Autostability of Automatic Presentations of Well Orders and Low Rank Linear Orders

The proposition that if $L$ is linear order then its $FC$-rank is finite was proved in Rubin's thesis (S. Rubin, Automatic Structures, A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy, The University of Auckland, 2004), Automatic Structures. We showed that every two automatic presentation of ordinal or linear order with $FC$-rank less then 2 are computable isomorphic. The example of automatic linear order with complicated structure was provided. Moreover it was showed that every automatic linear order is definable in appropriate automatic linear order with $FC$-rank 1 by the first order fomula with additional quantifier $\exists^\infty$.