Continuous functions on essential P-spaces: A model-theoretic analysis of some non-projectable lattice-ordered groups.

摘要

In the 1980's, Weispfenning studied the model theory of projectable l-groups [22]. In particular, he classified several collections of projectable l-groups up to elementary equivalence. In this thesis, we prove analogous results for a special class of non-projectable l-groups.; The set of all continuous functions on a topological space is an l-group under the usual pointwise operations. If the underlying space is completely regular, then the associated l-group is projectable just in case the space is basically disconnected. As a first step we investigate l-groups of continuous functions on P-spaces, a special subclass of the class of all basically disconnected spaces, and indicate their relevance to Weispfenning's work on projectable l-groups.; Motivated by our understanding of P-spaces we next look at a broader class of spaces called essential P-spaces, which are not necessarily basically disconnected. After pointing out some general facts, we restrict our attention to a special subclass PEPℵ0 of the class of all essential P-spaces. Results of Shen-Weispfenning [17] and Flothow [5] are used to show that model-theoretic information about l-groups of continuous functions on PEPℵ0 -spaces is completely determined by model-theoretic information about the zero-set lattices of such spaces. It is then shown that model-theoretic information about such lattices is completely determined by certain Boolean algebras with distinguished ideal. Finally, using work of Touraille [ 19], we obtain invariants that completely classify l-groups of continuous functions on PEPℵ0 -spaces up to elementary equivalence.