Intermediate arithmetic operations on ordinal numbers

There are two well-known ways of doing arithmetic with ordinal numbers: the "ordinary" addition, multiplication, and exponentiation, which are defined by transfinite iteration; and the "natural" (or "Hessenberg") addition and multiplication (denoted circle plus and circle times), each satisfying its own set of algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of multiplying ordinals (denoted x), defined by transfinite iteration of natural addition, as well as the notion of exponentiation defined by transfinite iteration of his multiplication, which we denote alpha(x beta). (Jacobsthal's multiplication was later rediscovered by Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this paper, we pick up where Jacobsthal left off by considering the notion of exponentiation obtained by transfinitely iterating natural multiplication instead; we shall denote this alpha(circle times beta). We show that alpha(circle times(beta circle plus gamma)) = (alpha(circle times beta)) circle times (alpha(circle times gamma)) and that alpha(circle times(beta x gamma)) = (alpha(circle times beta))(circle times gamma) ; note the use of Jacobsthal's multiplication in the latter. We also demonstrate the impossibility of defining a "natural exponentiation" satisfying reasonable algebraic laws. (C) 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim