Rigid chains admitting many embeddings

Abstract

Abstract. A chain (linearly ordered set) is rigid if it has no non-trivial automorphisms. The construction of dense rigid chains was carried out by Dushnik and Miller for subsets of R, and there is a rather different construction of dense rigid chains of cardinality κ, an uncountable regular cardinal, using stationary sets as 'codes', which was adapted by Droste to show the existence of rigid measurable spaces. Here we examine the possibility that, nevertheless, there could be many order-embeddings of the chain, in the sense that the whole chain can be embedded into any interval. In the case of subsets of R, an argument involving Baire category is used to modify the original one. For uncountable regular cardinals, a more complicated version of the corresponding argument is used, in which the stationary sets are replaced by sequences of stationary sets, and the chain is built up using a tree. The construction is also adapted to the case of singular cardinals.

Surjectively rigid chains

“…We return to the situation, but now at higher cardinalities. In [2], it was shown how to construct dense chains in any uncountable cardinality κ which are rigid but admit many embeddings. These ideas can be adapted to consider epimorphisms too.…”

“…These ideas can be adapted to consider epimorphisms too. We give two constructions of , as in [2], first the basic one which is rigid, and demonstrate that it is also epimorphism‐rigid, and the other which is still epimorphism‐rigid, but admits embeddings into every interval (so is far from embedding‐rigid). The chains are exactly the same as before, but for completeness we give an outline of their construction, concentrating on establishing epimorphism‐rigidity.…”

“…All other points arise at some intermediate stage, and since they do not lie in any copy of , must have the form for some ; these have cofinality , which they also retain in X . It was shown in [2] that X is (automorphism‐)rigid. We adapt that argument to show that it is also epimorphism‐rigid.…”

“…It is constructed by transfinite induction using the fact that the number of ‘potential’ order‐automorphisms is . In fact the chain that they construct has the stronger property that it has no non‐trivial order‐embeddings, and in [2], it was shown that one can construct dense subchains of which are rigid, but which nevertheless admit many embeddings.…”

“…§ 4 gives two results which hold in models in which is false, using the Fraenkel‐Mostowski method. In the final section we adapt the other part of [2], which treats dense chains of larger cardinalities, and we show that the same techniques, involving stationary sets, can be used to give parallel results for epimorphisms in this context. Since the models are just as in the earlier paper, we give fewer details, but concentrate on proving the additional rigidity properties asserted.…”