduce the notion of [Formula: see text]-invertibility for an element of [Formula: see text] and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra [Formula: see text] and an algebra [Formula: see text], we present sufficient conditions for [Formula: see text] to admit an inner action of [Formula: see text]. We also prove that if [Formula: see text] is a left [Formula: see text]-module algebra then [Formula: see text] acts innerly on the smash product [Formula: see text] if and only if [Formula: see text] is a quantum commutative weak Hopf algebra.
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Jakobsen, Hans Plesner. "Algebras of variable coefficient quantized differential operators." Journal of Mathematical Physics 63, no. 8 (2022): 081704. http://dx.doi.org/10.1063/5.0091631.
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In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, “quantized Hermitian symmetric spaces,” an obvious problem is to describe (quantum) holomorphically induced representations in terms of some manageable structures. An intimately related problem is to decide what the algebras of quantized differential operators with variable coefficients should be. It is an immediate point that even zeroth order operators, given as multiplications by polynomials, have to be specified as, e.g., left or right multiplication operators since the polynomial algebras are replaced by quadratic, non-commutative algebras. In the settings we are interested in, there are bilinear pairings that allow us to define differential operators as duals of multiplication operators. Indeed, there are different choices of pairings which lead to quite different results. We consider three different pairings while specializing to [Formula: see text]. The pairings are between quantized generalized Verma modules and quantized holomorphically induced modules. It is a natural demand that the corresponding representations can be expressed by (matrix valued) differential operators. We show that a quantum Weyl algebra [Formula: see text] introduced by Hayashi [Commun. Math. Phys. 127(1), 129–144 (1990)] plays a fundamental role. In fact, for one pairing, the algebra of differential operators, though inherently depending on a choice of basis, is precisely matrices over [Formula: see text]. We determine explicitly the form of the (quantum) holomorphically induced representations and determine, for the different pairings, if they can be expressed by differential operators.
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Brown, Lawrence G. "Close hereditary C*-subalgebras and the structure of quasi-multipliers." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, no. 2 (2017): 263–92. http://dx.doi.org/10.1017/s0308210516000172.
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We answer a question of Takesaki by showing that the following can be derived from the thesis of Shen: if A and B are σ-unital hereditary C*-subalgebras of C such that ‖p – q‖ < 1, where p and q are the corresponding open projections, then A and B are isomorphic. We give some further elaborations and counterexamples with regard to the σ-unitality hypothesis. We produce a natural one-to-one correspondence between complete order isomorphisms of C*-algebras and invertible left multipliers of imprimitivity bimodules. A corollary of the above two results is that any complete order isomorphism between σ-unital C*-algebras is the composite of an isomorphism with an inner complete order isomorphism. We give a separable counterexample to a question of Akemann and Pedersen; namely, the space of quasi-multipliers is not linearly generated by left and right multipliers. But we show that the space of quasi-multipliers is multiplicatively generated by left and right multipliers in the σ-unital case. In particular, every positive quasi-multiplier is of the form T*T for T a left multiplier. We show that a Lie theory consequence of the negative result just stated is that the map sending T to T*T need not be open, even for very nice C*-algebras. We show that surjective maps between σ-unital C*-algebras induce surjective maps on left, right, and quasi-multipliers. (The more significant similar result for multipliers is Pedersen's non-commutative Tietze extension theorem.) We elaborate the relations of the above with continuous fields of Hilbert spaces and in so doing answer a question of Dixmier and Douady. We discuss the relationship of our results to the theory of perturbations of C*-algebras.
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KOVAČEVIĆ, DOMAGOJ, and STJEPAN MELJANAC. "KAPPA-MINKOWSKI SPACETIME, KAPPA-POINCARÉ HOPF ALGEBRA AND REALIZATIONS." International Journal of Geometric Methods in Modern Physics 09, no. 06 (2012): 1261009. http://dx.doi.org/10.1142/s0219887812610099.
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The κ-Minkowski spacetime and Lorentz algebra are unified in unique Lie algebra. Introducing commutative momenta, a family of κ-deformed Heisenberg algebras and κ-deformed Poincaré algebras are defined. They are determined by the matrix depending on momenta. Realizations and star product are defined and analyzed in general. The relation among the coproduct of momenta, realization and the star product is pointed out. Hopf algebra of the Poincaré algebra, related to the covariant realization, is presented in unified covariant form. Left–right dual realizations and dual algebra are introduced and considered. The generalized involution and the star inner product are defined and analyzed. Partial integration and deformed trace property are obtained in general. The translation invariance of the star product is pointed out.
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KHARLAMPOVICH, OLGA, and ALEXEI MYASNIKOV. "UNDECIDABILITY OF THE FIRST ORDER THEORIES OF FREE NONCOMMUTATIVE LIE ALGEBRAS." Journal of Symbolic Logic 83, no. 3 (2018): 1204–16. http://dx.doi.org/10.1017/jsl.2017.80.
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AbstractLet R be a commutative integral unital domain and L a free noncommutative Lie algebra over R. In this article we show that the ring R and its action on L are 0-interpretable in L, viewed as a ring with the standard ring language $+ , \cdot ,0$. Furthermore, if R has characteristic zero then we prove that the elementary theory $Th\left( L \right)$ of L in the standard ring language is undecidable. To do so we show that the arithmetic ${\Bbb N} = \langle {\Bbb N}, + , \cdot ,0\rangle $ is 0-interpretable in L. This implies that the theory of $Th\left( L \right)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.
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Gumerov, R. N., and E. V. Lipacheva. "Topological Grading of Semigroup C*-Algebras." Herald of the Bauman Moscow State Technical University. Series Natural Sciences, no. 3 (90) (June 2020): 44–55. http://dx.doi.org/10.18698/1812-3368-2020-3-44-55.
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The paper deals with the abelian cancellative semigroups and the reduced semigroup C*-algebras. It is supposed that there exist epimorphisms from the semigroups onto the group of integers modulo n. For these semigroups we study the structure of the reduced semigroup C*-algebras which are also called the Toeplitz algebras. Such a C*-algebra can be defined for any non-abelian left cancellative semigroup. It is a very natural object in the category of C*-algebras because this algebra is generated by the left regular representation of a semigroup. In the paper, by a given epimorphism σ we construct the grading of a semigroup C*-algebra. To this aim the notion of the σ-index of a monomial is introduced. This notion is the main tool in the construction of the grading. We make use of the σ-index to define the linear independent closed subspaces in the semigroup C*-algebra. These subspaces constitute the C*-algebraic bundle, or the Fell bundle, over the group of integers modulo n. Moreover, it is shown that this grading of the reduced semigroup C*-algebra is topological. As a corollary, we obtain the existence of the contractive linear operators that are non-commutative analogs of the Fourier coefficients. Using these operators, we prove the result on the geometry of the underlying Banach space of the semigroup C*-algebra
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Zhang, James J. "Quadratic algebras with few relations." Glasgow Mathematical Journal 39, no. 3 (1997): 323–32. http://dx.doi.org/10.1017/s0017089500032249.
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Throughout V will be a finite dimensional vector space over a field k and T(V) will denote the tensor algebra over V. For simplicity the symbol ⊗ will be omitted in the writing of the elements of T(V). Let be a basis of V ordered by Xi<Xi+1 for all i. Then we order the non-commutative monomials and 1 ≤ is ≤ n for s = 1,…, l} lexicographically from the left. D. Anick [1, p. 652] defines the high term of an element b in T(V) to be the highest monomial appearing in b. As a consequence of [1,3.2], if the set of the high terms of homogeneous relations is combinatorically free in the sense of no overlap ambiguities, then the connected algebra has global dimension 2. The purpose of this note is to prove this result and more for quadratic algebras under other hypotheses on the relations.
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Lin, Hong-Yang, Marc Cahay, Badri N. Vellambi, and Dennis Morris. "A Generalization of Quaternions and Their Applications." Symmetry 14, no. 3 (2022): 599. http://dx.doi.org/10.3390/sym14030599.
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There are a total of 64 possible multiplication rules that can be defined starting with the generalized imaginary units first introduced by Hamilton. Of these sixty-four choices, only eight lead to non-commutative division algebras: two are associated to the left- and right-chirality quaternions, and the other six are generalizations of the split-quaternion concept first introduced by Cockle. We show that the 4×4 matrix representations of both the left- and right-chirality versions of the generalized split-quaternions are algebraically isomorphic and can be related to each other by 4×4 permutation matrices of the C2×C2 group. As examples of applications of the generalized quaternion concept, we first show that the left- and right-chirality quaternions can be used to describe Lorentz transformations with a constant velocity in an arbitrary spatial direction. Then, it is shown how each of the generalized split-quaternion algebras can be used to solve the problem of quantum-mechanical tunneling through an arbitrary one-dimensional (1D) conduction band energy profile. This demonstrates that six different spinors (4×4 matrices) can be used to represent the amplitudes of the left and right propagating waves in a 1D device.
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YAPTI ÖZKURT, Zeynep. "Bases of fixed point subalgebras on nilpotent Leibniz algebras." Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 26, no. 1 (2024): 272–78. http://dx.doi.org/10.25092/baunfbed.1332488.
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Let K be a field of characteristic zero, X={x_(1,) x_2,…,x_n} and R_m={r_(1,) ,…,r_m} be two sets of variables, F be the free left nitpotent Leibniz algebra generated by X, and K[R_m ] be the commutative polynomial algebra generated by R_m over the base field K. The fixed point subalgebra of an automorphism φ is the subalgebra of F consisting of elements that are invariant under the automorphism. In this work, we consider specific automorphisms of F and determine the fixed point subalgebras of these automorphisms. Then, we find bases of these fixed point subalgebras. In addition, we get generators of these subalgebras as a free K[R_m ] -module.
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Avila, Julio Cesar, Martín Eduardo Frías-Armenta, and Elifalet López-González. "A-Differentiability over Associative Algebras." Mathematics 13, no. 10 (2025): 1619. https://doi.org/10.3390/math13101619.
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The unital associative algebra structure A on Rn allows for defining elementary functions and functions defined by convergent power series. For these, the usual derivative has a simple form even for higher-order derivatives, which allows us to have the A-calculus. Thus, we introduce A-differentiability. Rules for A-differentiation are obtained: a product rule, left and right quotients, and a chain rule. Convergent power series are A-differentiable, and their A-derivatives are the power series defined by their A-derivatives. Therefore, we use associative algebra structures to calculate the usual derivatives. These calculations are carried out without using partial derivatives, but only by performing operations in the corresponding algebras. For f(x)=x2, we obtain dfx(v)=vx+xv, and for f(x)=x−1, dfx(v)=−x−1vx−1. Taylor approximations of order k and expansion by the Taylor series are performed. The pre-twisted differentiability for the case of non-commutative algebras is introduced and used to solve families of quadratic ordinary differential equations.
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BAI, LI, and SHUANHONG WANG. "TWISTED DRINFELD DOUBLES AND REPRESENTATIONS OF A HOPF ALGEBRA." Journal of Algebra and Its Applications 11, no. 06 (2012): 1250118. http://dx.doi.org/10.1142/s0219498812501186.
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In this paper, we consider a class of non-commutative and non-cocommutative Hopf algebras Hp(α, q, m) and then show that these Hopf algebras can be realized as a quantum double of certain Hopf algebras with respect to Hopf skew pairings (Ap(q, m), Bp(q, m), τα). Furthermore, using the Hopf skew pairing with appropriate group homomorphisms: ϕ : π → Aut (Ap(q, m)) and ψ : π → Aut (Bp(q, m)), we construct a twisted Drinfeld double D(Ap(q, m), Bp(q, m), τ; ϕ, ψ) which is a Turaev [Formula: see text]-coalgebra, where the group [Formula: see text] is a twisted semi-direct square of a group π. Then we obtain its quasi-triangular Turaev [Formula: see text]-coalgebra structure. We also study irreducible representations of Hp(1, q, m) and construct a corresponding R-matrix. Finally, we introduce the notion of a left Yetter–Drinfeld category over a Turaev group coalgebra and show that such a category is a Turaev braided group category by a direct proof, without center construction. As an application, we consider the case of the quasi-triangular Turaev [Formula: see text]-coalgebra structure on our twisted Drinfeld double.
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Stepanova, A. A. "S-acts over a Well-ordered Monoid with Modular Congruence Lattice." Bulletin of Irkutsk State University. Series Mathematics 35 (2021): 87–102. http://dx.doi.org/10.26516/1997-7670.2021.35.87.
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This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left $S$act over monoid $S$ is a set $A$ upon which $S$ acts unitarily on the left. In this paper, we consider $S$acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid $S$ is a linearly ordered set with a minimal element and with a binary operation $ \ max$, with respect to which $S$ is obviously a commutative monoid; a well-ordered monoid $S$ is a well-ordered set with a binary operation $ \ max$, with respect to which $S$ is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with M.S. Kazak, which describes $S$acts over linearly ordered monoids with a linearly ordered congruence lattice and $S$-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.
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Moldovyan, Dmitry, Alexander Moldovyan, and Denis Guryanov. "Blind signature protocols based on hidden discrete logarithm problem." Information and Control Systems, no. 3 (June 15, 2020): 71–78. http://dx.doi.org/10.31799/1684-8853-2020-3-71-78.
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Introduction: The progress in the development of quantum computing has raised the problem of constructing post-quantum two-key cryptographic algorithms and protocols, i.e. crypto schemes resistant to attacks from quantum computers. Based on the hidden discrete logarithm problem, some practical post-quantum digital signature schemes have been developed. The next step could be the development of post-quantum blind signature protocols. Purpose: To develop blind signature protocols based on the computational difficulty of the hidden discrete logarithm problem. Method: The use of blinding factors introduced by the client during the blind signature protocol when the parameters necessary for the blind signature formation are passed to the signatory. Results: It has been proposed to use blinding multipliers of two different types: left-sided and right-sided ones. With them, you can develop blind signature protocols on the base of schemes with a verification equation defined in non-commutative algebraic structures. New blind signature protocols have been developed, based on the computational difficulty of the hidden discrete logarithm problem. As the algebraic carrier for the developed protocols, finite non-commutative associative algebras of two types are used: 1) those with a global two-sided unit, and 2) those with a large set of global left units. Practical relevance: The proposed protocols have a high performance and can be successfully implemented either in software or in hardware.
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Singh, Tejinder P. "Gravitation, and quantum theory, as emergent phenomena." Journal of Physics: Conference Series 2533, no. 1 (2023): 012013. http://dx.doi.org/10.1088/1742-6596/2533/1/012013.
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Abstract There must exist a reformulation of quantum field theory, even at low energies, which does not depend on classical time. The octonionic theory proposes such a reformulation, leading to a pre-quantum pre-spacetime theory. The ingredients for constructing such a theory, which is also a unification of the standard model with gravitation, are : (i) the pre-quantum theory of trace dynamics – a matrix-valued Lagrangian dynamics, (ii) the spectral action principle of non-commutative geometry, (iii) the number system known as the octonions, for constructing a non-commutative manifold and for defining elementary particles via Clifford algebras, (iv) a Lagrangian with E 8 × E 8 symmetry. The split bioctonions define a sixteen dimensional space (with left-right symmetry) whose geometry (evolving in Connes time) relates to the four known fundamental forces, while predicting two new forces, SU(3) grav and U(1) grav . This latter interaction is possibly the theoretical origin of MOND. Coupling constants of the standard model result from left-right symmetry breaking, and their values are theoretically determined by the characteristic equation of the exceptional Jordan algebra of the octonions. The quantum-to-classical transition, precipitated by the entanglement of a critical number of fermions, is responsible for the emergence of classical spacetime, and also for the familiar formulation of quantum theory on a spacetime background.
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Ashraf, Mohammad, Aisha Jabeen та Mohd Akhtar. "Generalized Jordan triple (σ,τ)-higher derivation on triangular algebras". Filomat 33, № 8 (2019): 2285–94. http://dx.doi.org/10.2298/fil1908285.
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Let R be a commutative ring with unity, U = Tri(A,M,B) be a triangular algebra consisting of unital algebras A,B and (A,B)-bimodule M which is faithful as a left A-module and also as a right B-module. Let ? and ? be two automorphisms of U. A family ? = {?n}n?N of R-linear mappings ?n : U ? U is said to be a generalized Jordan triple (?,?)-higher derivation on A if there exists a Jordan triple (?,?)-higher derivation D = {dn}n?N on U such that ?0 = IU, the identity map of U and ?n(XYX) = ?i+j+k=n ?i(?n-i(X))dj(?k?i(Y))dk(?n-k(X)) holds for all X,Y ? U and each n ? N. In this article, we study generalized Jordan triple (?,?)-higher derivation on A and prove that every generalized Jordan triple (?,?)-higher derivation on U is a generalized (?,?)-higher derivation on U.
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Gryshchuk, Serhii. "Monogenic functions in commutative complex algebras of the second rank and the Lamé equilibrium system for some class of plane orthotropy." Ukrainian Mathematical Bulletin 16, no. 3 (2019): 345–46. http://dx.doi.org/10.37069/1810-3200-2019-16-3-3.
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We consider a class of plane orthotropic deformations of the form \(\varepsilon_{x} = \sigma_x + a_{12} \sigma_y\), \(\gamma_{xy} = 2 \left(p-a_{12}\right) \tau_{xy}\), \(\varepsilon_{y}= a_{12}\sigma_x+\sigma_y\), where \(\sigma_x\), \(\tau_{xy}\), \(\sigma_y\) and \(\varepsilon_{x}\), \(\frac{\gamma_{xy}}{2}\), \(\varepsilon_{y}\) are components of the stress tensor and the deformation tensor, respectively, real parameters \(p\) and \(a_{12}\) satisfy the inequalities: \(-1 \lt p \lt 1\), \(-1 \lt a_{12} \lt p\). A class of solutions of the Lamé equilibrium system for displacements is built in the form of linear combinations of components of ''analytic'' functions which take values in commutative and associative two-dimensional algebras with unity over the field of complex numbers.
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Alkenani, Ahmad N., Mohammad Ashraf та Aisha Jabeen. "Nonlinear generalized Jordan (σ, Γ)-derivations on triangular algebras". Special Matrices 6, № 1 (2017): 216–28. http://dx.doi.org/10.1515/spma-2017-0008.
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Abstract Let R be a commutative ring with identity element, A and B be unital algebras over R and let M be (A,B)-bimodule which is faithful as a left A-module and also faithful as a right B-module. Suppose that A = Tri(A,M,B) is a triangular algebra which is 2-torsion free and σ, Γ be automorphisms of A. A map δ:A→A (not necessarily linear) is called a multiplicative generalized (σ, Γ)-derivation (resp. multiplicative generalized Jordan (σ, Γ)-derivation) on A associated with a (σ, Γ)-derivation (resp. Jordan (σ, Γ)-derivation) d on A if δ(xy) = δ(x)r(y) + σ(x)d(y) (resp. σ(x<sup>2</sup>) = δ(x)r(x) + δ(x)d(x)) holds for all x, y Є A. In the present paper it is shown that if δ:A→A is a multiplicative generalized Jordan (σ, Γ)-derivation on A, then δ is an additive generalized (σ, Γ)-derivation on A.
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Bavula, V. V. "The Group of Automorphisms of the Algebra of one-Sided Inverses of a Polynomial Algebra. II." Proceedings of the Edinburgh Mathematical Society 58, no. 3 (2015): 543–80. http://dx.doi.org/10.1017/s0013091514000303.
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AbstractThe algebra of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra is isomorphic to the algebra of scalar integro-differential operators provided that char(K) = 0. Ignoring the non-Noetherian property, the algebra belongs to a family of algebras like the nth Weyl algebra An and the polynomial algebra P2n. Explicit generators are found for the group Gn of automorphisms of the algebra and for the group of units of (both groups are huge). An analogue of the Jacobian homomorphism AutK-alg (Pn) → K* is introduced for the group Gn (notice that the algebra is non-commutative and neither left nor right Noetherian). The polynomial Jacobian homomorphism is unique. Its analogue is also unique for n > 2 but for n = 1, 2 there are exactly two of them. The proof is based on the following theorem that is proved in the paper:
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GOULD, VICTORIA, and MIKLÓS HARTMANN. "Coherency, free inverse monoids and related free algebras." Mathematical Proceedings of the Cambridge Philosophical Society 163, no. 1 (2016): 23–45. http://dx.doi.org/10.1017/s0305004116000505.
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AbstractA monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is the non-additive notion corresponding to that for a ring R stating that every finitely generated submodule of every finitely presented right R-module is finitely presented. For monoids (and rings) right coherency is an important finitary property which determines, amongst other things, the existence of a model companion of the class of right S-acts (right R-modules) and hence that the class of existentially closed right S-acts (right R-modules) is axiomatisable.Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruškuc, have shown that (free) groups, free commutative monoids and free monoids have the same properties. It is then natural to ask whether other free algebras in varieties of monoids, possibly with an augmented signature, are right coherent. We demonstrate that free inverse monoids are not.Munn described the free inverse monoid FIM(Ω) on Ω as consisting of birooted finite connected subgraphs of the Cayley graph of the free group on Ω. Sitting within FIM(Ω) we have free algebras in other varieties and quasi-varieties, in particular the free left ample monoid FLA(Ω) and the free ample monoid FAM(Ω). The former is the free algebra in the variety of unary monoids corresponding to partial maps with distinguished domain; the latter is the two-sided dual. For example, FLA(Ω) is obtained from FIM(Ω) by considering only subgraphs with vertices labelled by elements of the free monoid on Ω.The main objective of the paper is to show that FLA(Ω) is right coherent. Furthermore, by making use of the same techniques we show that FIM(Ω), FLA(Ω) and FAM(Ω) satisfy (R), (r), (L) and (l), related conditions arising from the axiomatisability of certain classes of right S-acts and of left S-acts.
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Jespers, Eric, and Arne Van Antwerpen. "Left semi-braces and solutions of the Yang–Baxter equation." Forum Mathematicum 31, no. 1 (2019): 241–63. http://dx.doi.org/10.1515/forum-2018-0059.
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Abstract Let {r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra {K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions {r_{B}} that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so, we first describe such semi-braces and then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.
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Zhuchok, Anatolii. "Structure of relatively free n-tuple semigroups." Algebra and Discrete Mathematics 36, no. 1 (2023): 109–28. http://dx.doi.org/10.12958/adm2173.
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An n-tuple semigroup is an algebra defined on a set with n binary associative operations. This notion was considered by Koreshkov in the context of the theory of n-tuple algebras of associative type. The n>1 pairwise interassociative semigroups give rise to an n-tuple semigroup. This paper is a survey of recent developments in the study of free objects in the variety of n-tuple semigroups. We present the constructions of the free n-tuple semigroup, the free commutative n-tuple semigroup, the free rectangular n-tuple semigroup, the free left (right) k-nilpotent n-tuple semigroup, the free k-nilpotent n-tuple semigroup, and the free weakly k-nilpotent n-tuple semigroup. Some of these results can be applied to constructing relatively free cubical trialgebras and doppelalgebras.
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Gryshchuk, Serhii. "Monogenic functions in two dimensional commutative algebras to equations of plane orthotropy." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 32 (December 28, 2018): 18–29. http://dx.doi.org/10.37069/1683-4720-2018-32-3.
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Among all two-dimensional commutative and assosiative algebras of the second rank with the unity \(e\) over the field of complex numbers \(\mathbb{C}\) we find a semi-simple algebra \(\mathbb{B}_{0} := \{c_1 e+c_2 \omega: c_k\in\mathbb{C}, k=1,2\}\), \(\omega^2=e\), containing a basis \((e_1,e_2)\), such that \( e_1^4 + 2p e_1^2 e_2^2 + e_2^4 = 0 \) for any fixed \( p \) such that \(-1 \lt p \gt 1 \). A domain \(\mathcal{B}_{1}=\{(e_1,e_2)\}\), \(e_1=e\), is discribed in an explicit form. We consider an approach of \(\mathbb{B}_{0}\)-valued ''analytic'' functions \(\Phi(xe_1+ye_2) = U_{1}(x,y)e_1 + U_{2}(x,y)ie_1+ U_{3}(x,y)e_2 + U_{4}(x,y)ie_2\) (\((e_1,e_2)\in \mathcal{B}\), \(x\) and \(y\) are real variables) such that their real-valued components \(U_{k}\), \(k=\overline{1,4}\), satisfy the equation on finding the stress function \(u\) in the case of orthotropic plane deformations (with absence of body forses): \( \left(\frac{\partial^4}{\partial x^4} +2p\frac{\partial^4 }{\partial x^2\partial y^2}+ \frac{\partial^4 }{\partial y^4} \right) u(x,y)=0\) for every \((x,y)\in D\), where \(D\) is a domain of the Cartesian plane \(xOy\). A characterization of solutions \(u\) for this equation in a bounded simply-connected domain via real components \(U_{k}\), \(k=\overline{1,4}\), of the function \(\Phi\) is done in the following sense: let \(D\) be a bounded and simply-connected domain, a solution \(u\) is fixed, then \(u\) is a first component of monogenic function \(\Phi_{u}\). The variety of such \(\Phi_{u}\) is found in a complete form. We consider a particular case of \((e,e_2)\in \mathcal{B}_{1}\) for which \(\Phi_{u}\) can be found in an explicit form. For this case a function \(\Phi_{u}\) is obtained in an explicit form. Note, that in case of orthotropic plane deformations, when Eqs. of the stress function is of the form: \( \left(\frac{\partial^4}{\partial x^4} +2p\frac{\partial^4}{\partial x^2\partial y^2}+\frac{\partial^4 } {\partial y^4} \right) u(x,y)=0\), here \(p\) is a fixed number such that \(p>1\), a similar research is done in [Gryshchuk S. V. Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. I. Ukr. Mat. Zh. 2018. 70, No. 8. pp. 1058-1071 (Ukrainian); Gryshchuk S. V. Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. II. Ukr. Mat. Zh. 2018. 70, No. 10. pp. 1382-1389 (Ukrainian)].
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Kassem, M. S., and K. Rowlands. "The quasi-strict topology on the space of quasi-multipliers of a B*-algebra." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 3 (1987): 555–66. http://dx.doi.org/10.1017/s0305004100066913.
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The notion of a left (right, double) multiplier may be regarded as a generalization of the concept of a multiplier to a non-commutative Banach algebra. Each of these is a special case of a more general object, namely that of a quasi-multiplier. The idea of a quasi-multiplier was first introduced by Akemann and Pedersen in ([1], §4), where they consider the quasi-multipliers of a C*-algebra. One of the defects of quasi-multipliers is that, at least a priori, there does not appear to be a way of multiplying them together. The general theory of quasi-multipliers of a Banach algebra A with an approximate identity was developed by McKennon in [5], and in particular he showed that the quasi-multipliers of a considerable class of Banach algebras could be multiplied. McKennon also introduced a locally convex topology γ on the space QM(A) of quasi-multipliers of A and derived some of the elementary properties of (QM(A), γ).
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Dean, Samuel. "Duality and contravariant functors in the representation theory of artin algebras." Journal of Algebra and Its Applications 18, no. 06 (2019): 1950111. http://dx.doi.org/10.1142/s0219498819501111.
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We know that the model theory of modules leads to a way of obtaining definable categories of modules over a ring [Formula: see text] as the kernels of certain functors [Formula: see text] rather than of functors [Formula: see text] which are given by a pp-pair. This paper will give various algebraic characterizations of these functors in the case that [Formula: see text] is an artin algebra. Suppose that [Formula: see text] is an artin algebra. An additive functor [Formula: see text] preserves inverse limits and [Formula: see text] is finitely presented if and only if there is a sequence of natural transformations [Formula: see text] for some [Formula: see text] which is exact when evaluated at any left [Formula: see text]-module. Any additive functor [Formula: see text] with one of these equivalent properties has a definable kernel, and every definable subcategory of [Formula: see text] can be obtained as the kernel of a family of such functors. In the final section, a generalized setting is introduced, so that our results apply to more categories than those of the form [Formula: see text] for an artin algebra [Formula: see text]. That is, our results are extended to those locally finitely presented [Formula: see text]-linear categories whose finitely presented objects form a dualizing variety, where [Formula: see text] is a commutative artinian ring.