Uncountable cofinalities of automorphism groups of linear and partial orders

Uncountable cofinalities of automorphism groups of linear and partial orders

We demonstrate the uncountable cofinality of the automorphism groups of various linear and partial orders. We also relate this to the ‘Bergman’ property, and discuss cases where this may fail even though the cofinality

Automorphism Groups of Partial Orders

1. The automorphism group T(P) of a partial order P is the collection of all order preserving permutations (automorphisms) of P, a subgroup of the symmetric group on P. If P and g are partial orders then P x g becomes a partial order by reverse lexicography: (p, q) < (p', q') if q < q' or q = q' and p < p'. If ƒ is a function whose domain contains the element a, we use af to denote the image of...

Cofinalities of Linear Orders

Uncountable Linear Orders

Two important classes of linear orders that are not σ-scattered are the real types and the Aronszajn types: the real types are those uncountable dense linear orders that are separable; the Aronszajn types are those linear orders that are uncountable and yet have no uncountable suborders that are scattered or real types. This latter class was considered—and proved nonempty—long ago by Aronszajn ...

Automorphism Groups of Countably Categorical Linear Orders are Extremely Amenable

We show that the automorphism groups of countably categorical linear orders are extremely amenable. Using methods of Kechris, Pestov, and Todorcevic, we use this fact to derive a structural Ramsey theorem for certain families of finite ordered structures with finitely many partial equivalence relations with convex classes.