The Model Theory Of Dedekind Algebras :: Algebra Mathematics Essays

The Model Theory Of Dedekind Algebras ABSTRACT: A Dedekind algebra is an ordered pair (B, h) where B is a non-empty 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 signat ures to delineate homogeneous, universal and homogeneous-universal Dedekind algebras. These delineations establish various results about these classes of Dedekind algebras including existence and uniqueness. 1. INTRODUCTION One of the more striking accomplishments of foundational studies prior to 1930 was the characterization of various mathematical systems uniquely up to isomorphism (see Corcoran [1980]). Among the first systems to receive such a characterization is the sequence of the positive integers. Both Dedekind and Peano provided characterizations of this system in the late 1880's. Dedekind's characterization commenced by considering B, a non-empty set, and h, a "similar transformation" on B (i.e. an injective unary function on B). In deference to Dedekind, the ordered pair B = (B,h) is called a Dedekind algebra. While the study of Dedekind algebras can naturally be viewed as a continuation of Dedekind's work, the focus here is different. Rather than investigating whether a particular Dedekind algebra (the sequence of the positive integers) is characterizable, we proceed by investigating conditions on Dedekind algebras which imply that they are characterizable. In the following we review some of the results obtained in the model theory of Dedekind algebras and discuss some of their consequences. These results are stated without proofs. Weaver [1997a] and [1997b] provide the details of these proofs. Attention is restricted here to the model theory of the second order theories of Dedekind algebras. Weaver [1998] focuses on the model theory of the first order theories of these algebras. 2. CONFIGURATIONS Given a Dedekind algebra B = (B,hB), AB is the transitive closure of hB.