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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.

Hereditarily Odd–Even and combinatorial isols