Quasi-finitely characterizable and finitely characterizable Dedekind algebras
George E. Weaver, Benjamin R. George · 2002
A Dedekind algebra is an ordered pair (B,h) where h is a similarity transformation on the non-empty set B. Among these is the sequence of the positive integers. An algebra is flnitely characterizable provided some flnite subset of its second-order theory is categorical. Dedekind showed that the sequence of the positive integers is flnitely characterizable. Here a condition is presented that is both necessary and su‐cient for a Dedekind algebra to be flnitely characterizable. It is also shown that a weaker condition is both necessary and su‐cient for a Dedekind algebra to be quasi-flnitely characterizable (i.e. that some flnite subset of its second-order theory is categorical in all powers). Finally, it is shown that there is a number theoretic condition both necessary and su‐cient for a countably inflnite Dedekind algebra to be flnitely characterizable. Classiflcation Numbers: 03B15, 03C85