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.
