A survey of clones on infinite sets

A clone on a set X is a set of finitary operations on X which contains all projections and which is moreover closed under functional composition. Ordering all clones on X by inclusion, one obtains a complete algebraic lattice, called the clone lattice. We summarize what we know about the clone lattice on an infinite base set X and formulate what we consider the most important open problems.Comment: 37 page

Existence of optimal ultrafilters and the fundamental complexity of simple theories

In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the global theory, and introducing nonforking, seminal for the development of stability theory. Prior to the present paper, there had been no such characterization of an unstable class. In the present paper, we first establish the existence of so-called optimal ultrafilters on Boolean algebras, which are to simple theories as Keisler's good ultrafilters are to all theories. Then, assuming a supercompact cardinal, we characterize the simple theories in terms of saturation of ultrapowers. To do so, we lay the groundwork for analyzing the global structure of simple theories, in ZFC, via complexity of certain amalgamation patterns. This brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability.Comment: The revisions aim to separate the set theoretic and model theoretic aspects of the paper to make it accessible to readers interested primarily in one side. We thank the anonymous referee for many thoughtful comment

The number of countable models via Algebraic logic

Vaught's Conjecture states that if T is a complete First order theory in a countable language that has more than aleph_0 pairwise non isomorphic countable models, then T has 2^aleph_0 such models. Morley showed that if T has more than aleph_1 pairwise non isomorphic countable models, then it has 2^aleph_0 such models. In this paper, we First show how we can use algebraic logic, namely the representation theory of cylindric and quasi-polyadic algebras, to study Vaught's conjecture (count models), and we re-prove Morley's above mentioned theorem. Second, we show that Morley's theorem holds for the number of non isomorphic countable models omitting a countable family of types. We go further by giving examples showing that although this number can only take the values given by Morley's theorem, it can be different from the number of all non isomorphic countable models. Moreover, our examples show that the number of countable models omitting a family of types can also be either aleph_1 or 2 and therefore different from the possible values provided by Vaught's conjecture and by his well known theorem; in the case of aleph_1, however, the family is uncountable. Finally, we discuss an omitting types theorem of Shelah