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SUMMARY:Garrett Ervin (Eötvös Loránd University)
DTSTART:20260312T180000Z
DTEND:20260312T190000Z
DTSTAMP:20260423T021356Z
UID:OLS/195
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OLS/195/">Ne
 w arithmetic laws for order types</a>\nby Garrett Ervin (Eötvös Loránd 
 University) as part of Online logic seminar\n\n\nAbstract\nLet (LO\, +) de
 note the class of linear orders equipped with the operation of ordered sum
  (i.e. concatenation). Despite the enormous diversity of linear order type
 s\, arithmetic in (LO\, +) is surprisingly nice in certain respects: Linde
 nbaum showed that (LO\, +) satisfies a completely general Euclidean divisi
 on theorem\, and Aronszajn found an elegant structural characterization of
  the commuting pairs in (LO\, +). Yet although these theorems generalize b
 asic facts about sums of natural numbers\, the published proofs are somewh
 at difficult and ad hoc. \n\nIn 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. Usin
 g this approach\, we give new\, unified proofs of Lindenbaum’s and Arons
 zajn’s theorems. We then generalize this approach to semigroups acting b
 y convex embeddings on linear orders\, obtain an arithmetic characterizati
 on of commutativity in (LO\, +)\, and determine exactly the commutative se
 migroups that can be represented in (LO\, +). I will give an overview of o
 ur work\, outline some of the proofs\, and discuss future directions.\n
LOCATION:https://researchseminars.org/talk/OLS/195/
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