On Convex Subgroups of Groups with the Interpolation Property

Characteristics of groups with the interpolation relation (not necessarily directed) are considered. A necessary and sufficient condition for a partially ordered group to be an interpolation group is obtained. An almost orthogonality criterion for positive elements of an interpolation group is proved. Characteristics of minimal convex directed subgroups containing almost orthogonal elements are described. Properties of convex directed subgroups in the subclass of interpolation groups in which each element is a quotient of two almost orthogonal elements are investigated.     

The intertwining lifting theorem for ordered groups

Sz.-Nagy's dilation theorem for a contraction on a Hilbert space has been extended by Mlak to the case of a contraction semigroup whose indices are the positive elements of a totally ordered discrete group. We generalize to this case the intertwining lifting theorem of Sz.-Nagy and Foias. Some previous interpolation results on ordered groups are obtained as consequences.     

Holland’s theorem for pseudo-effect algebras

We give two variations of the Holland representation theorem for ℓ-groups and of its generalization of Glass for directed interpolation po-groups as groups of automorphisms of a linearly ordered set or of an antilattice, respectively. We show that every pseudo-effect algebra with some kind of the Riesz decomposition property as well as any pseudo MV-algebra can be represented as a pseudo-effect algebra or as a pseudo MV-algebra of automorphisms of some antilattice or of some linearly ordered set.     

Ordered *-Algebras with an Order Unit

We develop a theory of ordered *-algebras with an order unit. These are complex algebras equipped with a conjugate-linear vector space involution whose hermitian elements form an ordered real algebra with an order unit but not necessarily with a multiplicative identity. Our main result is a representation theorem for ordered *-algebras with an order unit. This can be viewed as the complex and the non-unital version of the representation theorem of Stone and Kadison for ordered real algebras containing an order unit which is a multiplicative identity. A key role in our approach is played by certain weighted function algebras.     

On a Generalization of the Notion of Orthogonality and on the Riesz Groups

A method for generalizing the notion of orthogonality to arbitrary partially ordered groups is considered. In the paper, the properties of almost orthogonal elements in Riesz groups are investigated. A series of results on convex directed subgroups associated with a pair of almost orthogonal elements of a Riesz group is obtained. For these subgroups, some theorems on homomorphisms are proved.     

More Spectral Theory in Ordered Banach Algebras

We continue our development of spectral theory for positive elements in an ordered Banach algebra. In particular we provide a suitable version of the Krein-Rutman theorem, obtain some results concerning the peripheral spectrum of a positive element and provide a characterisation of positive quasi inessential elements, in the context of an ordered Banach algebra.     

Some remarks on almost l-groups

The divisibility group of every Bézout domain is an abelian l-group. Conversely, Jaffard, Kaplansky, and Ohm proved that each abelian l-group can be obtained in this way, which generalizes Krull’s theorem for abelian linearly ordered groups. Dumitrescu, Lequain, Mott, and Zafrullah [3] proved that an integral domain is almost GCD if and only if its divisibility group is an almost l-group. Then they asked whether the Krull-Jaffard-Kaplansky-Ohm theorem on l-groups can be extended to the framework of almost l-groups, and asked under what conditions an almost l-group is lattice-ordered [3, Questions 1 and 2]. This note answers the two questions. Received: 29 April 2008     

Completeness and interpolation of almost‐everywhere quantification over finitely additive measures

We give an axiomatization of first‐order logic enriched with the almost‐everywhere quantifier over finitely additive measures. Using an adapted version of the consistency property adequate for dealing with this generalized quantifier, we show that such a logic is both strongly complete and enjoys Craig interpolation, relying on a (countable) model existence theorem. We also discuss possible extensions of these results to the almost‐everywhere quantifier over countably additive measures.     

有限級有序加羣及有序環的表現

在另一文中,我们讨论了由全体2维實向量所成的有序环,在该文最後並说當维数n>2时(n为有限)也可类似地作初步讨论.为了显示这种向量环的用途,我们考虑用向量环来表现一般有序环的问题.在本文中我们证明:任一“n级的”(见以下定义)有序环都能与一个由若干n维實向量所组成的有序环同构.(主要在於证出关於n级有序加羣的类似结果.)我们希望有较好的结果,即:任一n级有序环都能与由全体n维實向量所成的一个有序环的一个子环同构,但未能证明或否定.     

适合无限维实零点定理的序域之结构Ⅰ

本文的目的是建立适合无限维实零点定理的序域的结构定理．作为预备工作,文章的第一部分研究一类无秩为d的裂缝的序群,这里d是无限基数．藉助于Hahn嵌入定理,本文给出了无秩为d的裂缝的序群的结构．     

适合无限维实零点定理的序域之结构Ⅰ

本文的目的是建立适合无限维实零点定理的序域的结构定理．作为预备工作,文章的第一部分研究一类无秩为d的裂缝的序群,这里d是无限基数．藉助于Hahn嵌入定理,本文给出了无秩为d的裂缝的序群的结构．     

Freef-modules

Let R be a directed ring whose cone of positive elements is strict and satisfies Ore's condition. The main result: there exists a freef-module over an o-module_RM with cone p_m if and only if p_M is half-closed (this generalizes Vainberg's theorem for ordered Abelian groups). In this connection various characterizations off-modules with strict cones are given.     

The fixed point property in ordered sets of width two

We consider the fixed point property (FPP) in an ordered set of width two (every antichain contains at most two elements). The necessary condition of the FPP and a number of equivalent conditions to the FPP in such sets is established. The product theorem is proved, as well.     

Completion of Linearly Ordered Metabelian Groups

We prove a theorem saying that in finitely generated linearly ordered metabelian groups there exists a finite system of normal convex subgroups satisfying orderability conditions for groups, and an embedding theorem for linearly ordered metabelian groups whose initial linear orders extend to -divisible linearly ordered metabelian ones. As a consequence, it is stated that orderable metabelian groups are embedded, with extension of all their linear orders, in -divisible orderable metabelian groups.     

Existence of positive pseudo almost periodic solutions to a class of neutral integral equations

In this work, a new fixed point theorem in partially ordered Banach spaces is established, and then used to prove the existence of positive pseudo almost periodic solutions to a class of neutral integral equations. Our existence theorem extends some recent results due to [E. Ait Dads, P. Cieutat, L. Lhachimi, Positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations, Mathematical and Computer Modelling 49 (2009) 721–739] to a more general class of integral equations.     

The interpolation theorem for narrow quadrilateral isoparametric finite elements

Summary. The interpolation theorem for convex quadrilateral isoparametric finite elements is proved in the case when the condition is not satisfied, where is the diameter of the element and is the radius of an inscribed circle in . The interpolation error is in the -norm and in the -norm provided that the interpolated function belongs to . In the case when the long sides of the quadrilateral are parallel the constants appearing in the estimates are evaluated. Received September 1993 / Revised version received March 6, 1995     

Pseudo almost periodic in distribution solutions and optimal solutions to impulsive partial stochastic differential equations with infinite delay

In this paper, we study a general class of impulsive partial stochastic differential equations with infinite delay and pseudo almost periodic coefficients in Hilbert spaces. Firstly, a more appropriate concept of pseudo almost periodic in distribution for stochastic processes of infinite class is introduced. Secondly, the existence of pseudo almost periodic in distribution mild solutions is investigated by utilizing the interpolation theory, the stochastic analysis techniques and fixed point theorem. The existence of optimal mild solutions of the systems is also proved. Finally, an example is provided to show the effectiveness of the theoretical results.     

On interpolation of bilinear operators

In this paper we study interpolation of bilinear operators between products of Banach spaces generated by abstract methods of interpolation in the sense of Aronszajn and Gagliardo. A variant of bilinear interpolation theorem is proved for bilinear operators from corresponding weighted c₀ spaces into Banach spaces of non-trivial the periodic Fourier cotype. This result is then extended to the spaces generated by the well-known minimal and maximal methods of interpolation determined by quasi-concave functions. In the case when a maximal construction is generated by Hilbert spaces, we obtain a general variant of bilinear interpolation theorem. Combining this result with the abstract Grothendieck theorem of Pisier yields further results. The results are applied in deriving a bilinear interpolation theorem for Calderón-Lozanovsky, for Orlicz spaces and an embedding interpolation formula for (E,p)-summing operators.     

On archimedean extensions ofAO-groups

The concept of almost orthogonality is a variant for extending the concept of the orthogonal elements of lattice-ordered groups to arbitrary partially ordered groups. In the present paper, the notion of an Archimedean extension of anAO-groups is studied. Some results are obtained concerning interpolationAO-groups. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.