Submission date: 20 December 2006. Revised on 7 September 2008.

Abstract:

We consider structures A consisting of an abelian group with a subgroup A(P) distinguished by a 1-ary relation symbol P, and complete theories T of such structures. Such a theory T is (k,m)-categorical if T has models A of cardinality m with A(P) of cardinality k, and given any two such models A, B with A(P) = B(P), there is an isomorphism from A to B which is the identity on A(P). We state all true theorems of the form: If T is (k,m)-categorical then T is (k',m')-categorical. We classify the A of finite order m with A(P) of order k which are (k,m)-categorical.

Mathematics Subject Classification: 03C35, 20K01, 20K35

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Full text: pdf (Version of 20 December 2006: pdf).