Identities and indiscernibility (Q1095894)

A structure \({\mathfrak A}\) is indiscernible in a theory T if there exists a model \({\mathfrak B}\) of T and a one-to-one mapping \(f: A\to B\) such that for all \(n\in \omega\) and all pairs of sequences \(<a_ 1,...,a_ n>\) and \(<a_ 1',...,a_ n'>\) in \(A^ n:\) if they satisfy the same quantifier- free formulas in \({\mathfrak A}\) then \(<f(a_ 1),...,f(a_ n)>\) and \(<f(a_ 1'),...,f(a_ n'>\) satisfy the same type in \({\mathfrak B}\). Then we define a hierarchy between complete theories as follows: \(T_ 1\leq T_ 2\) iff for all structures \({\mathfrak A}:\) if \({\mathfrak A}\) is indiscernible in \(T_ 1\) then \({\mathfrak A}\) is indiscernible in \(T_ 2.\) We use Shelah's identities to investigate the above defined relation between complete theories. First we prove some facts about identities and connections with indiscernibility. This allows us to translate problems about indiscernibility to problems about identities. Using this method we prove among others that increasing and decreasing chains have a least upper bound and a greatest lower bound, respectively.