Siblings of Direct Sums of Chains

TL;DR

This paper proves Thomassé's conjecture for countable direct sums of chains, showing they have a limited number of siblings, and extends results to direct sums of any cardinality, revealing their sibling structure.

Contribution

It establishes the number of siblings for countable and uncountable direct sums of chains, confirming Thomassé's conjecture in the countable case and generalizing to larger cardinalities.

Findings

Countable direct sums of chains have 1, countably many, or continuum many siblings.

Direct sums of any cardinality have either one or infinitely many siblings.

The results confirm Thomassé's conjecture for countable structures.

Abstract

We prove that a countable direct sum of chains has either one, countably many or else continuum many isomorphism classes of siblings. This proves Thomass\'e's conjecture for such structures. Further, we show that a direct sum of chains of any cardinality has one or infinitely many siblings, up to isomorphism.