Intermediate arithmetic operations on ordinal numbers

TL;DR

This paper explores a new form of ordinal exponentiation obtained by transfinitely iterating natural multiplication, analyzing its algebraic properties and limitations compared to existing ordinal operations.

Contribution

It introduces and studies a novel ordinal exponentiation based on natural multiplication, extending Jacobsthal's work and establishing its algebraic laws and limitations.

Findings

Proves distributive-like laws for the new exponentiation.

Shows the algebraic laws for natural exponentiation cannot be fully satisfied.

Extends Jacobsthal's framework with new transfinite operations.

Abstract

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 $\oplus$ and $\otimes$), each satisfying its own set of algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of multiplying ordinals (denoted $\times$), defined by transfinite iteration of natural addition, as well as the notion of exponentiation defined by transfinite iteration of his multiplication, which we denote $\alpha^{\times\beta}$. (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 will denote this $\alpha^{\otimes\beta}$. We show that $\alpha^{\otimes(\beta\oplus\gamma)} = (\alpha^{\otimes\beta}) \otimes(\alpha^{\otimes\gamma})$ and that $\alpha^{\otimes(\beta\times\gamma)}=(\alpha^{\otimes\beta})^{\otimes\gamma}$; note the use of Jacobsthal's multiplication in the latter. We also demonstrate the impossibility of defining a "natural exponentiation" satisfying reasonable algebraic laws.