Core-EP decomposition and related relations revisited

On sets of elements in Rickart rings induced by partial orders

Let (S, *) be a ring with involution *. The involution is positive definite if, for all finite subsets {r,-} of S, £ w * = 0 implies all the r{ are zero. Then the set of self-adjoint elements of S, denoted S, possesses a natural partial ordering, with positive cone consisting of elements of the form Z l w * ; with this ordering, S is a directed partially ordered abelian group. Let Sb denote the set of bounded elements, that is, the set of elements s such that ss* is less than an integral multiple of 1 in this ordering. Then Sb is a *-subalgebra of S whenever S is an algebra over the rationals. We will be studying the objects Sb9 S , and (Sb) , from the point of view of their ordered structures. For instance, suppose S is a field, and * is the identity. Then S is formally real, and Sb must be Prufer domain, all of whose residue fields are themselves formally real (and in fact, are embeddable in the reals). Viewing Sb as a partially ordered abelian group with order unit 1 (indeed, Sb is the convex subgroup of S generated by 1), Sb has the Riesz decomposition property, and its normalized extremal states are precisely the ring homomorphisms into the reals. There is a natural mapping from the collection of total orderings of S to the set of extremal states of Sb9 and this in turn maps to Spec Sb (the prime ideal space of Sb); when S is even a real algebra much more can be said. If either S is a field and * is not the identity, or S is a quaternionic division algebra with the natural involution, essentially the same properties hold, with the appropriate modifications. A useful tool here is an involutory version of the Artin Schreier Theorem, about the existence of sufficiently many total orderings finer than the natural ordering. Studies are made of several specific bounded subrings. For instance, if S is the rational function field in one variable over the reals, then Sb is a Dedekind domain with class group of order 2, with spectrum the unit circle (in the point-open topology), and all of its maximal ideals are not principal. Expanding the scope of S somewhat, we next allow S to be a division

We study orders in proper $*$-rings that are derived from the core-nilpotent decomposition. The notion of the C-N-star partial order and the S-star partial order is extended from $M_ {n} ( \mathbb{C)}$, the set of all $n \times n$ complex matrices, to the set of all Drazin invertible elements in proper $*$-rings with identity. Properties of these orders are investigated and their characterizations are presented. For a proper $*$-ring $\mathcal{A}$ with identity, it is shown that on the set of all Drazin invertible elements $a \in \mathcal{A}$ where the core part of $a$ is an EP element, the C-N-star partial order implies the star partial order.

JB-algebras of rank zero

A bipartite analogue of Dilworth’s theorem for multiple partial orders

This paper presents first results toward the extension of possibilistic logic when the total order on formulas is replaced by a partial preorder. Few works have dealt with this matter in the past but they include some by Halpern, and Benferhat et al. Here we focus on semantic aspects, namely the construction of a partial order on interpretations from a partial order on formulas and conversely. It requires the capability of inducing a partial order on subsets of a set from a partial order on its elements. The difficult point lies in the fact that equivalent definitions in the totally ordered case are no longer equivalent in the partially ordered one. We give arguments for selecting one approach extending comparative possibility and its preadditive refinement, pursuing some previous works by Halpern. It comes close to non-monotonic inference relations in the style of Kraus Lehmann and Magidor. We define an intuitively appealing notion of closure of a partially ordered belief base from a semantic standpoint, and show its limitations in terms of expressiveness, due to the fact that a partial ordering on subsets of a set cannot be expressed by means of a single partial order on the sets of elements. We also discuss several existing languages and syntactic inference techniques devised for reasoning from partially ordered belief bases in the light of this difficulty. The long term purpose is to find a proof method adapted to partially ordered formulas, liable of capturing a suitable notion of semantic closure.

For a row finite directed graph E, Kumjian, Pask, and Raeburn proved that there exists a universal C * -algebra C * (E) generated by a Cuntz-Krieger E-family.In this paper we consider two density problems of invertible elements in graph C * -algebras C * (E), and it is proved that C * (E) has stable rank one, that is, the set of all invertible elements is dense in C * (E) (or in its unitization when C * (E) is nonunital) if and only if no loop of E has an exit.We also prove that for a locally finite directed graph E with no sinks if the graph C *algebra C * (E) has real rank zero (RR(C * (E)) = 0), that is, the set of invertible self-adjoint elements is dense in the set of all self-adjoint elements of C * (E) then E satisfies a condition (K) on loop structure of a graph, and that the converse is also true for C * (E) with finitely many ideals.In particular, for a Cuntz-Krieger algebra O A , RR(O A ) = 0 if and only if A satisfies Cuntz's condition (II).

We approach the problem of discovering interesting orders in data. In many applications, it is more important to find interesting partial orders since there is often no clear ordering between certain sets of elements. Furthermore, a partial order is more robust against partially erroneous data. We present the notion of fundamental partial orders (FPO), and argue that any partial order that satisfies this property is an interesting partial order. To mine such partial orders, we present a two-stage methodology that first finds an interesting total order, and then discovers a partial order satisfying FPO using this total order. To illustrate, we focus on {0,1} data. This is an important problem with many applications, e.g., in paleontology, where we chronologically order fossil sites by minimizing Lazarus counts. We present the experimental results of our method on paleontological data, and show that it outperforms existing approaches. The techniques developed here are general and can be abstracted for mining partial orders in any setting.

Let A be a Rickart *-ring and let ≤*,≤♯,≤⊕, and ≤⊕ denote the star, the sharp, the core, and the dual core partial orders in A, respectively. The sets of all b∈A such that a≤b, along with the sets of all b∈A such that b≤a, are characterized, where a∈A is given and where ≤ is one of the partial orders: ≤*, or ≤♯, or ≤⊕, or ≤⊕. Such sets of elements that are above or below a given element under the minus partial order ≤− in a Rickart ring A are also studied. Some recent results of Cvetković-Ilić et al. on partial orders in B(H), the algebra of all bounded linear operators on a Hilbert space H, are thus generalized.

Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica Alberto Calderon; Argentina

A semigroup S is said to be regular if for each a E S, a = aba for some b E S. A partial transformation alpha on a set is said to be almost identical if x alpha is not equal to x for at most a finite number of elements x in the domain of alpha. Let X be a partially ordered set. Let PTOP(X), TOP(X), LOP(X), UOP(X), VOP(X) and WOP(X) denote the order-preserving partial transformation semigroup on X, the full order-preserving transformation semigroup on X, the order-preserving 1-1 partial transformation semigroup on X, the semigroup of all order-preserving almost identical partial transformations of X, the semigroup of all order-preserving almost identical transformations of X and the semigroup of all order-preserving almost identical 1-1 partial transformations of X, respectively. Let Z and R denote the set of integers and the set of real numbers, respectively. In this abstract, the partial order on any subset of R is the usual partial order on R. The main results of this research are as follows: Theorem 1. If X is a chain which is order-isomorphic to a subset of Z, then TOP(X) is regular. Theorem 2. For any interval X of R, TOP(X) is regular if and only if X is closed and bounded. Theorem 3. If X is a chain, then all of PTOP(X), IOP(X), UOP(X), VOP(X) and WOP(X) are regular. Theorem 4. Let X be a partially ordered set which is not a chain and let S be one of PTOP(X), IOP(X), UOP(X) and WOP(X). Then S is regular if and only if X is isolated. Theorem 5. If X is a partially ordered set containing (i) disjoint components C1 and C2 with /C1/>1 or a subposet of the forms... Theorem 6. Let X be a partially ordered set and M(X) and m(X) denote the set of all maximal elements of X and the set of all minimal elements of X, respectively. If (i) X = M(X) union m(X) and (ii) for x E m(X) and y E M(X), x>y, then TOP(X) is regular. Theorem 7. Let X be a partially ordered set. If X has a maximum element alpha and a minimum element b such that for all distinct x, y E X-{a, b}, x and y are not comparable, then TOP(X) is regular.

:In this paper we show that the set of all Koliha-Drazin invertible elements in a complex unit al Banach algebra forms a regularity as defined by Kordula and Müller, and we explore the properties of the set as a regularity. We also use this result to simplify the proof that the set of all Drazin invertible elements of a complex unital Banach algebra forms a regularity.

Maximal sized antichains in partial orders

We present a weighted-averaging frequency-domain finite-element method for an accurate and efficient 2D elastic wave modeling technique. Our method introduces three kinds of supplementary element sets in addition to a basic element set that is used in the standard finite-element method. By constructing global stiffness and mass matrices for four kinds of element sets and then averaging them with weighting coefficients, we obtain a new global stiffness and mass matrix. With optimal weighting coefficients determined by a Marquardt–Levenberg method to minimize grid dispersion and grid anisotropy, we can reduce the number of nodal points per shear wavelength from 33.3 (using the standard finite-element method) and 20 (using the eclectic method) to 5, with the errors of group velocities no larger than 1%. By reducing the number of grid points per wavelength, we achieve a 97% and 75% reduction of computer memory required to store the complex impedance matrix for a band-type matrix solver and a nested dissection method, respectively, compared with those of the eclectic method. Our method gives approximate solutions compatible with exact solutions for an infinite homogeneous, a semi-infinite homogeneous (Lamb9s problem), and a horizontal two-layer model with fewer grid points than the standard and the eclectic method. A major advantage of the weighted-averaging finite-element method for the elastic wave equation is that it provides solutions very close to correct solutions for Lamb9s problem economically, unlike most of the displacement approaches. In addition, our scheme makes the complex impedance matrix symmetric, which satisfies reciprocity. Seismic forward modeling techniques that satisfy reciprocity are of critical importance in seismic imaging and inversion because we can economically calculate a Jacobian matrix using the reciprocity. Successful simulation of a large-size model shows that our method can be used for the simulation of wave propagation in the geological model needed in the reverse-time migration or seismic inversion.

In this paper we consider classical shop problems:n jobs have to be processed onm machines. The processing timep i,j of jobi on machinej is given for all operations (i, j). Each machine can process at most one job at a time and each job can be processed at most on one machine at a given time. The machine orders are fixed (job-shop) or arbitrary (open-shop). We have to determine a feasible combination of machine and job orders, a so-called sequence, which minimizes the makespan. We introduce a partial order on the set of sequences with the property that there exists at least one optimal sequence in the set of minimal elements of this partial order independent of the given processing times. The set of minimal elements (set of irreducible sequences) can be in detail described in the case of the two machine open-shop problem. The cardinality is calculated. We will show which sequences are generated by the well-known polynomial algorithms for the construction of optimal schedules. Furthermore, we investigate the problemO∥C max on an operation set with spanning tree structure.