Surjectively rigid chains

Professor John Truss, University of Leeds. Part of the Logic Seminar Series.

A structure is said to be rigid if its only automorphism is the identity map. Similar definitions apply to embeddings, epimorphisms, and endomorphisms. We study rigidity properties of linearly ordered sets (chains) in these cases. A classical result of Dushnik and Miller provides a dense subchain of the real numbers which is (automorphism-) rigid. We modify this to give other examples of dense subchains of the real numbers. One of these for instance is epimorphism-rigid but admits non-identity embeddings. A completely different method is used to construct dense chains of uncountable (regular) cardinalities, using stationary sets as 'codes' for points to prevent there being non-identity automorphisms (or epimorphisms). The (easy) proof that if there is a non-identity epimorphism there is also a non-identity embedding requires blatant appeal to the axiom of choice, but if we drop AC, this may be false: a Fraenkel--Mostowski model is given which contains an embedding-rigid dense chain which however admits a non-identity epimorphism.

Some of this work is joint with Professor Dr Manfred Droste and Mayra Ballesteros-Montalvo.