The convexity lattice of a poset

The authors investigate the lattice Co(P) of convex subsets of a general partially ordered set P. In particular, they determine the conditions under which Co(P) and Co(Q) are isomorphic; and give necessary and sufficient conditions on a lattice L so that L is isomorphic to Co(P) for some P.     

TOPOLOGICAL DECOMPOSITIONS OF THE DUALS OF LOCALLY CONVEX VECTOR SEQUENCE SPACES

ABSTRACT

Let Λ be a scalar sequence space which is endowed with a normal locally convex topology. For a separated locally convex space E we denote by Λ(E) the vector space of all sequences g in E for which (>g(i),a<) ε Λ for all a ε E'. We define a locally convex topology ζ on Λ(E) and then characterize the dual of the ζ-closure (denoted by Λ_c (E)) of the finite sequences in Λ(E). We demonstrate the existence of a continuous projection from Λ(E)' onto a subspace of Λ(E)' which is isomorphic to Λ_c(E)'. Furthermore, we find a topological decomposition of Λ^α _c (E)”, where one of the factors is isomorphic to Λ;^α(E). These results are then applied to find necessary and sufficient conditions for Λ^α(E) to be semi-reflexive. A parallel development yields the same results for the space Λ(E') of all sequences f in E' for which (>x, f(i)<) ε Λ; for all x ε E, when E is barrelled. We conclude the paper by application of the results on vector sequence spaces to spaces of operators—including for instance, necessary and sufficient conditions for L_b (E,Λ;) and L_b (Λ,E) to be semi-reflexive.     

Embedding inverse semigroups in coset semigroups

The setK(G) of all cosets X of a group G, modulo all subgroups of G, forms an inverse semigroup under the multiplication X*Y=smallest coset that constains XY. In this note we show that each inverse semigroup S can be embedded in some coset semigroupK(G). This follows from a result which shows that symmetric inverse semigroups can be embedded in the coset semigroups of suitable symmetric groups. We also give necessary and sufficient conditions on an inverse semigroup S in order that it should be isomorphic to someK(G). This research was supported by a grant from the National Science Foundation.     

Isomorphism conditions for completely decomposable torsion-free abelian groups with isomorphic rings of endomorphisms

Assuming the generalized continuum hypothesis, we obtain necessary and sufficient conditions for completely decomposable torsion-free abelian groups with isomorphic rings of endomorphisms to be isomorphic.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 403–408, April, 1972.I. H. Bekker has devoted much attention to this work and the author is deeply grateful to him.     

完全单半群及完全正则半群的逆断面

指出完全单半群Ｓ的任何一个Ｆ－类是逆断面,且为Ｑ－逆断面,而Ｓ的任何一个逆断面必是一个Ｆ－类,因而所有逆断面同构。并且给出完全正则半群的逆断面存在的充要条件。     

Interassociates of the bicyclic semigroup

An interassociate of a semigroup \((S,\cdot )\) is a semigroup \((S, *)\) such that for all \(a, b, c \in S\), \(a\cdot (b*c)=(a\cdot b) *c\) and \(a*(b\cdot c)=(a*b) \cdot c\). We investigate the bicyclic semigroup C and its interassociates. In particular, if p and q are the generators of the bicyclic semigroup and m and n are fixed nonnegative integers, the operation \(a*_{m,n} b= aq^mp^n b\) is known to be an interassociate. We show that for distinct pairs (m, n) and (s, t), the interassociates \((C, *_{m,n})\) and \((C, *_{s,t})\) are not isomorphic. We also generalize a result regarding homomorphisms on C to homomorphisms on its interassociates.     

Existence of the uniformly minimum risk equivariant estimators of parameters in a class of normal linear models

In this paper, we study the existence of the uniformly minimum risk equivariant (UMRE) estimators of parameters in a class of normal linear models, which include the normal variance components model, the growth curve model, the extended growth curve model, and the seemingly unrelated regression equations model, and so on. The necessary and sufficient conditions are given for the existence of UMRE estimators of the estimable linear functions of regression coefficients, the covariance matrixV and (trV)^α, where α > 0 is known, in the models under an affine group of transformations for quadratic losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model, the conclusions given in literature for estimating regression coefficients can be derived by applying the general results in this paper, and the sufficient conditions for non-existence of UMRE estimators ofV and tr(V) are expanded to be necessary and sufficient conditions. In addition, the necessary and sufficient conditions that there exist UMRE estimators of parameters in the variance components model are obtained for the first time.     

Representation of partially ordered sets

It is well known [1] that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, (–) such that the supremum and the infimum of any finite setF ofp correspond, respectively, to the union and intersection of the images of the elements ofF. Here necessary and sufficient conditions are given for similar isomorphic representation of a poset where however the supremum and infimum of also infinite subsetsI correspond to the union and intersection of images of elements ofI. Presented by R. Freese.     

On the role of effective representations of Lie groupoids

In this paper, we undertake the study of the Tannaka duality construction for the ordinary representations of a proper Lie groupoid on vector bundles. We show that for each proper Lie groupoid G, the canonical homomorphism of G into the reconstructed groupoid T(G) is surjective, although — contrary to what happens in the case of groups — it may fail to be an isomorphism. We obtain necessary and sufficient conditions in order that G may be isomorphic to T(G) and, more generally, in order that T(G) may be a Lie groupoid. We show that if T(G) is a Lie groupoid, the canonical homomorphism G→T(G) is a submersion and the two groupoids have isomorphic categories of representations.     

Local edge colorings that are global

A local (H, k)-coloring of K_n is a coloring of its edges in which each of its subgraphs isomorphic to H is colored by at most k colors. A necessary condition on H (conjectured to be sufficient) is found such that each local (H, k)-coloring is a k-coloring. Furthermore, necessary and sufficient conditions are given for each local (H, k)-coloring to be a coloring by a bounded number of colors.     

A system of generalized variational inclusions involving generalized H(·, ·)-accretive mapping in real q-uniformly smooth Banach spaces

In this paper, we consider a class of accretive mappings called generalized H(·, ·)-accretive mappings in Banach spaces. We prove that the proximal-point mapping of the generalized H(·, ·)-accretive mapping is single-valued and Lipschitz continuous. Further, we consider a system of generalized variational inclusions involving generalized H(·, ·)-accretive mappings in real q-uniformly smooth Banach spaces. Using proximal-point mapping method, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as a refinement and improvement of some known results in the literature.     

Subdivision Extendibility

Let H be a multigraph and G a graph containing a subgraph isomorphic to a subdivision of H, with S ⊂ V(G) (the ground set) the image of V(H) under the isomorphism. We consider connectivity and minimum degree or degree sum conditions sufficient to imply there is a spanning subgraph of G isomorphic to a subdivision of H on the same ground set S. These results generalize a number of theorems in the literature.     

Hyperbolic unitals in the Hall planes

A new transformation method for incidence structures was introduced in [8],an open problem is to characterize classical incidence structures obtained by transformation of others. In this work we give some, sufficient conditions to transform, with the procedure of [8],a unital embedded in a projective plane into another one. As application of this result we construct unitals in the Hall planes by transformation of the hermitian curves and we give necessary and sufficient conditions for the constructed unitals to be projectively equivalent. This allows to find different classes of not projectively equivalent Buekenhout's unitals, [2],and to find the class of unitals descovered by Grüning, [4],easily proving its embeddability in the dual of a Hall plane. Finally we prove that the affine unital associated to the unital of [4]is isomorphic to the affine hyperbolic hermitian curve.Work performed under the auspicies of G.N.S.A.G.A. and supported by 40% grants of M.U.R.S.T.     

拟相似算子的右本质谱的连通分支

设算子S和T拟相似,通过引进RT类算子、R类算子和RR类算子的概念,给出右本质谱σre(S)的连通分支与σre(T)相交的充分条件和必要条件以及σre(S)的连通分支与本质谱盯σe(T)的某些子集的相交关系,并给出算子是属于RT类算子和RR类算子的充分条件和必要条件.     

Limit theorems for convolutions of probabilities on non abelian semigroups

Let S be a locally compact semigroup. We study the sequence (λⁿ) of the convolution powers of a probability measure λ on S and their shifts by a probability measure η on S. We shall give sufficient conditions for lim ‖λⁿ−η*λⁿ‖ = 0 (where ‖.‖ denotes the norm). In particular we consider the case the η is a point measure and we study the subsemigroup L_O(λ) = {x ∈ S : lim ‖λⁿ−δ_X*λⁿ‖ = 0}. We shall give necessary and sufficient conditions for L_o(λ)=S. In this case we want to treat the problem of the convergence of the sequence (λⁿ).     

Pseudo-unitarizable weight modules over generalized Weyl algebras

We define a notion of pseudo-unitarizability for weight modules over a generalized Weyl algebra (of rank one, with commutative coefficient ring R), which is assumed to carry an involution of the form X^∗=Y, R^∗⊆R. We prove that a weight module V is pseudo-unitarizable iff it is isomorphic to its finitistic dual V^?. Using the classification of weight modules by Drozd, Guzner and Ovsienko, we obtain necessary and sufficient conditions for an indecomposable weight module to be isomorphic to its finitistic dual, and thus to be pseudo-unitarizable. Some examples are given, including U_q(sl₂) for q a root of unity.     

Lattice-ordered triangular matrix algebras

In this article we give some necessary and sufficient conditions for a lattice-ordered semigroup algebra to be isomorphic to a lattice-ordered triangular matrix algebra.     

An Extremal Problem with Excluded Subposet in the Boolean Lattice

Let 2[n] be the poset of all subsets of {1,...,n} ordered by inclusion. A poset P is said to be contained in a subposet F of 2[n] as a subposet if there exists a subposet P' of F that is isomorphic to P. In this paper we will give an estimate on the size of a maximally sized subposet of 2[n] under the assumption that the subposet does not contain Y(u,v) = ({a1,...,au, b1,...,bv}, {a1< ··· < au, au< b1, au< b2,...,au< bv}) as a subposet.     

On the fundamental group of real toric varieties

LetX (Δ) be the real toric variety associated to a smooth fan Δ. The main purpose of this article is: (i) to determine the fundamental group and the universal cover ofX (Δ), (ii) to give necessary and sufficient conditions on Δ under which π₁(X(Δ)) is abelian, (iii) to give necessary and sufficient conditions on Δ under whichX(Δ) is aspherical, and when Δ is complete, (iv) to give necessary and sufficient conditions forC _Δ to be aK (π, 1) space whereC _Δ is the complement of a real subspace arrangement associated to Δ.     

On the representation of partially ordered sets

It is well known that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, ⊆) such that the supremum and the infimum of any finite setF ofP correspond, respectively to the union and intersection of the images of the elements ofF. Here necessary and sufficient conditions are given for similar isomophic representation of a poset where however the supremum and infimum of also infinite subsetsI correspond to the union and intersection of images of elements ofI.