Several problems that pertain to certain arithmetically well-behaved countable subsemirings of Λ, the semiring of isols, are discussed. This is relevant to the present volume memorializing the late John Myhill, in that Myhill was an early co-developer of the theory of Λ.

In this paper we discuss the following contributions to recursion theory made by John Myhill: two sets are recursively isomorphic iff they are one-one equivalent; two sets are recursively isomorphic iff they are recursively equivalent and their complements are also recursively equivalent; every two creative sets are recursively isomorphic; the recursive analogue of the Cantor–Bernstein theorem; the notion of a combinatorial function and its use in the theory of recursive equivalence types.

A universal Horn sentence in the language of polynomial-time computable combinatorial functions of natural numbers is true for the natural numbers if, and only if, it is true for PETs of p-time p-isolated sets with functions induced by fully p-time combinatorial operators.