Hereditarily odd-even and combinatorial isols

In this paper we study some of the arithmetic structure that is found in a special kind of semi-ring in the isols. These are the semi-rings [D (Y), +, .] that were introduced by J. C. E. Dekker, and that were later shown by E. Ellentuck to model the true universal recursive statements of arithmetic when Y is a regressive isol and is hyper-torre ( = hereditarily odd-even = HOE). When Y is regressive and HOE, we further reflect on the structure of D (Y). In addition, a new variety of regressive isol is introduced, called combinatorial. When Y is such an isol, then it is also HOE, and more, and the arithmetic of D (Y) is shown to have a richer structure.