While the usual approach using Cauchy sequences is not powerful enough to give completions of uniform spaces in general, if handled appropriately Cauchy sequences suffice for completions of uniform locales. We describe the construction and talk about how this approach might be applied more...

Given a \( \sigma \)-space \( X \) and its lattice of open sets \( \mathcal{O}(X) \), we will study congruences on the \( \sigma \)-frame \( \mathcal{O}(X) \) representing the subspaces of \( X \) (referred to as induced congruences). We will show that when \( X \) is a \( T_D \) \(...

A concept of a proper projective semimodule is introduced and proper projective resolutions are used to construct derived functors of additive functors from the category of cancellative semimodules to the category of cancellative semimodules. We investigate exactness of the long sequence of...

The field of machine learning has experienced a strong expansion in recent years, mostly driven by the development of new computational methods and the unprecedented surge in the volume of data accessible for processing and analysis. This dynamic has opened up new frontiers for exploration...