\Lambda \Lambda

this paper we are interested in those structures in which the basic computations can be performed by Turing machines.

This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC-509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995

2.1 Notation from model theory................... 4 2.2 F

this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25

We give a straightforward computable-model-theoretic definition of a property of ∆0 2 sets, called order-computability. We then prove various results about these sets which suggest that, simple though the definition is, the property defies any characterization in pure computability theory. The most striking example is the construction of two computably isomorphic c.e. sets, one of which is order-computable and the other not. 1

We construct a strongly minimal (and thus uncountably categorical) but not totally categorical theory in a finite language of binary predicates whose only constructive (or recursive) model is the prime model.

Abstract. Two properties of the Co-Spectrum of the Joint Spectrum of finitely many abstract structures are presented- a Minimal Pair type theorem and the existence of a Quasi-Minimal degree with respect to the Joint Spectrum of the structures. 1

Abstract. The Turing degree spectrum of a countable structure A is the set of all Turing degrees of isomorphic copies of A. The Turing degree of the isomorphism type of A, if it exists, is the least Turing degree in its degree spectrum. We show there are countable fields, rings, and torsion-free abelian groups of arbitrary rank, whose isomorphism types have arbitrary Turing degrees. We also show that there are structures in each of these classes whose isomorphism types do not have Turing degrees. 1.

A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model A, i.e., the elementary diagram D e (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree. 1