Completions of orthomodular lattices II

If is a variety of orthomodular lattices generated by a set of orthomodular lattices having a finite uniform upper bound om the length of their chains, then the MacNeille completion of every algebra in again belongs to .The author gratefully acknowledges the support of NSERC.     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.     

On the convergence of powers of interval matrices (2)

Summary Let be a real irreduciblen×n interval matrix. Then a necessary and sufficient condition is given for the sequence of the powers of an interval matrix to converge to a matrix which is not the null matrix. In addition a criterion for is proved to decide whether the limit matrix satisfies the condition of symmetry .     

On the dimension of the cartesian product of relations and orders

It is known that for incidence structures and , max , wheref dim stands for Ferrers relation. We shall show that under additional assumptions on and , both bounds can be improved. Especially it will be shown that the square of a three-dimensional ordered set is at least four-dimensional.     

Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

Lattice Ordered Polynomial Algebras

In this paper we define a lattice order on a set F of binary functions. We then provide necessary and sufficient conditions for the resulting algebra F to be a distributive lattice or a Boolean algebra. We also prove a Cayley theorem for distributive lattices by showing that for every distributive lattice , there is an algebra F of binary functions, such that is isomorphic to F and we show that F is a distributive lattice iff the operations and are idempotent and cummutative, showing that this result cannot be generalized to non-distributive lattices or quasilattices without changing the definitions of and . We also examine the equational properties of an Algebra for which , now defined on the set of binary -polynomials is a lattice or Boolean algebra.     

Series-parallel posets and relative Ockham lattices

A study is undertaken of order-reversing maps on series-parallel posets and structural characterisations are obtained of various subclasses of such ordered sets. The results are applied to complete the authors' earlier investigation of classes of finite relate lattices, where is a variety of Ockham lattices and the distributive lattice duals of the algebras in are required to be series-parallel.     

Kinematik in der Laguerre-Ebene II: 97-0197-0197-01-Bewegungen mit Polkurven

This second part of kinematics in the Laguerre-plane contains differential geometric properties of -motions. In this paper we consider only such -motions which are instantaneous products of a rotation and a homothetic transformation with common center. The centers, called poles of a -motion, generate a curve on a ruled surface consisting of fixed lines. First we investigate the dependence of a -motion on the pair of strips along the pole curves which contact each other. Furthermore the points with isotropic tangents of their paths lie on a cone at each time of a -motion. On this cone there exist curves of some interest like the curves of the cycles of inflection and of the vertical cycles and the Bresse-curves. The isotropic tangents of the paths in the cycles of inflection define a socalled hypercycle of Blaschke.     

On periodic groups of automorphisms of extremal groups

It is proved that if a periodic group has an extremal normal divisor , determining a complete abelian factor group , then the center of the group contains a complete abelian subgroup , satisfying the relation and intersecting on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group is a finite extension of a contained in it subgroup of inner automorphisms of the group .Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 91–96, July, 1968.     

Positive elements in the algebra of the quantum moment problem

Summary Let denote the extended Weyl algebra, , the Weyl algebra. It is well known that every element of of the formA=B _k ^* B _k is positive. We prove that the converse implication also holds: Every positive elementA in has a quadratic sum factorization for some finite set of elements (B _k ) in . The corresponding result is not true for the subalgebra . We identify states on which do not extend to states on . It follows from a result of Powers (and Arveson) that such states on cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on , and this property is not in general shared by representations generated by states defined only on the subalgebra .Work supported in part by the NSF     

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

Homogeneous algebraic varieties defined by Jordan pairs

Every Jordan pair defines an algebraic varietyX containing as a dense open subset.X is projective (affine) if and only if is separable (radical). The Picard group ofX is generated by the irreducible factors of the generic norm of . If is separable then the automorphism group ofX is the projective group of .     

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

Varieties Defined by Permutations

We continue to study interrelations between permutative varieties and the cyclic varieties defined by cycles of the form . A criterion is given determining whether a cyclic variety is interpretable in . For a permutation without fixed elements, it is stated that a set of primes for which is interpretable in in the lattice is finite. It is also proved that for distinct primes , the Helly number of a type in coincides with dimension of the dual type and equals .     

Minimax-robust prediction of discrete time series

Summary We discuss a robust approach for predicting a weakly stationary discrete time series whose spectral density f is not exactly known. We assume that we know that f ), where is a convex set of spectral densities fulfilling some not too stringent conditions. We proof the existence of a most indeterministic density f ⁰ in , and we show that the classical optimal linear predictor for a time series with spectral density f ⁰ is mini-max-robust in the sense that it minimizes the maximal possible prediction error.We investigate some special models , and, in doing so, we illustrate a generally applicable method for determining the robust predictor. In particular, we discuss model sets which are defined by conditions on a finite part of the autocovariance sequence of the corresponding time series. These examples are of particular interest as the most indeterministic density is an autoregressive one, i.e. the robust predictor is finite. We discuss connections between this type of model set and maximum entropy and generalized maximum entropy spectral estimates.     

Gray codes from antimatroids

We show three main results concerning Hamiltonicity of graphs derived from antimatroids. These results provide Gray codes for the feasible sets and basic words of antimatroids.For antimatroid (E, ), letJ( ) denote the graph whose vertices are the sets of , where two vertices are adjacent if the corresponding sets differ by one element. DefineJ( ;k) to be the subgraph ofJ( )² induced by the sets in with exactlyk elements. Both graphsJ( ) andJ( ;k) are connected, and the former is bipartite.We show that there is a Hamiltonian cycle inJ( )×K ₂. As a consequence, the ideals of any poset % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFpepuaaa!414C!\[\mathcal{P}\] may be listed in such a way that successive ideals differ by at most two elements. We also show thatJ( ;k) has a Hamilton path if (E, ) is the poset antimatroid of a series-parallel poset.Similarly, we show thatG( )×K ₂ is Hamiltonian, whereG( ) is the basic word graph of a language antimatroid (E, ). This result was known previously for poset antimatroids.Research supported in part by NSERC.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A3379.     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .