Cohomology of partially ordered sets and local cohomology of section rings

We study local cohomology of rings of global sections of sheafs on the Alexandrov space of a partially ordered set. We give a criterion for a splitting of the local cohomology groups into summands determined by the cohomology of the poset and the local cohomology of the stalks. The face ring of a rational pointed fan can be considered as the ring of global sections of a flasque sheaf on the face poset of the fan. Thus we obtain a decomposition of the local cohomology of such face rings. Since the Stanley-Reisner ring of a simplicial complex is the face ring of a rational pointed fan, our main result can be interpreted as a generalization of Hochster's decomposition of local cohomology of Stanley-Reisner rings.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

Generalized contractions in partially ordered metric spaces

We present some fixed point results for monotone operators in a metric space endowed with a partial order using a weak generalized contraction-type assumption.     

Totally ordered sets and the prime spectra of rings

Let T be a totally ordered set and let D(T) denotes the set of all cuts of T. We prove the existence of a discrete valuation domain O_v such that T is order isomorphic to two special subsets of Spec(O_v). We prove that if A is a ring (not necessarily commutative), whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set U?Spec(A) such that the prime spectrum of A is order isomorphic to D(U). We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view.     

The fractional weak discrepancy of a partially ordered set

In this paper we introduce the notion of the fractional weak discrepancy of a poset, building on previous work on weak discrepancy in [J.G. Gimbel and A.N. Trenk, On the weakness of an ordered set, SIAM J. Discrete Math. 11 (1998) 655-663; P.J. Tanenbaum, A.N. Trenk, P.C. Fishburn, Linear discrepancy and weak discrepancy of partially ordered sets, ORDER 18 (2001) 201-225; A.N. Trenk, On k-weak orders: recognition and a tolerance result, Discrete Math. 181 (1998) 223-237]. The fractional weak discrepancywd_F(P) of a poset P=(V,?) is the minimum nonnegative k for which there exists a function f:V→R satisfying (1) if a?b then f(a)+1?f(b) and (2) if a∥b then |f(a)-f(b)|?k. We formulate the fractional weak discrepancy problem as a linear program and show how its solution can also be used to calculate the (integral) weak discrepancy. We interpret the dual linear program as a circulation problem in a related directed graph and use this to give a structural characterization of the fractional weak discrepancy of a poset.     

An abstract sub-supersolution method in partially ordered spaces

We present a general framework in which sub-supersolution techniques can be applied. Specifically, we consider a normed vector space endowed with a partial order. This allows us to introduce a notion of sub- and supersolutions relative to an equation driven by operators defined from the space to its topological dual. We provide sufficient conditions to guarantee that the interval determined by an ordered pair of sub-supersolutions contains a solution of the initial problem. We also study existence of extremal positive or negative solutions and location of nodal solutions. Examples of applications to quasilinear boundary value problems are given.     

Existentially closed ordered difference fields and rings

We describe classes of existentially closed ordered difference fields and rings. We show an Ax‐Kochen type result for a class of valued ordered difference fields (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Directed ideals in partially ordered vector spaces

For a linear subspace I$I$ of a Riesz space there are various well-known properties that are equivalent to I$I$ being an ideal, such as I$I$ is a full Riesz subspace, I$I$ is a solid subspace, I$I$ is a Riesz subspace and the kernel of a positive linear map, I$I$ is the kernel of a Riesz homomorphism. Generalizations of these properties to partially ordered vector spaces are considered and their relations are investigated. It is shown that for directed subspaces all these generalizations are equivalent, just as in the case of Riesz spaces.     

The toric -vectors of partially ordered sets

An explicit formula for the toric -vector of an Eulerian poset in terms of the -index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag -vector. For this, another proof based on Fine's algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, , is the only equation relating the -vectors of posets and their duals. A result on the -vectors of oriented matroids is given. A simple formula for the -index in terms of the flag -vector is derived.

     

Quasimartingales on partially ordered sets

The concept of a quasimartingale, and therefore also of a function of bounded variation, is extended to processes with a regular partially ordered index set V and with values in a Banach space. We show that quasimartingales can be described by their associated measures, defined on an inverse limit space $\text{S × Ω}$ containing $\text{V × Ω}$, furnished with the σ-algebra $\text{P}$ of the predictable sets. With the help of this measure, a Rao-Krickeberg and a Riesz decomposition is obtained, as well as a convergence theorem for quasimartingales. For a regular quasimartingale X it is proven that the spaces ($\text{S × Ω}$, $\text{P}$) and the measures associated with X are unique up to isomorphisms. In the case V = $\text{R}$₊ⁿ we prove a duality between classical (right-) quasimartingales and left-quasimartingales.     

Gabriel filters in real closed rings

Real closed rings arise in semi-algebraic geometry and topology as well as in the investigation of partially ordered rings. It is shown that localizations of real closed rings with respect to Gabriel filters, or more generally: multiplicative filters, are again real closed. Thus, real closedness is preserved under a large number of important ring theoretic constructions. For a few particularly simple cases the multiplicative filters are classified and the localizations are determined. Received: August 26, 1996     

A short note on band projections in partially ordered vector spaces

Consider an Archimedean partially ordered vector space $X$ with generating cone (or, more generally, a pre-Riesz space $X$). Let $P$ be a linear projection on $X$ such that both $P$ and its complementary projection $I?P$ are positive; we prove that the range of $P$ is a band. This shows that the well-known concept of band projections on vector lattices can, to a certain extent, be transferred to the framework of ordered vector spaces.     

Coupled fixed point results for mappings without mixed monotone property in partially ordered G-metric spaces

In this paper, we prove coupled fixed point results for mappings without mixed monotone property in partially ordered G-metric spaces. Also we showed that if $(X\text{,}G)$ is regular, these fixed point results holds.     

Fully ordered groups in partially ordered linear algebras

No abstract. Received September 2, 1999; accepted in final form October 23, 2000.     

Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces

We introduce the concept of a mixed g-monotone mapping and prove coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces. Presented theorems are generalizations of the recent fixed point theorems due to Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393] and include several recent developments.     

偏序半群的偏序扩张

引入了偏序半群(S,·,≤)上的半拟序σ及模σ半拟链的概念.通过模σ半拟链,将S的偏序≤扩张为≤*,讨论了(S,*,≤*)是偏序半群的充分条件,并获得了若干理想的结果.特别地,得到了SPO(S)到PO(S)的两个半格同态定理.最后,还给出了S的满足某些给定条件的有限子集在≤*下成链的充要条件.     

Compressions on partially ordered abelian groups

If is a C*-algebra and is a self-adjoint idempotent, the mapping is called a compression on . We introduce effect-ordered rings as generalizations of unital C*-algebras and characterize compressions on these rings. The resulting characterization leads to a generalization of the notion of compression on partially ordered abelian groups with order units.

     

Group rings which are G-Dedekind prime rings

Let R be a commutative Noetherian domain and A be a polycyclic-by-finite group. In this paper, it is determined, in terms of properties of R and A when the group ring R[A] is a G-Dedekind prime ring.