Multiplicative Isomorphism Research Articles

Multiplicative isomorphisms and derivations on axial algebras

In this paper, we show that the multiplicative derivations on J(α)-axial algebras, with α≠1,0,12, are additive under suitable conditions, which nowadays are called Martindale-type conditions. Besides, with proper assumptions, we proceed to study the additivity of multiplicative isomorphisms and derivations in the context of M(α,β)-axial algebras, except for multiplicative derivations when β=12. In this case, we mention a research question at the end.

Multiplicative maps on generalized \(n\)-matrix rings

Let \(\mathfrak{R}\) and \(\mathfrak{R}'\) be two associative rings (not necessarily with identity elements). A bijective map \(\varphi\) of \(\mathfrak{R}\) onto \(\mathfrak{R}'\) is called an \textit{\(m\)-multiplicative isomorphism} if {\(\varphi (x_{1} \cdots x_{m}) = \varphi(x_{1}) \cdots \varphi(x_{m})\)} for all \(x_{1}, \dotsc ,x_{m}\in \mathfrak{R}.\) In this article, we establish a condition on generalized matrix rings, that assures that multiplicative maps are additive. And then, we apply our result for study of \(m\)-multiplicative isomorphisms and \(m\)-multiplicative derivations on generalized matrix rings.

Multiplicative maps of matrices over rings

Let R be an arbitrary associative ring (not necessarily with an identity). A sufficient condition is given to determine when a multiplicative isomorphism on the matrix ring is additive without the assumption of idempotents. We formulate a version for matrix rings over rings with identities, and then present two applications. One is a new characterization of the automorphisms of the matrix algebra over an arbitrary field by means of multiplicative preservers, and the other one is a new characterization of topologically isomorphisms of finite dimensional real Banach spaces with dimensions no less than 2 by the existence of multiplicative isomorphisms between their rings of all bounded linear operators.

Isomorphic Multiplicative Transitivity for Intuitionistic and Interval-Valued Fuzzy Preference Relations and Its Application in Deriving Their Priority Vectors

Intuitionistic fuzzy preference relations (IFPRs) are used to deal with hesitation, while interval-valued fuzzy preference relations (IVFPRs) are for uncertainty in multicriteria decision making (MCDM). This paper aims to explore the isomorphic multiplicative transitivity for IFPRs and IVFPRs, which builds the substantial relationship between hesitation and uncertainty in MCDM. To do that, the definition of the multiplicative transitivity property of IFPRs is established by combining the multiplication of intuitionistic fuzzy sets and Tanino's multiplicative transitivity property of fuzzy preference relations. It is proved to be isomorphic to the multiplicative transitivity of IVFPRs derived via Zadeh's extension principle. The use of the multiplicative transitivity isomorphism is twofold: 1) to discover the substantial relationship between IFPRs and IVFPRs, which will bridge the gap between hesitation and uncertainty in MCDM problems; and 2) to strengthen the soundness of the multiplicative transitivity property of IFPRs and IVFPRs by supporting each other with two different reliable sources, respectively. Furthermore, based on the existing isomorphism, the concept of multiplicative consistency for IFPRs is defined through a strict mathematical process, and it is proved to satisfy the following several desirable properties: weak transitivity, max–max transitivity, and center-division transitivity. A multiplicative consistency-based multiobjective programming (MOP) model is investigated to derive the priority vector from an IFPR. This model has the advantage of not losing information, as the priority vector representation coincides with that of the input information, which was not the case with the existing methods, where crisp priority vectors were derived as a consequence of the modeling transitivity just for the intuitionistic membership function and not for the intuitionistic nonmembership function. Finally, a numerical example concerning green supply selection is given to validate the efficiency and practicality of the proposed multiplicative consistency MOP model.

A sharp representation of multiplicative isomorphisms of uniformly continuous functions

Let X, Y be complete metric spaces, U(X), U(Y) the spaces of uniformly continuous functions with real values defined on X and Y. We will show the form that every multiplicative isomorphism T:U(Y)→U(X) has: for x outside a uniformly isolated subset S⊂X,Tf(x)=sign(f(τ(x)))|f(τ(x))|1+p(x), where τ:X→Y is a uniform homeomorphism and p:X∖S→R is such that p⋅h⋅log⁡h is uniformly continuous for every h∈U(X,(2,∞)).

Multiplicative isomorphisms at invertible matrices

Multiplicative isomorphisms at invertible matrices

SMOOTH PATHS OF CONDITIONAL EXPECTATIONS

Let [Formula: see text] be a von Neumann algebra with a finite trace τ, represented in [Formula: see text], and let [Formula: see text] be sub-algebras, for t in an interval I (0 ∈ I). Let [Formula: see text] be the unique τ-preserving conditional expectation. We say that the path t ↦ Et is smooth if for every [Formula: see text] and [Formula: see text], the map [Formula: see text] is continuously differentiable. This condition implies the existence of the derivative operator [Formula: see text] If this operator satisfies the additional boundedness condition, [Formula: see text] for any closed bounded subinterval J ⊂ I, and CJ > 0 a constant depending only on J, then the algebras [Formula: see text] are *-isomorphic. More precisely, there exists a curve [Formula: see text], t ∈ I of unital, *-preserving linear isomorphisms which intertwine the expectations, [Formula: see text] The curve Gt is weakly continuously differentiable. Moreover, the intertwining property in particular implies that Gt maps [Formula: see text] onto [Formula: see text]. We show that this restriction is a multiplicative isomorphism.

Characterization of Lie multiplicative isomorphisms between nest algebras

Let \(\mathcal{N}\) and \(\mathcal{M}\) be nests on Banach spaces X and Y over the real or complex field \(\mathbb{F}\), respectively, with the property that if \(M \in \mathcal{M}\) such that M− = M, then M is complemented in Y. Let \(Alg\mathcal{N}\) and \(Alg\mathcal{M}\) be the associated nest algebras. Assume that \(\Phi :Alg\mathcal{N} \to Alg\mathcal{M}\) is a bijective map. It is proved that, if dimX = ∞ and if there is a nontrivial element in \(\mathcal{N}\) which is complemented in X, then Φ is Lie multiplicative (i.e. Φ([A,B]) = [Φ(A), Φ(B)] for all \(A,B \in Alg\mathcal{N}\)) if and only if Φ has the form Φ(A) = TAT−1 + τ(A) for all \(A \in Alg\mathcal{N}\) or Φ(A) = −TA*T−1 + τ(A) for all \(A \in Alg\mathcal{N}\), where T is an invertible linear or conjugate linear operator and \(\tau :Alg\mathcal{N} \to \mathbb{F}I\) is a map with τ([A,B]) = 0 for all \(A,B \in Alg\mathcal{N}\). The Lie multiplicative maps are also characterized for the case dimX < ∞.

Weakly-peripherally multiplicative conditions and isomorphisms between uniform algebras

Suppose that A and B are uniform algebras on compact Hausdorff spaces X and Y , respectively. Let ; : ¥E iae A and S; T : ¥E iae B be mappings on a nonempty set ¥E. Suppose that (¥E); (¥E) and S(¥E); T(¥E) are closed under multiplications and contain expA and expB respectively and that S(e1) io S(¥E)..1, T(e2) io T(¥E)..1 with |S(e1)T(e2)| = 1 on Ch(B) for some fixed e1; e2 io A1 with (e1) = (e2) = 1. If  (S(f)T(g)) iu  ((f) (g)) .= . for all f; g io ¥E and there exists a first-countable dense subset DB in Ch(B), or a first-countable dense subset DA in Ch(A), then there exists an algebra isomorphism eS : A iae B such that eS((f)) = S(e1)..1S(f) and eS( (f)) = T(e2)..1T(f) for every f io ¥E.

Additivity of Lie multiplicative maps on triangular algebras

Let 𝒜 and ℬ be unital algebras over a commutative ring ℛ, and ℳ be a (𝒜, ℬ)-bimodule, which is faithful as a left 𝒜-module and also as a right ℬ-module. Let 𝒰 = Tri(𝒜, ℳ, ℬ) be the triangular algebra and 𝒱 any algebra over ℛ. Assume that Φ : 𝒰 → 𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST − TS) = Φ(S)Φ(T) − Φ(T)Φ(S) for all S, T ∈ 𝒰. Then Φ(S + T) = Φ(S) + Φ(T) + Z S,T for all S, T ∈ 𝒰, where Z S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T.

Characterization of ξ-Lie multiplicative isomorphisms

Characterization of ξ-Lie multiplicative isomorphisms

Multiplicative Lie Isomorphisms Between Prime Rings

Let ℛ be a prime ring with 1 containing a nontrivial idempotent E, and let ℛ′ be another prime ring. If Φ:ℛ → ℛ′ is a multiplicative Lie isomorphism, then Φ(T + S) = Φ(T) + Φ(S) + Z′ T,S for all T, S ∈ ℛ, where Z′ T,S is an element in the center 𝒵′ of ℛ′ depending on T and S.

Multiplicative Mappings of Rings

Let ℛ and \({\fancyscript S}\) be arbitrary associative rings. A mapping φ of ℛ onto \({\fancyscript S}\) is called a multiplicative isomorphism if φ is bijective and satisfies φ(xy) = φ(x)φ(y) for all x, y ∈ ℛ. In this short note, we establish a condition on ℛ, in the case where ℛ may not contain any non–zero idempotents, that assures that φ is additive, which generalizes the famous Martindale's result. As an application, we show that under a mild assumption every multiplicative isomorphism from the radical of a nest algebra onto an arbitrary ring is additive.

Multiplicative mappings of operator algebras

Nest algebras are the natural analogues of upper triangular matix algebras in an infinite dimensional Hilbert space. In this paper, we study multiplicative isomorphisms of subalgebras of nest algebras which contain all finite rank operators but might contain no idempotents. We prove that such multiplicative mappings are automatically additive and linear (or conjugate linear).

Constructions of Self-complementary Circulants with No Multiplicative Isomorphisms

The main topic of the paper is the question of the existence of self-complementary Cayley graphs Cay(G, S) with the property Sσ≠=G#\\S for all σ∈Aut(G). We answer this question in the positive by constructing an infinite family of self-complementary circulants with this property. Moreover, we obtain a complete classification of primes p for which there exist self-complementary circulants of order p2with this property.

Multipliers between invariant subspaces of the backward shift

Contained in the Hardy space H2 on the unit disk in the complex plane are certain Hubert spaces which are invariant under the adjoint of the shift. One such space H(b) is associated with each function in the closed unit ball of H°°. In the special case where is an inner function, 7ί(b) is just the subspace of H2 orthogonal to the shift-invariant subspace bH2. It is proven here that for any functions b and b2 in the closed ball of H°°, the spaces U(bι) and 7ί(62) are isometrically isomorphic under a multiplication operator if and only if there is a disk automorphism r such that 62 = τobi. In this case, the multiplicative isomorphism is determined explicitly and uniquely. This motivates an investigation of multipliers between Ή,(b\) and 9^(62)9 that is, multiplication operators acting bijectively but not necessarily isometrically. Restricting to the case where b and 62 ar e inner functions, it is shown that a multiplier between given spaces is unique up to multiplication by a nonzero constant, and several theorems are proven concerning the existence of such multipliers. Finally, consideration is given to the implications of these results for the characterization of the invariant subspaces in H2 on an annulus.

Invariance of minimal prime ideals of nonassociative rings under multiplicative isomorphisms

It is proved that the image of a minimal prime ideal under a multiplicative isomorphism in a nonassociative ring is also a minimal prime ideal and congruence with respect to its modulus is preserved.

MULTIPLICATIVE CLASSIFICATION OF ASSOCIATIVE RINGS

Let be a ring, and the left and right annihilators of the element , the two-sided ideal in called the additive controller, and let be an -isomorphism (i.e., multiplicative isomorphism) and its defect. An ideal in the ring is called an -ideal if for all -isomorphisms , is an ideal in and if and only if . It is shown that Very general sufficient conditions are given that a multiplicative isomorphism of subsemigroups of multiplicative semigroups of rings be extendible to the isomorphism of the subrings generated by them. Minimal prime ideals and the prime radical of a ring are -ideals. The strongly regular and regular rings that have unique addition are characterized.Bibliography: 29 titles.

On a theorem of A. Borel

trivially on the fibre F. In what follows, cohomology will always mean singular (cubical) cohomology; the standard notation, C*, Z*, etc., for cochains, cocycles, etc., will be used, and coefficients will always be taken in a commutative ring K with a unit element. Throughout, it will be assumed that the cohomology algebra H*(F; K) is a graded tensor product of monogenic, possibly truncated, polynomial algebras with transgressive generators of positive degrees, and that H*(F; K) is of finite type (i.e., has a finite (additive) K-basis in each dimension) with IGH°(F; K). My aim here is, as in [3], to study the relation between the cohomology groups of E and those of B and F. I recall that in [3] details were given of the method of Hirsch in singular cohomology theory, in which under suitable conditions, a coboundary operator d is proved to exist on the tensor product C*(B; K)®KH*(F; K), together with a map u:C*(B; K)®KH*(F; K) —>C*(E; K) which induces isomorphisms on cohomology groups. The results proved here below show that, in fact, under the conditions given above, d and u, in the Hirsch method, can be defined by explicit formulae in terms of cup- and cup-l-products obtained from the generators of H*(F; K). These explicit formulae of course generalize the explicit formulae given in [3] up to dimension 2m — 1 for d and u, in case F is a space of type X(tr, n). Collected together, the results may be regarded as a weak generalization of a theorem of A. BoreK1), referred to in the title. I use the adjective weak because whereas Borel proves the existence of a multiplicative isomorphism

On the Rationality of the Zeta Function of an Algebraic Variety

Let p be a prime number, a2 the completion of the algebraic closure of the field of rational p-adic numbers and let A be the residue class field of Q. The field A is the algebraic closure of its prime subfield and is of characteristic p. If T* is the set of all roots of unity in a2 of order prime to p then the restriction of the residue class map to T* is a multiplicative isomorphism of T* onto the multiplicative group of R. The elements of T T* U {O} form the Teichmiiller representatives of A in Q2 and for each x C A the representative of x in Q will be understood to be the element of T in the class x. The non-archimedean valuation of 0 will be denoted by the ordinal function, abbreviated , and normalized by the condition ord p = 1.