Our paper contains three
theorems on regressive isols that are hereditarily odd-even. Two are characterizations
of hereditarily odd-even isols in terms of a parity property of the isol and a property
on the comparability of summands of the isol. In the third theorem, we show that if a
regressive isol has a special comparability of summands property, then it has a
predecessor that is hereditarily odd-even.