The Model Theory Of Dedekind Algebras

A Dedekind algebra is an ordered pair (B, h) where B is a nonempty set and h is a "similarity transformation" on B. Among the Dedekind algebras is the sequence of positive integers. Each Dedekind algebra can be decomposed into a family of disjointed, countable subalgebras which are called the configurations of the algebra. There are many isomorphic types of configurations. Each Dedekind algebra is associated with a cardinal value function called the confirmation signature which counts the number of configurations in each isomorphism type occurring in the decomposition of the algebra. Two Dedekind algebras are isomorphic if their configuration signatures are identical. I introduce conditions on configuration signatures that are sufficient for characterizing Dedekind algebras uniquely up to isomorphisms in second order logic. I show Dedekind's characterization of the sequence of positive integers to be a consequence of these more general results, and use configuration signatures to delineate homogeneous, universal and homogeneous-universal Dedekind algebras. These delineations establish various results about these classes of Dedekind algebras including existence and uniqueness.