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In what follows, I will limit myself to those contexts in which interpretations are 'underdetermined' at best only up to isomorphism.
Since I am (artificially) restricting myself to those cases in which the 'underdetermination' of referenc is at best up to isomorphism, non-isomorphic models on the same domain should b assumed to separate.
It is interesting to establish which groups, up to isomorphism, are the full automorphism group of a combinatorial design of a given type.
In [13] the last three authors determined a formula enumerating all 1-rotational HCS(2n + 1) up to isomorphism.