The Model Theory Of Dedekind Algebras :: Algebra Mathematics Essays

The Model Theory Of Dedekind AlgebrasABSTRACT A Dedekind algebra is an sighted pair (B, h) where B is a non-empty set and h is a analogy break on B. Among the Dedekind algebras is the sequence of positive integers. separately Dedekind algebra can be decomposed into a family of disjointed, countable subalgebras which ar 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 Dedekinds characterization of the sequence of positive integers to be a consequence of these more general result s, and use configuration signatures to delineate homogeneous, universal and homogeneous-universal Dedekind algebras. These delineations establish non-homogeneous results about these classes of Dedekind algebras including existence and uniqueness. 1. INTRODUCTIONOne 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 fill such a characterization is the sequence of the positive integers. Both Dedekind and Peano provided characterizations of this system in the late 1880s. Dedekinds 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 of course be viewed as a continuation of Dedekinds work, the focus here is different. Rathe r than probe 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. weaverbird 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. CONFIGURATIONSGiven a Dedekind algebra B = (B,hB), AB is the transitive closure of hB.