Bipartite Divisor Graphs for Integer Subsets

Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph Γ(X) and the prime vertex graph Δ(X), for a set X of positive integers, we define the bipartite divisor graph B(X), and show that many of these connections flow naturally from properties of B(X). In particular we establish links between parameters of these three graphs, such as number and diameter of components, and we characterise bipartite graphs that can arise as B(X) for some X. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of B(X), for one of Γ(X) or Δ(X) to contain a complete subgraph of size 3 or 4.     

Regular centralizers of idempotent transformations

Denote by T(X) the semigroup of full transformations on a set X. For ε∈T(X), the centralizer of ε is a subsemigroup of T(X) defined by C(ε)={α∈T(X):αε=εα}. It is well known that C(id _X )=T(X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(id _X ) contains all non-invertible transformations in C(id _X ).     

Representation of martingale additive functionals on Banach spaces

Assume thatB is a separable real Banach space andX(t) is a diffusion process onB. In this paper, we will establish the representation theorem of martingale additive functionals ofX(t).     

Additive Set of Idempotents in Rings

Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, f ∈ I(R) (e ≠ f), e + f ∈ I(R), and M(R) is additive in I(R) if for all e, f ∈ M(R)(e ≠ f), e + f ∈ I(R). In this article, the following points are shown: (1) I(R) is additive if and only if I(R) is multiplicative and the characteristic of R is 2; M(R) is additive in I(R) if and only if M(R) is orthogonal. If 0 ≠ ef ∈ I(R) for some e ∈ M(R) and f ∈ I(R), then ef ∈ M(R), (2) If R has a complete set of primitive idempotents, then R is a finite product of connected rings if and only if I(R) is multiplicative if and only if M(R) is additive in I(R).     

The Algebra of Flows in Graphs

We define a contravariant functorKfrom the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graphX, an abelian groupB, and a nonnegative integerj, an element of Hom(K^j(X), B) is a coherent family ofB-valued flows on the set of all graphs obtained by contracting some (j − 1)-set of edges ofX; in particular, Hom(K¹(X), ) is the familiar (real) “cycle-space” ofX. We show thatK ^· (X) is torsion-free and that its Poincaré polynomial is the specializationt^{n − k}T_X(1/t, 1 + t) of the Tutte polynomial ofX(hereXhasnvertices andkcomponents). Functoriality ofK ^· induces a functorial coalgebra structure onK ^· (X); dualizing, for any ringBwe obtain a functorialB-algebra structure on Hom(K ^· (X), B). WhenBis commutative we present this algebra as a quotient of a divided power algebra, leading to some interesting inequalities on the coefficients of the above Poincaré polynomial. We also provide a formula for the theta function of the lattice of integer-valued flows inX, and conclude with 10 open problems.     

Nonlinear Preserver Problems on B（H）

Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤* B by A*A = A*B and AA* = AB*. Then (B(H), “≤*”) is a partially ordered set and the relation “≤*” is called the star order on B(H). Denote by Bs(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B _s (H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.     

Binary relations as lattice isomorphisms

Summary In 1963, Zaretskiį established a one-to-one correspondence between the setB _X of binary relations on a set X and the set of triples of the form (W, ϕ, V) where W and V are certain lattices and ϕ: W→V is an isomorphism. We provide a multiplication for these triples making the Zaretskiį correspondence a semigroup isomorphism. In addition, we consider faithful representations ofB _X by pairs of partial transformations and also as the translational hull of its rectangular relations. Using these triples, we study idempotents, regular and completely regular elements and relationsH-equivalent to some relations with familiar properties such as reflexivity, transitivity, etc. Entrata in Redazione il 14 aprile 1998.     

Study of Hopficity in Certain Classes of Rings

The main results proved in this paper are:

1. For any non-zero vector space V _Dover a division ring D, the ring R= End(V _D) is hopfian as a ring

2. Let Rbe a reduced π-regular ring &; B(R) the boolean ring of idempotents of R. If B(R) is hopfian so is R.The converse is not true even when Ris strongly regular.

3. Let Xbe a completely regular spaceC(X) (resp. C ^?(X)) the ring of real valued (resp. bounded real valued) continuous functions on X. Let Rbe any one of C(X) or C ^?(X). Then Ris an exchange ring if &; only if Xis zero dimensional in the sense of Katetov. for any infinite compact totally disconnected space X C(X) is an exchange ring which is not von Neumann regular.

4. Let Rbe a reduced commutative exchange ring. If Ris hopfian so is the polynomial ring R[T ₁,…,T _n] in ncommuting indeterminates over Rwhere nis any integer ≥ 1.

5. Let Rbe a reduced exchange ring. If Ris hopfian so is the polynomial ring R[T].     

Points of joint continuity and large oscillations

For topological spaces X and Y and a metric space Z, we introduce a new class N( X ×Y, Z ) \mathcal{N}\left( {X \times Y,\,Z} \right) of mappings f: X × Y → Z containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping f from this class and any countable-type set B in Y, the set C _B (f) of all points x from X such that f is jointly continuous at any point of the set {x} × B is residual in X: We also prove that if X is a Baire space, Y is a metrizable compact set, Z is a metric space, and f ? N( X ×Y, Z ) f \in \mathcal{N}\left( {X \times Y,\,Z} \right) , then, for any ε > 0, the projection of the set D ^ε (f) of all points p ∈ X × Y at which the oscillation ω _f (p) ≥ ε onto X is a closed set nowhere dense in X.     

Numerical peak holomorphic functions on Banach spaces

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let Ab(BX:X) be the Banach space of all bounded continuous functions f on the unit ball BX of a Banach space X and their restrictions to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense Gδ-subset of Ab(BX:X). We also prove that if X is a smooth Banach space with the Radon-Nikodým property, then the set of all numerical strong peak functions is dense in Ab(BX:X). In particular, when X=Lp(μ)(1<p<∞) or X=?₁, it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense Gδ-subset of Ab(BX:X). As an application, the existence and properties of numerical boundary of Ab(BX:X) are studied. Finally, the numerical peak function in Ab(BX:X) is characterized when X=C(K) and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.     

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.     

A Grothendieck-type theorem for the space of totally measurable functions

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let X be a real Banach space and let X^* stand for the Banach dual of X. Let B(Σ, X) be the Banach space of Σ-totally measurable functions f: Ω → X, and let B(Σ, X)^* and B(Σ, X)^** denote the Banach dual and the Banach bidual of B(Σ, X) respectively. Let bvca(Σ, X^*) denote the Banach space of all countably additive vector measures ν: Σ → X^* of bounded variation. We prove a form of generalized Vitali-Hahn-Saks theorem saying that relative σ(bvca(Σ, X^*), B(Σ, X))-sequential compactness in bvca(Σ, X^*) implies uniform countable additivity. We derive that if X reflexive, then every relatively σ(B(Σ, X)^*, B(Σ, X))-sequentially compact subset of B(Σ, X)_c^~ (= the σ-order continuous dual of B(Σ, X)) is relatively σ(B(Σ, X)^*, B(Σ, X)^**)-sequentially compact. As a consequence, we obtain a Grothendieck type theorem saying that σ(B(Σ, X)^*, B(Σ, X))-convergent sequences in B(Σ, X)_c^~ are σ(B(Σ, X)^*, B(Σ, X)^**)-convergent.     

Logarithmic residues, generalized idempotents, and sums of idempotents in Banach algebras

In a commutative Banach algebraB the set of logarithmic residues (i.e., the elements that can be written as a contour integral of the logarithmic derivative of an analyticB-valued function), the set of generalized idempotents (i.e., the elements that are annihilated by a polynomial with non-negative integer simple zeros), and the set of sums of idempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition and multiplication. Counterexamples show that the commutativity condition onB is essential. The results extend to logarithmic residues of meromorphicB-valued functions.     

Biorder Structure of the Bilinear Form Semigroup

If X and Y are two-finite dimensional vector spaces over a field K and B: X × Y K is a bilinear form, in [6], we have established that the set of adjoint pairs of linear maps forms a regular subsemigroup of L(X) × L(Y)^op. We call it the bilinear form semigroup. Here, we see that the maximal class in this semigroup is a Completely Simple Orthodox semigroup. Again, we note that given any idempotent. We can find a maximal covering chain containing it and ending at a maximal idempotent. We use this to obtain an estimate for the degeneracy of the bilinear form in terms of the rank of maximal idempotents.AMS Subject Classification (2000): 20M17, 15A04     

The State Space of K 0 of Exchange Rings

ABSTRACT

For a directly finite exchange ring R which satisfies general comparability, we construct all extreme points of the state space S(V(R),? R?), where V(R) denotes the monoid of all isomorphic classes of finitely generated projective R-modules. From this, we further prove that S((K ₀(R),[R])) is affinely homeomorphic to M ₁ ⁺(BS(R)), where BS(R) denotes the spectrum of the Boolean algebra B(R) of all central idempotents in R, and M ⁺ ₁(BS(R)) the set of all probability measures on BS(R). These generalize the corresponding results on regular rings. Particularly, all of our results hold for exchange rings with all the idempotents central.     

On finite <Emphasis Type="Italic">X</Emphasis>-semilattices of unions

Using a characteristic family of sets, a characteristic mapping, and basis sources of an X-semilattice of unions D, we characterize the class Σ(X, m) consisting of all finite X-semilattices of unions that are isomorphic to a semilattice D given in advance. For a finite set X, the number of elements in the considered class is found. Commutative semigroups of idempotents are known to play a significant role in semigroup theory (see [25, 26]). Moreover, any commutative idempotent semigroup is isomorphic to some X-semilattice of unions (see [26]), whereas X-semilattices play an especially important role in studying many abstract properties of complete semigroups of binary relations (see [1–4, 7–24]). __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 27, Algebra and Geometry, 2005.     

Smooth Schubert varieties in G/B and B-submodules of {{\mathfrak{g}} \left/ {\mathfrak{b}} \right.}

Let G be a semisimple linear algebraic group over \mathbbC \mathbb{C} without G ₂-factors, B a Borel subgroup of G and T ⊂ B a maximal torus. The flag variety G/B is a projective G-homogeneous variety whose tangent space at the identity coset is isomorphic, as a B-module, to \mathfrakg

First Alexandroff Decomposition Theorem for Topological Lattice Group Valued Measures

Let (X, F) be an Alexandroff space, let A(F) be the Boolean subalgebra of 2 ^X generated by F, let G be a Hausdorff commutative topological lattice group and let rba_F(A(F), G) denote the set of all order bounded F-inner regular finitely additive set functions from A(F) into G. Using some special properties of the elements of rba_F(A(F), G), we extend to this setting the first decomposition theorem of Alexandroff.     

Ordered representations of spaces of integrable functions

Let X be a Banach space, (Ω,Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L¹(m). In this paper we analyze the set of (countably additive) vector measures n satisfying that L¹(n) = L¹(m). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation of the space L¹(m) associated to downward directed subsets of the set of all the representations. This research has been partially supported by La Junta de Andalucía. The support of D.G.I. under project MTM2006–11690–C02 (M.E.C. Spain) and FEDER is gratefully acknowledged.