Varieties whose skeletons are lattices

Translated from Algebra i Logika, Vol. 31, No. 1, pp. 74–82, January–February, 1992.     

Two topological properties of topological linear spaces

A topological and a geometrical-topological property, previously known only for normed linear spaces, are established here for much more general classes of topological linear spaces. This research was conducted at the University of Washington in 1963 when the first author was visiting there. The work of both authors was supported in part by the National Science Foundation, U. S. A. (NSF-GP-378).     

Connectedness in topological linear spaces

The following properties, well known for normed linear spaces of dimension ≧2, are established for an arbitrary topological linear space of dimension ≧2: (a) every neighborhood of 0 contains one whose complement is connected; (b) the complement of a bounded set has exactly one unbounded component. Research supported by the National Science Foundation, U.S.A. (NSF-GP-378).     

Graded left modular lattices are supersolvable

We provide a direct proof that a finite graded lattice with a maximal chain of left modular elements is supersolvable. This result was first established via a detour through EL-labellings in [MT] by combining results of McNamara [Mc] and Liu [Li]. As part of our proof, we show that the maximum graded quotient of the free product of a chain and a single-element lattice is finite and distributive.Received May 24, 2004; accepted in final form October 12, 2004.     

Compact Hausdorff fuzzy topological spaces are topological

Many examples of compact fuzzy topological spaces which are highly non topological are known [5, 6]. Equally many examples of Hausdorff fuzzy topological spaces which are highly non topological can be given. In this paper we show that the two properties - compact and Hausdorff - combined however necessarily imply that the fuzzy topological space is topological. This at once solves some open questions with regard to the compactification of fuzzy topological spaces [8]. It also emphasizes once more the particular role played by compact Hausdorff topological spaces not only in the category of topological spaces but even in the category of fuzzy topological spaces.     

On existence of cone-maximal points in real topological linear spaces

A sufficient condition for a convex coneC in a Hausdorff topological linear space is given in order to ensure the existence of cone-maximal points. The condition becomes a necessary one in a topological linear space with a countable local base, that is, if the space is pseudometrizable. The paper extends known results to infinite dimensions and we answer Corley’s question in the affirmative with the exception of a pathological case.     

On a fixed-point theorem on locally convex linear topological spaces

Iff is a self mapping on a closed convex subsetK of a separated quasicomplete locally convex linear topological spaceE such that (i)E is strictly convex, (ii)f (K) is contained in a compact subset ofK and (iii)f satisfies a contraction condition, then it is shown that for eachxK, the sequence of {U ⁿ (x)} _n =1 of iterates, whereU KK is defined byU (y)=f(y)+(1-) y, yK, converges to a fixed point off.     

On a fixed-point theorem on locally convex linear topological spaces

Iff is a self mapping on a closed convex subsetK of a separated quasicomplete locally convex linear topological spaceE such that (i)E is strictly convex, (ii)f (K) is contained in a compact subset ofK and (iii)f satisfies a contraction condition, then it is shown that for eachxK, the sequence of {U ⁿ (x)} _n =1 of iterates, whereU KK is defined byU (y)=f(y)+(1-) y, yK, converges to a fixed point off.     

On subseries convergent series and m-quasi-bases in topological linear spaces

We prove that the algebraic dimension of the linear span of sums of a subseries convergent series in a Hausdorff topological linear space is either finite or equals 2°. This result is applied to represent every infinite-dimensional metrizable linear spaee of cardinality 2° as the direct sum of a sequence of dense subspaces with a strange summability property. Moreover, we show that every infinite-dimensional separable metrizable linear space admits an m-quasi-basis which is not a quasi-basis.Some results of this paper were presented at the 9th Winter School on Abstract Analysis in Srni (Czechoslovakia), January 1981