Generalized sums of linear orders

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.

💡 Research Summary

The paper investigates binary operations on linear orders that behave like a “sum”: for any two orders A and B the result A⊕B contains a copy of A and a copy of B as disjoint suborders. While the usual concatenation A + B and its dual A +* B satisfy regularity and associativity, they are not coproducts in the category of linear orders and they are not commutative. The authors ask whether other regular, associative sums exist and how they can be characterized.

Two main constructions are introduced. First, a sum‑generating class C ⊆ LO is a collection of orders that determines a canonical decomposition of any linear order X into a longest initial segment X_L belonging to C and the remaining final segment X_R. Using a fixed choice of either + or +* for two “piecewise” operations (denoted ∘ and ⋄), they define \