On Rigid Chains Admitting Many Embeddings

J. K. Truss, University of Leeds

A chain is a linearly ordered set. A structure is said to be rigid if its only automorphism is the identity. It is possible for a rigid structure to admit many embeddings to itself (which will of course, not be onto), an easy example being any infinite ordinal as a chain. A stronger property, which (for chains with more than one element) implies that the ordering is dense without endpoints, is that any finite partial automorphism should extend to an embedding. When this occurs, we say that the chain admits many embeddings.

Since the theory of dense linear orderings without endpoints is aleph-0-categorical, and clearly the rationals as a chain is far from rigid, any (non-trivial) rigid chain admitting many embeddings is uncountable. I shall present two methods for constructing such chains, each of which is a modification of a standard construction of rigid chains. The first adapts a classical 'diagonal' construction by Dushnik and Miller of a rigid subchain of the reals, whereby each possible non-trivial automorphism is destroyed in turn. The extra twist here, to ensure that there are many embeddings, uses a simple argument involving Baire category, and some Cantor sets.

The second construction, which applies in any regular uncountable cardinality kappa, involves 'coding' each point of the desired chain by a stationary set in the chain having that point as limit. If we use Solovay's result that any stationary set can be written as the disjoint union of kappa pairwise disjoint stationary sets, then there are enough 'codes' available. The proof therefore consists in determining how to keep track of performing all the encoding, together with the need to ensure that there are nevertheless 'many' embeddings, in the sense described above.