A fine structure in the theory of isols

In this paper we introduce a collection of isols having some interesting properties. Imagine a collection W of regressive isols with the following features: (1) u, v is an element of W implies that u less than or equal to v or v less than or equal to u, (2) u less than or equal to v and v is an element of W imply u is an element of W, (3) W contains N = {0, 1, 2,...} and some infinite isols, and (4) u is an element of W, u infinite, and u + v regressive imply u + v is an element of W. That such a collection Mr exists is proved in our paper. It has many nice features. It also satisfies (5) u, v is an element of W, u less than or equal to v and u infinite imply v less than or equal to g(Lambda)(u) for some recursive combinatorial function g, and (6) each u is an element of W is hereditarily odd-even and is hereditarily recursively strongly torre. The collection W that we obtain may be characterized in terms of a semi-ring of isols D(c) introduced by J. C. E. Dekker in [5]. We will show that W = D(c), where c is an infinite regressive isol that is called completely torre.