Generalized sums of linear orders

TL;DR

This paper explores generalized associative and commutative sums of linear orders, introducing sum-generating classes and complex classes to understand their algebraic and structural properties, including new sums on ordinals.

Contribution

It introduces sum-generating classes and complex classes of linear orders, expanding the understanding of associative and commutative sums beyond traditional definitions.

Findings

Existence of diverse associative sums different from the usual sum

Canonical decompositions of linear orders via sum-generating classes

Characterization of associative and commutative sums on ordinals

Abstract

We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that there is a rich array of associative sums different from the usual sum $+$ and its dual. The simplest of these sums arise from what we call sum-generating classes of linear orders. These classes determine canonical decompositions of every linear order into left and right halves. We study the structural and algebraic properties of these classes along with the sums they generate. We then turn our attention to commutative sums on various subclasses of the linear orders. For this, we introduce the notion of a complicated class of linear orders and show that over such classes sums can be constructed in a very flexible way. Using this construction, we prove the existence of associative sums lacking the structural properties of the usual sum. Along the way, we characterize the associative and commutative sums on the ordinals.