New arithmetic laws for order types

Abstract: Let (LO, +) denote the class of linear orders equipped with the operation of ordered sum (i.e. concatenation). Despite the enormous diversity of linear order types, arithmetic in (LO, +) is surprisingly nice in certain respects: Lindenbaum showed that (LO, +) satisfies a completely general Euclidean division theorem, and Aronszajn found an elegant structural characterization of the commuting pairs in (LO, +). Yet although these theorems generalize basic facts about sums of natural numbers, the published proofs are somewhat difficult and ad hoc.

In recent work with Eric Paul, we develop a systematic approach to the arithmetic of (LO, +) by adapting a structure theory for group actions on linear orders due to McCleary and others. Using this approach, we give new, unified proofs of Lindenbaum’s and Aronszajn’s theorems. We then generalize this approach to semigroups acting by convex embeddings on linear orders, obtain an arithmetic characterization of commutativity in (LO, +), and determine exactly the commutative semigroups that can be represented in (LO, +). I will give an overview of our work, outline some of the proofs, and discuss future directions.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic