Lattices in generative classes

Abstract

We define and study lattices in generative classes associated with generic structures. It is shown that these lattices can be non-distributive and, moreover, arbitrary enough. Heights and wights of the lattices are described. A model-theoretic criterion for the linear ordering is proved and these linear orders are described. Boolean algebras generated by the considered lattices are also described.

Topological models, computer science semantics and applications of logic in AI

“…Using these calculi, in Section 5, we introduce the notion of meeting of cardinality contradiction and, for given generative class, characterize the existence of a generic structure. In section 6 [34] we define and study lattices in generative classes associated with generic structures. It is shown that these lattices can be non-distributive and, moreover, sufficiently arbitrary enough.…”