Some questions of general topology and Tikhonov semifields. II

This article is, in the opinion of its authors, a natural continuation of the survey [4] (with a similar title). And if the main question in [4] was whether every ring homomorphism $\psi\colon R^\Delta\to R^{\Delta''}$ is continuous, so one of the main questions in this paper is whether every sequentially continuous map $f\colon R^\Delta\to R^{\Delta''}$ is continuous. This circle of questions is closely connected with works of Mazur, Keisler and Tarski, with [4], and with others, and leads us to consider new and very wide classes of cardinals. In particular, we consider the classes of sequential, strictly sequential, measurable, and compact cardinals.
We also clarify some of the connections between properties of the Tikhonov semifields $R^\Delta$ and the lattices $Z^\Delta$, $N^\Delta$ contained in them, with various questions of axiomatic and combinatorial set theory (see [18], [25], [75].)