$\mathcal{P}$-kernel normal systems for $\mathcal{P}$-inversive semigroups

As a generalization of Preston’s kernel normal systems, P\mathcal{P}-kernel normal systems for P\mathcal{P}-inversive semigroups are introduced, and strongly regular P\mathcal{P}-congruences on P\mathcal{P}-inversive semigroups in terms of their P\mathcal{P}-kernel normal systems are characterized. These results generalize the corresponding results for P\mathcal{P}-regular semigroups and P\mathcal{P}-inversive semigroups.     

[C(E) ,Ca(E)]中的唯一原子

对于集合X上的任一非平凡等价关系E,本文考察了半群T_E(x)上的同余C_*(E),并证明了C_*(E)是T_E(X)的同余格的完全子格[C(E),C_a(E)]中的唯一原子.     

Embeddings ofC(Δ) andL 1[0, 1] in Banach lattices

It is proved that ifE is a separable Banach lattice withE′ weakly sequentially complete,F is a Banach space andT:E→F is a bounded linear operator withT′F′ non-separable, then there is a subspaceG ofE, isomorphic toC(Δ), such thatT _G is an isomorphism, whereC(Δ) denotes the space of continuous real valued functions on the Cantor discontinuum. This generalizes an earlier result of the second-named author. A number of conditions are proved equivalent for a Banach latticeE to contain a subspace order isomorphic toC(Δ). Among them are the following:L ¹ is lattice isomorphic to a sublattice ofE′;C(Δ)′ is lattice isomorphic to a sublattice ofE′; E contains an order bounded sequence with no weak Cauchy subsequence;E has a separable closed sublatticeF such thatF′ does not have a weak order unit. The research of both authors was partially supported by the National Science Foundation, NSF Grant No MPS 71-02839 A04.     

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

On a class of isometric subgraphs of a graph

In a graphG, which has a loop at every vertex, a connected subgraphH=(V(H),E(H)) is a retract if, for anya, b ∈V(H) and for any pathsP, Q inG, both joininga tob, and satisfying |Q|≧ ≧|P|, thenP ⫅V(H) wheneverQ ⫅V(H). As such subgraphs can be described by a closure operator we are led to the investigation of the corresponding complete lattice of “closed” subgraphs. For example, in this complete lattice every element is the infimum of an irredundant family of infimum irreducible elements. The work presented here was supported in part by N.S.E.R.C. Operating Grant No. A4077.     

On the Connection Between the Projection and the Extension of a Parallelotope

This paper presents a result concerning the connection between the parallel projection P _v,H of a parallelotope P along the direction v (into a transversal hyperplane H) and the extension P + S(v), meaning the Minkowski sum of P and the segment S(v) = {λv | −1 ≤ λ ≤ 1}. A sublattice L _v of the lattice of translations of P associated to the direction v is defined. It is proved that the extension P + S(v) is a parallelotope if and only if the parallel projection P _v,H is a parallelotope with respect to the lattice of translations L _v,H , which is the projection of the lattice L _v along the direction v into the hyperplane H.     

On linear isometries of Banach lattices in <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>(Ω)-spaces

Consider the space C0(Ω) endowed with a Banach lattice-norm ‖ · ‖ that is not assumed to be the usual spectral norm ‖ · ‖_∞ of the supremum over Ω. A recent extension of the classical Banach-Stone theorem establishes that each surjective linear isometry U of the Banach lattice (C ₀(Ω), ‖ · ‖) induces a partition Π of Ω into a family of finite subsets S ⊂ Ω along with a bijection T: Π → Π which preserves cardinality, and a family [u(S): S ∈ Π] of surjective linear maps u(S): C(T(S)) → C(S) of the finite-dimensional C*-algebras C(S) such that

Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized cluster complex Δ^m(Φ) was introduced by S. Fomin and N. Reading. It was conjectured by these authors that Δ^m(Φ) is shellable and by V. Reiner that it is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part Δ₊^m(Φ) of Δ^m(Φ). An explicit homotopy equivalence is given between Δ₊^m(Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong.     

Differential operators associated with zonal polynomials. II

Summary LetC _κ(S) be the zonal polynomial of the symmetricm×m matrixS=(s_ij), corresponding to the partition κ of the non-negative integerk. If ∂/∂S is them×m matrix of differential operators with (i, j)th entry ((1+δ_ij)∂/∂s_ij)/2, δ being Kronecker's delta, we show that C_k(∂/∂S)C_λ(S)=k!δ_λkC_k(I), where λ is a partition ofk. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients in the zonal polynomial expansion of homogenous symmetric polynomials.     

Extending a Partially Ordered Set: Links with its Lattice of Ideals

A well-known result of Bonnet and Pouzet bijectively links the set of linear extensions of a partial order P with the set of maximal chains of its lattice of ideals I(P). We extend this result by showing that there is a one-to-one correspondence between the set of all extensions of P and the set of all sublattices of I(P) which are chain-maximal in the sense that every chain which is maximal (for inclusion) in the sublattice is also maximal in the lattice.We prove that the absence of order S as a convex suborder of P is equivalent to the absence of I(S) as a convex suborder of I(P). Let S be a set of partial orders and let us call S-convex-free any order that does not contain any order of S as a convex suborder. We deduce from the previous results that there is a one-to-one correspondence between the set of S-convex-free extensions of P and the set of I(S)-convex-free chain-maximal sublattices of I(P). This can be applied to some classical classes of orders (total orders and in the finite case, weak orders, interval orders, N-free orders). In the particular case of total orders this gives as a corollary the result of Bonnet and Pouzet.     

-Decompositions of and D v ∪P where P is a 2-Regular Subgraph of D v

In this paper, we extend the study of C₄-decompositions of the complete graph with 2-regular leaves and paddings to directed versions. Mainly, we prove that if P is a vertex-disjoint union of directed cycles in a complete digraph D_v, then and D_v∪P can be decomposed into directed 4-cycles, respectively, if and only if v(v−1)−|E(P)|≡0(mod 4) and v(v−1)+|E(P)|≡0(mod 4) where |E(P)| denotes the number of directed edges of P, and v≥8.     

Osservazioni su alcuni operatori ibridi iperbolico-ipoellittici

Riassunto Viene dimostrata l'equivalenza di alcune condizioni sul polinomio a coefficienti costantiP(ζ). La prima (si vedaH?rmander [3]), garantisce l'esistenza di una soluzione fondamentale diP(D) con supporto singolare in un cono convesso; l'ultima esprime il carattere ibrido diP(ζ). Viene infine costruita una soluzione fondamentale ultradistribuzione per un operatore 1/β-iperbolico-1/α-ipoellittico estendendo un precedente risultato [5] dell'autore.

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Summary The equivalence of some conditions on the polynomialP(ζ) with constant coefficients is proved. The first one (seeH?rmander [3]), implies the existence of a fundamental solution forP(D) with singular support contained in a convex cone; the last one shows thatP(ζ) is an hyperbolic-hypoelliptic polynomial. Finally an ultradistribution fundamental solution for a 1/β-hyperbolic-1/α-hypoelliptic operator is constructed, so extending a previous result [5] of the author.

     

Reflexive operators with domain in Köthe spaces

It is proved that if a K?the space λ₁(A) is distinguished and E is an arbitrary Fréchet space then every reflexive map T: λ₁(A)→E (i.e., T maps bounded sets into relatively weakly compact ones) factorizes through a reflexive Fréchet space. An analogous result is proved for Montel maps (i.e., which map bounded sets into relatively compact ones). The result is a consequence of the fact proved also in this paper that, for a distinguished λ₁(A) space, the spaces of reflexive maps R(λ₁(A), C(K)) and of Montel maps M(λ₁(A), C(K)) are the Mackey completions of the spaces of weakly compact and compact maps, respectively. Consequences for spaces of vector-valued (weakly) continuous functions are also obtained. Received: 24 November 1997 / Revised version: 14 May 1998     

Clifford martingale Φ-Equivalence betweenS(f) andf*

TheL ²-norm equivalence between a Clifford martingalef and its square functionS(f) plays an important role in the proof of theL ²-boundedness of Cauchy integral operators on Lipschitz graphs and the CliffordT(b) Theorem [2, 4]. This note generalises the result to the Φ-equivalence between the maximal functionf* andS(f), where Φ is a nondecreasing and continuous function fromIR ⁺ toIR ⁺, of the moderate growth Φ(2u)≤C ₁Φ(u) and satisfies Φ(0)=0.     

Differential operators associated with zonal polynomials. I

Summary Associated with each zonal polynomial,C _k(S), of a symmetric matrixS, we define a differential operator ∂_k, having the basic property that ∂_kC_λδ_kλ, δ being Kronecker's delta, whenever κ and λ are partitions of the non-negative integerk. Using these operators, we solve the problems of determining the coefficients in the expansion of (i) the product of two zonal polynomials as a series of zonal polynomials, and (ii) the zonal polynomial of the direct sum,S⊕T, of two symmetric matricesS andT, in terms of the zonal polynomials ofS andT. We also consider the problem of expanding an arbitrary homogeneous symmetric polynomial,P(S) in a series of zonal polynomials. Further, these operators are used to derive identities expressing the doubly generalised binomial coefficients ( _P ^λ ),P(S) being a monomial in the power sums of the latent roots ofS, in terms of the coefficients of the zonal polynomials, and from these, various results are obtained.     

A Property About Minimum Edge- and Minimum Clique-Cover of a Graph

 Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ ⁿ :Ex≤1,x≥0}, P(C)={x∈ℜ ⁿ :Cx≤1,x≥0}, α _E (G)=max{1 ^T x subject to x∈P(E)}, and α _C (G)= max{1 ^T x subject to x∈P(C)}. In this paper we prove that if α _E (G)=α _C (G), then γ(G)=θ(G). Received: May 20, 1998?Final version received: April 12, 1999