Surjective Homomorphism Research Articles

Epimorphism between Fine and Ferguson’s Matrices for Angell’s AC

Angell’s logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. It is shown that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. Some candidate 7-valued matrices are ruled out as characteristic of AC. Whether matrices with fewer than 9 values exist remains an open question. The results were obtained with the help of the MUltlog system for investigating finite-valued logics; the results serve as an example of the usefulness of techniques from computational algebra in logic. A tableau proof system for NC is also provided. .

A new realisation of the i-quantum group U(n)

We follow the approach developed in [3] and modified in [7] to investigate a new realisation for the i -quantum groups U ȷ ( n ) , building on the multiplication formulas discovered in [2, Lem. 3.2] . This allows us to present U ȷ ( n ) via a basis and multiplication formulas by generators. We also establish a surjective algebra homomorphism from a Lusztig type form U Z ȷ ( n ) of U ȷ ( n ) to the integral q -Schur algebras S Z ȷ ( n , r ) of type B . Thus, base change to a field k allows us to relate representations of the i -quantum hyperalgebras U k ȷ ( n ) to representations of finite orthogonal groups of odd degree in non-defining characteristics. This generalises part of Dipper–James' type A theory to the type B case.

Third homology of perfect central extensions

For a central perfect extension of groups $A \rightarrowtail G \twoheadrightarrow Q$, first we study the natural image of $H_3(A,\mathbb{Z})$ in $H_3(G, \mathbb{Z})$. As a particular case, we show that if the extension is universal this image is 2-torsion. Moreover when the plus-construction of the classifying space of $Q$ is an $H$-space, we also study the kernel of the surjective homomorphism $H_3(G,\mathbb{Z}) \to H_3(Q, \mathbb{Z})$.

The SHAI property for the operators on Lp

A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for Lp(0,1) for 1<p<∞, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1<p<∞.

Surjections of unit groups and semi-inverses

Given a surjective ring homomorphism, we study when the induced group homomorphism on unit groups is surjective. To this end, we introduce notions of generalized inverses and units, as well as a class of rings such that the set of closed points in the spectrum is a closed set. It is shown that any surjection out of such a ring induces a surjection on unit groups.

The Walker Abel\u2013Jacobi map descends

For a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths’ Abel–Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel–Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel–Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we determine the image of the l-adic Bloch map restricted to algebraically trivial cycle classes in terms of the coniveau filtration.

A Certain Structure of Bipolar Fuzzy Subrings

The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define (α,β)-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.

Perturbations of surjective homomorphisms between algebras of operators on Banach spaces

A remarkable result of Moln\'ar [Proc. Amer. Math. Soc., 126 (1998), 853-861] states that automorphisms of the algebra of operators acting on a separable Hilbert space is stable under "small" perturbations. More precisely, if $\phi,\psi$ are endomorphisms of $\mathcal{B}(\mathcal{H})$ such that $\|\phi(A)-\psi(A)\|<\|A\|$ and $\psi$ is surjective then so is $\phi$. The aim of this paper is to extend this result to a larger class of Banach spaces including $\ell_p$ and $L_p$ spaces ($1<p<+\infty$). En route to the proof we show that for any Banach space $X$ from the above class all faithful, unital, separable, reflexive representations of $\mathcal B (X)$ which preserve rank one operators are in fact isomorphisms.

On Fundamental Theorems of Fuzzy Isomorphism of Fuzzy Subrings over a Certain Algebraic Product

In this study, we define the concept of an ω-fuzzy set ω-fuzzy subring and show that the intersection of two ω-fuzzy subrings is also an ω-fuzzy subring of a given ring. Moreover, we give the notion of an ω-fuzzy ideal and investigate different fundamental results of this phenomenon. We extend this ideology to propose the notion of an ω-fuzzy coset and develop a quotient ring with respect to this particular fuzzy ideal analog into a classical quotient ring. Additionally, we found an ω-fuzzy quotient subring. We also define the idea of a support set of an ω-fuzzy set and prove various important characteristics of this phenomenon. Further, we describe ω-fuzzy homomorphism and ω-fuzzy isomorphism. We establish an ω-fuzzy homomorphism between an ω-fuzzy subring of the quotient ring and an ω-fuzzy subring of this ring. We constitute a significant relationship between two ω-fuzzy subrings of quotient rings under the given ω-fuzzy surjective homomorphism and prove some more fundamental theorems of ω-fuzzy homomorphism for these specific fuzzy subrings. Finally, we present three fundamental theorems of ω-fuzzy isomorphism.

Role colouring graphs in hereditary classes

A locally surjective homomorphism from a graph G to a graph H is called an H-role colouring of G. Deciding the existence of such a colouring with |H|=k is known to be NP-hard even under substantial restrictions on the input graph G. We study the family of hereditary classes for which the problem can be solved efficiently. Our main result is the first boundary class for this problem; that is, a minimal obstruction to solving the problem efficiently in a finitely defined hereditary class. We also give the first boundary class for the related k-coupon colouring problem, in which H is a complete graph with each vertex incident to a loop, when k is a prime power. As additional results, we show that 2-role colouring 2K2-free graphs and 2-coupon colouring cographs can be done in linear time.

Factorizations of surjective maps of connected quandles

We consider the problem of when one quandle homomorphism will factor through another, restricting our attention to the case where all quandles involved are connected. We provide a complete solution to the problem for surjective quandle homomorphisms using the structure theorem for connected quandles of Ehrman et al. (2008) and the factorization system for surjective quandle homomorphsims of Bunch et al. (2010) as our primary tools. The paper contains the substantive results obtained by an REU research group consisting of the first four authors under the mentorship of the fifth, and was supported by National Science Foundation, grant DMS-1659123.

Totally Acyclic Approximations

Let $$Q \rightarrow R$$ be a surjective homomorphism of Noetherian rings such that Q is Gorenstein and R as a Q-bimodule admits a finite resolution by modules which are projective on both sides. We define an adjoint pair of functors between the homotopy category of totally acyclic R-complexes and that of Q-complexes. This adjoint pair is analogous to the classical adjoint pair of functors between the module categories of R and Q. As a consequence, we obtain a precise notion of approximations of totally acyclic R-complexes by totally acyclic Q-complexes.

The continuous d-open homomorphism images and subgroups of [formula omitted]-factorizable paratopological groups

In this paper, we prove the following results: (1) Let f:G→H be a continuous d-open surjective homomorphism; if G is an R-factorizable paratopological group, then so is H. This improves a result by Peng and Zhang [4, Theorem 1.7], and (2) let G be a regular (so G is completely regular by a theorem of Banakh and Ravsky) R-factorizable paratopological group; then every subgroup H of G is R-factorizable if and only if H is z-embedded in G. This result gives a partial answer to a question of M. Sanchis and M. Tkachenko [8, Problem 5.3].

Surjective Homomorphisms from Algebras of Operators on Long Sequence Spaces are Automatically Injective

Abstract We study automatic injectivity of surjective algebra homomorphisms from $\mathscr{B}(X)$, the algebra of (bounded, linear) operators on X, to $\mathscr{B}(Y)$, where X is one of the following long sequence spaces: c0(λ), $\ell_{\infty}^c(\lambda)$, and $\ell_p(\lambda)$ ($1 \leqslant p \lt \infty$) and Y is arbitrary. En route to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the ‘sequential strong operator topology’.

Some remarks on Prüfer rings with zero-divisors

Let A be the fiber product R × T B , where B → T is a surjective ring homomorphism with regular kernel and R ⊆ T is a ring extension where T is an overring of R . In this paper we provide a characterization of when A has distinguished Prüfer-like properties and new constructions of Prüfer rings with zero-divisors. Furthermore we give examples of homomorphic images of Prüfer rings that are Prüfer without assuming that the kernel of the surjection is regular. Finally we provide some remarks on the ideal theory of pre-Prüfer rings.

The third homology of [formula omitted

We calculate the structure of H3(SL2(Q),Z[12]). Let H3(SL2(Q),Z)0 denote the kernel of the (split) surjective homomorphism H3(SL2(Q),Z)→K3ind(Q). Each prime number p determines an operator 〈p〉 on H3(SL2(Q),Z) with square the identity. We prove that H3(SL2(Q),Z[12])0 is the direct sum of the (−1)-eigenspaces of these operators. The (−1)-eigenspace of 〈p〉 is the scissors congruence group, over Z[12], of the field Fp, which is a cyclic group whose order is the odd part of p+1.

SUZUKI FUNCTOR AT THE CRITICAL LEVEL

In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t = 0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings induced by the functor and deduce that every simple module over the rational Cherednik algebra lies in its image. Our homomorphism between the two centres gives rise to a closed embedding of the Calogero–Moser space into the space of opers on the punctured disc. We give a partial geometric description of this embedding.

On the Unbounded Picture of KK-Theory

In the founding paper on unbounded $KK$-theory it was established by Baaj and Julg that the bounded transform, which associates a class in $KK$-theory to any unbounded Kasparov module, is a surjective homomorphism (under a separability assumption). In this paper, we provide an equivalence relation on unbounded Kasparov modules and we thereby describe the kernel of the bounded transform. This allows us to introduce a notion of topological unbounded $KK$-theory, which becomes isomorphic to $KK$-theory via the bounded transform. The equivalence relation is formulated entirely at the level of unbounded Kasparov modules and consists of homotopies together with an extra degeneracy condition. Our degenerate unbounded Kasparov modules are called spectrally decomposable since they admit a decomposition into a part with positive spectrum and a part with negative spectrum.

Quotients of groups of birational transformations of cubic del Pezzo fibrations

We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.

The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer

This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra SSym of permutations, mapped onto QSym by taking descents of permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions.We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra HSym, linearly spanned by signed permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a permutation from a signed permutation defines a Hopf algebra surjection form HSym to SSym and taking a suitable descent from a signed permutation defines a linear surjection from HSym to RQSym. The notion of weak P-partitions from signed permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from SSym and RQSym to QSym, we obtain a commutative diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, permutations and signed permutations.