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 prove a convenient equivalent criterion for monotone completeness of ordered fields of generalized power series $\left[\left[F^G\right]\right]$ with exponents in a totally ordered Abelian group G and coefficients in an ordered field F. This enables us to provide examples of such fields (monotone complete or otherwise) with or without integer parts, i.e. discrete subrings approximating each element within 1. We include a new and more straightforward proof that $\left[\left[F^G\right]\right]$ is always Scott complete. In contrast, the Puiseux series field...

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 $𝔹$ of all Borel measurable subsets of $R$ is a sequential ring completion of the subring ${𝔹}_0$ of all finite unions of half-open intervals; the two completions are not categorical. We study ${ℒ}_0^{*}$-rings of maps and develop a completion theory covering the two examples. In particular, the $\sigma$-fields of sets form an epireflective...