On enveloping type-definable structures

Abstract

We observe simple links between equivalence relations, groups, fields and groupoids (and between preorders, semi-groups, rings and categories), which are type-definable in an arbitrary structure, and apply these observations to the particular context of small and simple structures. Recall that a structure is small if it has countably many n-types with no parameters for each natural number n. We show that a ∅-type-definable group in a small structure is the conjunction of definable groups, and extend the result to semi-groups, fields, rings, categories, groupoids and preorders which are ∅-type-definable in a small structure. For an A-type-definable group G_A (where the set A may be infinite) in a small and simple structure, we deduce that

1. if G_A is included in some definable set X such that boundedly many translates of G_A cover X, then G_A is the conjunction of definable groups.

2. for any finite tuple \bar{g} in G_A, there is a definable group containing \bar{g}.