In  T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ1-elementary substructure. We gave a detailed ordinal arithmetical analysis (see ) of the ordinal structure based on Σ1-elementarity as defined in . This involved the development of an appropriate ordinal arithmetic that is based on a system of classical ordinal notations derived from Skolem hull operators, see . In the present paper we establish an effective order isomorphism between the classical and the new system of ordinal notations using the results from  and . Moreover, on the basis of a concept of relativization we develop mutual (relatively) elementary recursive assignments which are uniform with respect to the underlying relativization.
"Assignment of ordinals to patterns of resemblance." J. Symbolic Logic 72 (2) 704 - 720, June 2007. https://doi.org/10.2178/jsl/1185803630