Rigid homogeneous chains

摘要: Classifying (unordered) sets by the elementary (first order) properties of their automorphism groups was undertaken in (7), (9) and (11). For example, if Ω is a set whose group, S (Ω), satisfies then has cardinality at most ℵ 0 conversely (see (7)). We are interested classifying homogeneous totally ordered (homogeneous chains, for short) groups. (Note that we use ‘homogeneous’ here to mean group transitive.) This study begun (4) (5). any Ω, (Ω) primitive (i.e. no congruences). However, chain need not be o -primitive it may have convex Fortunately, ‘ -primitive’ property can captured first order sentence automorphisms chains. Hence our general problem falls naturally into two parts. The classify chains -primitive; second determine how components related arbitrary elementarily equivalent.