Completely simple topological commutative rings

The paper considers a generalization to topological algebras of the concept of algebraical simplicity (see, definitions 1 and 1' below). Such topological algebras are called completely simple. Completely simple topological commutative rings and Abelian groups are described. As an appendix, a new proof is obtained for Kowalsky's theorem on fields with topologies that cannot be weakened.Translated from Matematicheskie Zametki, Vol. 5, No. 2, pp. 161–171, February, 1969.In conclusion, the author wishes to express his gratitude to his scientific director, L. A. Skornyakov.     

Completely unimodal numberings of a simple polytope

A completely unimodal numbering of the m vertices of a simple d-dimensional polytope is a numbering 0, 1, …,m−1 of the vertices such that on every k-dimensional face (2≤k≤d) there is exactly one local minimum (a vertex with no lower-numbered neighbors on that face). Such numberings are abstract objective functions in the sense of Adler and Saigal [1]. It is shown that a completely unimodal numbering of the vertices of a simple polytope induces a shelling of the facets of the dual simplicial polytope. The h-vector of the dual simplicial polytope is interpreted in terms of the numbering (with respect to using a local-improvement algorithm to locate the vertex numbered 0). In the case that the polytope is combinatorially equivalent to a d-dimensional cube, a ‘successor-tuple’ for each vertex is defined which carries the crucial information of the numbering for local-improvement algorithms. Combinatorial properties of these d-tuples are studied. Finally the running time of one particular local-improvement algorithm, the Random Algorithm, is studied for completely unimodal numberings of the d-cube. It is shown that for a certain class of numberings (which includes the example of Klee and Minty [8] showing that the simplex algorithm is not polynomial and all Hamiltonian saddle-free injective pseudo-Boolean functions [6]) this algorithm has expected running time that is at worst quadratic in the dimension d.     

Completely simple semigroups with nilpotent structure groups

In this paper we find simple characterizations of completely simple semigroups with H-classes nilpotent of class ≤c, and of completely simple semigroups whose core has H-classes nilpotent of class ≤c. The notion of w-marginal completely regular semigroups is introduced, generalizing the concept of central semigroups. A law characterizing [x ₁,x ₂,…,x _c+1]-marginal completely simple semigroups is obtained. Additionally, the least congruences corresponding to these classes are described. Our results extend the corresponding results obtained by Petrich and Reilly in the abelian case. The author was supported by the Ministry of Higher Education, Science and Technology of Slovenia.     

When Summands of Completely Decomposable Modules Are Completely Decomposable

We examine when summands of completely decomposable modules over a domain R are again completely decomposable. We show that this is the case if R is an h-local Prüfer domain. If R is 1-dimensional Noetherian, then the problem reduces locally if almost all localizations are integrally closed. If R is 1-dimensional Noetherian and local, then the integral closure of R must have at most two maximal ideals.     

Completely lexsegment ideals

In this paper we study ideals which are generated by lexsegments of monomials. In contrast to initial lexsegments, the shadow of an arbitrary lexsegment is in general not again a lexsegment. An ideal generated by a lexsegment is called completely lexsegment, if all iterated shadows of the set of generators are lexsegments. We characterize all completely lexsegment ideals and describe cases in which they have a linear resolution. We also prove a persistence theorem which states that all iterated shadows of a lexsegment are again lexsegments if the first shadow has this property.

     

Completely regular designs

We study a class of t-designs which enjoy a high degree of regularity. These are the subsets of vertices of the Johnson graph which are completely regular, in the sense of Delsarte [Philips Res. Reports Suppl. 10 (1973)]. After setting up the basic theory, we describe the known completely regular designs. We derive very strong restrictions which must hold in order for a design to be completely regular. As a result, we are able to determine which symmetric designs are completely regular and which Steiner systems with t = 2 are completely regular. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 261–273, 1998     

A simple derivation of the MacWilliams identity for linear ordered codes and orthogonal arrays

In (Can J Math 51(2):326–346, 1999), Martin and Stinson provide a generalized MacWilliams identity for linear ordered orthogonal arrays and linear ordered codes (introduced by Rosenbloom and Tsfasman (Prob Inform Transm 33(1):45–52, 1997) as “codes for the m-metric”) using association schemes. We give an elementary proof of this generalized MacWilliams identity using group characters and use it to derive an explicit formula for the dual type distribution of a linear ordered code or orthogonal array.      

Completely normal matrices

Normal matrices in which all submatrices are normal are said to be completely normal. We characterize this class of matrices, determine the possible inertias of a particular completely normal matrix, and show that real matrices in this class are closed under (general) Schur complementation. We provide explicit formulas for the Moore–Penrose inverse of a completely normal matrix of size at least four. A result on irreducible principally normal matrices is derived as well.     

Completely nonmeasurable unions

Assume that no cardinal κ < 2 ^ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal $ \mathbb{I} $ \mathbb{I} of X contains uncountably many pairwise disjoint subfamilies $ \mathbb{I} $ \mathbb{I} -Bernstein unions ∪ $ \mathbb{I} $ \mathbb{I} -Bernstein if A and X \ A meet each Borel $ \mathbb{I} $ \mathbb{I} -positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].     

Completely regular semigroups

No abstract. August 22, 2001     

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。     

Completely representable lattices

It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be \({{\bf jCRL} \cap {\bf mCRL}}\) . We prove

${\bf CRL} \subset {\bf biCRL} = {\bf mCRL} \cap {\bf jCRL} \subset {\bf mCRL} \neq {\bf jCRL} \subset {\bf DL}$

where the marked inclusions are proper.

Let L be a DL. Then \({L \in {\bf mCRL}}\) iff L has a distinguishing set of complete, prime filters. Similarly, \({L \in {\bf jCRL}}\) iff L has a distinguishing set of completely prime filters, and \({L \in {\bf CRL}}\) iff L has a distinguishing set of complete, completely prime filters.Each of the classes above is shown to be pseudo-elementary, hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary.