We show that each sequentially continuous (with respect to the pointwise convergence) normed measure on a bold algebra of fuzzy sets (Archimedean $MV$-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...

We investigate free groups over sequential spaces. In particular, we show that the free $k$-group and the free sequential group over a sequential space with unique limits coincide and, barred the trivial case, their sequential order is ${\omega}_{1}$.

The ring $B\left(R\right)$ of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring $C\left(R\right)$ of all continuous functions and, similarly, the ring $\mathbb{B}$ of all Borel measurable subsets of $R$ is a sequential ring completion of the subring ${\mathbb{B}}_{0}$ of all finite unions of half-open intervals; the two completions are not categorical. We study ${\mathcal{L}}_{0}^{*}$-rings of maps and develop a completion theory covering the two examples. In particular, the $\sigma $-fields of sets form an epireflective...

We present a categorical approach to the extension of probabilities, i.e. normed $\sigma $-additive measures. J. Novák showed that each bounded $\sigma $-additive measure on a ring of sets $\mathbb{A}$ is sequentially continuous and pointed out the topological aspects of the extension of such measures on $\mathbb{A}$ over the generated $\sigma $-ring $\sigma \left(\mathbb{A}\right)$: it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space $X$ over its Čech-Stone compactification $\beta X$ (or as the extension of continuous...

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...

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.

It is known that the ring $B\left(\mathbb{R}\right)$ of all Baire functions carrying the pointwise convergence yields a sequential completion of the ring $C\left(\mathbb{R}\right)$ of all continuous functions. We investigate various sequential convergences related to the pointwise convergence and the process of completion of $C\left(\mathbb{R}\right)$. In particular, we prove that the pointwise convergence fails to be strict and prove the existence of the categorical ring completion of $C\left(\mathbb{R}\right)$ which differs from $B\left(\mathbb{R}\right)$.

Download Results (CSV)