$\Sigma$-Admissible Families over Linear Orders

Admissible sets of the form $\operatorname{HYP}(\mathfrak M)$, where $\mathfrak M$ is a recursively saturated system, are treated. We provide descriptions of subsets $\mathfrak M$, which are $\Sigma_*$-sets in $\operatorname{HYP}(\mathfrak M)$, and of families of subsets $\mathfrak M$, which form $\Sigma$-regular families in $\operatorname{HYP}(\mathfrak M)$, in terms of the concept of being fundamental couched in the article. Fundamental subsets and families are characterized for models of dense linear orderings.