Ideals in isols containing a set of regressive isols (Q1175483)

Let \(\mathbb{N}\) denote the set of natural numbers, \(\Lambda\), \(\Lambda_ R\) be the sets of isols and regressive isols, respectively. Let \(\mathcal K\) denote the family of all almost combinatorial recursive functions, \(\mathcal R\) the family of all recursive relations, and let \[ {\mathcal K}^{(n)}=\{f\in{\mathcal K}\mid f: \mathbb{N}^ n\to\mathbb{N}\},\qquad{\mathcal R}^{(n)}=\{R\in{\mathcal R}\mid R\subseteq\mathbb{N}^ n\}. \] A set \(\Gamma\subseteq\Lambda\) is called an ideal if 1) \(A,B\in\Gamma\to A+B\in\Gamma\); 2) \(A\subseteq B \& B\in\Gamma\to A\in\Gamma\); and a \(\mathcal K\)-ideal if in addition 3) \(A\in\Gamma^ n \& f\in{\mathcal K}^{(n)}\to f_ \Lambda(A)\in\Gamma\). Let \(\Lambda_ \omega=\{Y\mid \exists n\exists X\in\Lambda^ n_ R\) \((Y=X_ 0+\cdots+X_{n-1})\}\). Theorem 1. \(\Lambda_ \omega\) is the least ideal containing \(\Lambda_ R\). But \(\Lambda_ \omega\) is no \(\mathcal K\)-ideal. The systems \({\mathcal N}(\Lambda_ \Sigma,{\mathcal K},{\mathcal R})\) and \({\mathcal N}(\Lambda_ p,{\mathcal K},{\mathcal R})\) are defined and the following results proved. Theorem 3. The system \({\mathcal N}(\Lambda_ \Sigma,{\mathcal K},{\mathcal R})\) is a Nerode extension. Theorem 5. The system \({\mathcal N}(\Lambda_ p,{\mathcal K},{\mathcal R})\) is a total Nerode extension.