The aim of this paper is to define and study cardinal (direct) and ordinal operations of addition, multiplication, and exponentiation for $n$-ary relational systems. $n$-ary ordered sets are defined as special $n$-ary relational systems by means of properties that seem to suitably generalize reflexivity, antisymmetry, and transitivity from the case of $n=2$ or 3. The class of $n$-ary ordered sets is then closed under the cardinal and ordinal operations.

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence ${\varphi }_{ℳ}$ if ${\varphi }_{ℳ}$ has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if ${\aleph }_{\beta }$ is characterized by a Scott sentence, then $2^{{\aleph }_{\beta +\beta ₁}}$ is (homogeneously) characterized by a Scott sentence, for all 0 < β₁ < ω₁....

It is well-known that compacta (i.e. compact Hausdorff spaces) are maximally resolvable, that is every compactum $X$ contains $\Delta \left(X\right)$ many pairwise disjoint dense subsets, where $\Delta \left(X\right)$ denotes the minimum size of a non-empty open set in $X$. The aim of this note is to prove the following analogous result: Every compactum $X$ contains ${\Delta }_{\delta }\left(X\right)$ many pairwise disjoint $G_{\delta }$-dense subsets, where ${\Delta }_{\delta }\left(X\right)$ denotes the minimum size of a non-empty $G_{\delta }$ set in $X$.