# The Arithmetic of Linear Orders

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There are two natural arithmetic operations on the class of linear orders: the sum + and lexicographic product x. These operations generalize the sum and product of ordinals.

The arithmetic laws obeyed by the sum were uncovered in the pre-forcing days of set theory and are surprisingly nice. For example, while the left cancellation law A + X \cong B + X => A \cong B is not true in general, its failure can be completely characterized: a linear order X fails to cancel in some such isomorphism if and only if there is a non-empty order R such that R + X \cong X. Right cancellation is symmetrically characterized.

Tarski and Aronszajn characterized the commuting pairs of linear orders, i.e. the pairs X and Y such that X + Y \cong Y + X.

Lindenbaum showed that X + X \cong Y + Y implies X \cong Y for linear orders X and Y. More generally, the finite cancellation law nX \cong nY => X \cong Y holds. Lindenbaum showed that the sum even satisfies the Euclidean algorithm!

On the other hand, the arithmetic of the lexicographic product is much less well understood. The lone totally general classical result is due to Morel, who characterized when the right cancellation law A x X \cong B x X => A \cong B holds. Morel showed that an order X fails to cancel in some such isomorphism if and only if there is a non-singleton order R such that R x X \cong X, in analogy with the additive case.

In this talk we focus on the question of whether Morel’s cancellation theorem is true on the left. We’ll show that, while the literal left-sided version of Morel’s theorem is false, an appropriately reformulated version is true. Our results suggest that a complete characterization of left cancellation in lexicographic products is possible. We’ll also discuss how our work might help in proving multiplicative versions of Tarski’s, Aronszajn’s, and Lindenbaum’s additive laws.

This is joint work with Eric Paul.