On the ring of the variety of algebras over a ring
On congruences in direct sums of algebras
Two notes on locally finite cylindric algebras
Biequivalence vector spaces in the alternative set theory
As a counterpart to classical topological vector spaces in the alternative set theory, biequivalence vector spaces (over the field of all rational numbers) are introduced and their basic properties are listed. A methodological consequence opening a new view towards the relationship between the algebraic and topological dual is quoted. The existence of various types of valuations on a biequivalence vector space inducing its biequivalence is proved. Normability is characterized in terms of total...
Biequivalences and topology in the alternative set theory
Archimedean and geodetical biequivalences
Cuts of real classes
Borel classes in AST. Measurability, cuts and equivalence
Some connections between measure, indiscernibility and representation of cuts
Arithmetic of cuts and cuts of classes
Professor Tibor Katriňák will be seventy next year
Axiomatization and undecidability results for linear betweenness relations
Indiscernibles and dimensional compactness
This is a contribution to the theory of topological vector spaces within the framework of the alternative set theory. Using indiscernibles we will show that every infinite set in a biequivalence vector space , such that for distinct , contains an infinite independent subset. Consequently, a class is dimensionally compact iff the -equivalence is compact on . This solves a problem from the paper [NPZ 1992] by J. Náter, P. Pulmann and the second author.
Dimensional compactness in biequivalence vector spaces
The notion of dimensionally compact class in a biequivalence vector space is introduced. Similarly as the notion of compactness with respect to a -equivalence reflects our nonability to grasp any infinite set under sharp distinction of its elements, the notion of dimensional compactness is related to the fact that we are not able to measure out any infinite set of independent parameters. A fairly natural Galois connection between equivalences on an infinite set and classes of set functions ...
Some Ramsey type theorems for normed and quasinormed spaces
We prove that every bounded, uniformly separated sequence in a normed space contains a “uniformly independent” subsequence (see definition); the constants involved do not depend on the sequence or the space. The finite version of this result is true for all quasinormed spaces. We give a counterexample to the infinite version in for each 0 < p < 1. Some consequences for nonstandard topological vector spaces are derived.
Professor Ivan Korec died
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