The Priestley Separation Axiom for Scattered Spaces

Let R be a quasi-order on a compact Hausdorff topological space X. We prove that if X is scattered, then R satisfies the Priestley separation axiom if and only if R is closed in the product space X×X. Furthermore, if X is not scattered, then we show that there is a quasi-order on X that is closed in X×X but does not satisfy the Priestley separation axiom. As a result, we obtain a new characterization of scattered compact Hausdorff spaces.     

Graded Witt ring and unramified cohomology

It is proved that for a smooth affine curveX over a local ring or global field, the graded Witt ring ofX is isomorphic to the graded unramified cohomology ring ofX. IfX is projective and has a rational point, the same result holds if and only if every quadratic space defined on the complement of a rational point extends toX. Such an extension is possible, for instance, if the canonical line bundle onX is a square in PicX.     

ech-complete spaces and the upper topology

Let X be a topological space and let be the set of all compact subsets of X. The purpose of this note is to prove the following: if X is regular and q-space, then X is Lindelöf and ech-complete if and only if there exists a continuous map f from a Lindelöf and ech-complete space Y to the space endowed with the upper topology, such that f(Y) is cofinal in . This result extends the following result of Saint Raymond and Christensen: if X is separable metrizable, then X is a Polish space if and only if the space endowed with the Vietoris topology is the continuous image of a Polish space.     

Cc （X） Spaces with X Locally Compact

In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc （X） has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc （X） contains a dense （LM） subspace and if and only if X is a-compact.     

Torsion Elements in Incidence Algebras

Abstract

Let I(X, F) denote the incidence algebra of the locally finite partially ordered set, X, defined over the field, F. This paper considers when the set of torsion elements of I(X, F) forms a group. If F is of finite characteristic, it is shown that the torsion elements of I(X, F) form a group if and only if X is bounded, while if F is of characteristic 0, the torsion elements of the incidence algebra form a group if and only if X is an antichain.     

On c-realcompact spaces

Abstract

The c-realcompact spaces are fully studied and most of the important and well-known properties of realcompact spaces are extended to these spaces. For a zero-dimensional space X, the space υ₀X, which is the counterpart of υX, the Hewitt realcompactification of X, is introduced and studied. It is shown that υ₀X, which is the smallest c-realcompact space between X and β₀X, plays the same role (with respect to C_c(X)) as υX does in the context of C(X). It is proved for strongly zero-dimensional spaces, c-realcompact spaces, realcompact spaces and N-compact spaces coincide. In particular, if X is a strongly zero-dimensional space, then υX = υ₀X. It is obsesrved that a zero-dimensional space X is pseudocompact if and only if C_c(X) = C*_c(X), or equivalently if and only if υ₀X = β₀ X. In particular, a zero-dimensional pseudocompact space is compact if and only if it is c-realcompact. It is shown that Lindelöf spaces, subspaces of the one-point compactification (resp., Lindelöffication) of a discrete space with a nonmeasurable cardinal, are c-realcompact space. If X is a pseudocompact space, it is observed that C(X) = C_c(X) if and only if βX is scattered. Finally, the simplest possible proof (with reasoning) among the known proofs, of the well-known fact that discrete spaces of cardinality less than or equal to that of the continuum are realcompact, is given.     

Commuting Nilpotent Operators and Maximal Rank

Let X, [(X)\tilde]{\widetilde X} be commuting nilpotent matrices over k with nilpotency p ^t , where k is an algebraically closed field of positive characteristic p. We show that if X- [(X)\tilde]{X- \widetilde X} is a certain linear combination of products of pairwise commuting nilpotent matrices, then X is of maximal rank if and only if [(X)\tilde]{\widetilde X} is of maximal rank.     

Decidability of Equational Theories of Coverings of Semigroup Varieties

For every proper semigroup variety X, there exists a semigroup variety Y satisfying the following three conditions: (1) Y covers X, (2) if X is finitely based then so is Y, and (3) the equational theory of X is decidable if and only if so is the equational theory of Y. If X is an arbitrary semigroup variety defined by identities depending on finitely many variables and such that all periodic groups of X are locally finite, then one of the following two conditions holds: (1) all nilsemigroups of X are locally finite and (2) X includes a subvariety Y whose equational theory is undecidable and which has infinitely many covering varieties with undecidable equational theories.     

Orthogonally Complemented Subspaces in Banach Spaces

In this paper, we extend the Moreau (Riesz) decomposition theorem from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dim X ≥ 3 (dim X ≤ 2) is strongly orthognally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.     

Homotopy Properties of the Space If(X) of Idempotent Probability Measures

A subspace I_f(X) of the space of idempotent probability measures on a given compact space X is constructed. It is proved that if the initial compact space X is contractible, then I_f(X) is a contractible compact space as well. It is shown that the shapes of the compact spaces X and I_f(X) are equal. It is also proved that, given a compact space X, the compact space I_f(X) is an absolute neighborhood retract if and only if so is X.

     

On power series rings over valuation domains

Let V be a valuation domain and V[[X]] be the power series ring over V. In this paper, we show that if V[[X]] is a locally finite intersection of valuation domains, then V is an SFT domain and hence a discrete valuation domain. As a consequence, it is shown that the power series ring V[[X]] is a Krull domain if and only if V[[X]] is a generalized Krull domain if and only if V[[X]] is an integral domain of Krull type (or equivalently, a PvMD of finite t-character) if and only if V is a discrete valuation domain with Krull dimension at most one.     

The order bidual of <Emphasis Type="Italic">f</Emphasis> -algebras revisited

It is shown that the order bidual X ^~~ of an Archimedean semiprime f -algebra X has a unit element for the Arens multiplication if and only if every positive linear functional on X extends to a positive linear functional on the f -algebra Orth (X) of all orthomorphisms on X.     

On geometric properties of functors of positive-homogenous and semiadditive functionals

We investigate the functor OH of positive-homogenous functionals and the functor OS of semiadditive functionals. We prove that OH(X) is an absolute retract if and only if X is an open-generated compactum, and OS(X) is an absolute retract if and only if X is an opengenerated compactum of weight ≤ ω₁. We investigate the softness of mappings of multiplication of monads generated by these functors.     

Determining classes and independence

A class of functions of a random element X is sufficiently rich to determine the distribution of X if and only if it is sufficiently rich to determine whether X is independent of other random functions.     

Algebraicity of analytic maps to a hyperbolic variety

Let X be a complex algebraic variety. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over the complex numbers, every holomorphic map from S to X is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine complex algebraic curve C, every holomorphic map from C to X is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.     

Properties of measures on totally ordered spaces

The aim of this paper is to study the properties of measures on a totally ordered spaceX. It is shown that if the cardinal of every closed discrete subset ofX is of measure zero then every non-atomic measure onX is τ-additive. It is also proved that a continuous Borel measure μ onX is Radon if and only if μ is perfect, τ-additive and the support of μ is almost a totally ordered space.

     

Equitable Partitions of Flat Association Schemes

Let (X, G) be an association scheme. We say that (X, G) is flat if it is homogeneous and if any two distinct points have at most one common g-neighbor for each g ∈ G. In this paper we prove that any nondiscrete equitable partition of (X, G) has at most one singleton if (X, G) is flat, and {X} is the unique equitable partition without any singleton if (X, G) is flat and |X| is a prime. This work was supported for two years by Korea Research Foundation Grant (KRF-2006-003-C00010) and Pusan National University Research Grant. Received: January 31, 2007. Final version received: Novmeber 14, 2007.     

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i _X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i _X −1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i _X ≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρ_X> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.     

Sum and intersection of summand ideals in C(X)

Summand sum property (SSP) and summand intersection property (SIP) of modules are studied in [8] and [15] respectively. In this paper we give some topological characterizations of these properties in C(X). It is shown that the ring C(X) has SIPif and only if every intersection of closed-open subsets of Xhas a closed interior. This characterization then shows that for a large class of topological spaces, such as locally connected spaces and extremally disconnected spaces, the ring C(X) has SIP. It is also shown that C(X) has SSPif and only if the space Xhas only finitely many components. Finally, using summand ideals of C(X), we will give several algebraic characterizations of some disconnected spaces.     

Quotient spaces determined by algebras of continuous functions

We prove that if X is a locally compact σ-compact space, then on its quotient, γ(X) say, determined by the algebra of all real valued bounded continuous functions on X, the quotient topology and the completely regular topology defined by this algebra are equal. It follows from this that if X is second countable locally compact, then γ(X) is second countable locally compact Hausdorff if and only if it is first countable. The interest in these results originated in [1] and [7] where the primitive ideal space of a C*-algebra was considered.