Convergent sequences in
Łukasiewicz tribes are absolutely sequentially closed bold algebras
We show that each sequentially continuous (with respect to the pointwise convergence) normed measure on a bold algebra of fuzzy sets (Archimedean -algebra) can be uniquely extended to a sequentially continuous measure on the generated Łukasiewicz tribe and, in a natural way, the extension is maximal. We prove that for normed measures on Łukasiewicz tribes monotone (sequential) continuity implies sequential continuity, hence the assumption of sequential continuity is not restrictive. This yields...
On -sequentially regular spaces
On plane topologies with high sequential order
Further note on Fréchet spaces
A note on Fréchet spaces
Regularity and extension of mappings in sequential spaces
Extension of sequentially continuous mappings
Cauchy sequences in -groups
Products of coarse convergence groups
-groups versus -groups
We investigate free groups over sequential spaces. In particular, we show that the free -group and the free sequential group over a sequential space with unique limits coincide and, barred the trivial case, their sequential order is .
Statistical maps: a categorical approach
Rings of maps: sequential convergence and completion
The ring of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring of all continuous functions and, similarly, the ring of all Borel measurable subsets of is a sequential ring completion of the subring of all finite unions of half-open intervals; the two completions are not categorical. We study -rings of maps and develop a completion theory covering the two examples. In particular, the -fields of sets form an epireflective...
Rationals with exotic convergences
On convergence spaces and groups
Rationals with exotic convergences. II.
Duality for generalized events
Extension of measures: a categorical approach
We present a categorical approach to the extension of probabilities, i.e. normed -additive measures. J. Novák showed that each bounded -additive measure on a ring of sets is sequentially continuous and pointed out the topological aspects of the extension of such measures on over the generated -ring : it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space over its Čech-Stone compactification (or as the extension of continuous...
Relatively coarse sequential convergence
We generalize the notion of a coarse sequential convergence compatible with an algebraic structure to a coarse one in a given class of convergences. In particular, we investigate coarseness in the class of all compatible convergences (with unique limits) the restriction of which to a given subset is fixed. We characterize such convergences and study relative coarseness in connection with extensions and completions of groups and rings. E.g., we show that: (i) each relatively coarse dense group precompletion...
Sequential convergences on Boolean algebras defined by systems of maximal filters
We study sequential convergences defined on a Boolean algebra by systems of maximal filters. We describe the order properties of the system of all such convergences. We introduce the category of 2-generated convergence Boolean algebras and generalize the construction of Novák sequential envelope to such algebras.
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