Monotone infinitary operations on ordinals (extended version)

TL;DR

This paper introduces a new omega-ary monotone operation on ordinals, compares it with Hessenberg sums, and explores its order-theoretic and combinatorial properties.

Contribution

It defines a novel infinitary ordinal operation, analyzes its monotonicity, and provides order-theoretic and game-theoretic characterizations, extending understanding of ordinal sums.

Findings

The operation is strictly monotone in many cases.

It is comparable to Hessenberg natural sum and its generalizations.

Order-theoretic and combinatorial characterizations are established.

Abstract

We define and study an $ \omega $-ary operation on the class of the ordinals, which is strictly monotone in many significant cases (by an elementary argument, there is no fully strictly monotone infinitary operation on ordinals). We compare the operation with the finitary Hessenberg natural sum, which is the smallest finitary strictly monotone operation on each argument. We also compare it with other infinitary generalizations of Hessenberg sum. We provide order-theoretical characterizations of our operation, both as the rank of sequences in an appropriate well-founded order, and as a mixed (or shuffled) sum of the ordinals in the sequence. The latter means that such an infinitary sum is the largest realization as an order-preserving disjoint union of copies of the summands, under some boundedness restriction. The former characterization can be recast in terms of combinatorial games, leading to the problem whether the operation can be extended to the class of Conway surreal numbers.