Infinitely Definable Structures in Small Theories

We observe simple links between preorders, semi-groups, rings and categories (and between equivalence relations, groups, elds and groupoids), which are in nitely de nable in an arbitrary structure, and apply these observations to small structures. Recall that a structure is small if it has countably many pure n-types for each integer n. A category de ned by a pure n-type in a small structure is the conjunction of de nable categories. For a group GA de ned by an n-type over some arbitrary set A in a small and simple structure, we deduce that 1) if GA is included in some de nable set such that boundedly many translates of GA cover X, then GA is the conjunction of de nable groups. 2) for any nite tuple g in GA, there is a de nable group containing g. In a universe M, an in nitely A-de nable set, instead of being de ned by a formula, is the conjunction of in nitely many formulae with parameters in some set A. An in nitely A-de nable structure in M is any structure whose domain, functions and relations are in nitely A-de nable in M. De nition. Let L be a language, S a set of L-structures, and A an element of S which is in nitely de nable in M. We say that M loosely envelopes A with respect to S if A is contained in some de nable structure belonging to S. We say that M envelopes A with respect to S if A is the conjunction of de nable structures in S. In the sequel, the set S will consist either of groups, semi-groups, elds, rings, preorders, equivalence relations, categories or groupoids and will be obvious from the context. For instance, we shall say that a structure envelopes an in nitely de nable group G to say that G is the conjunction of de nable groups. Note that being enveloped is strictly stronger that being loosely enveloped. A stable structure is known to envelope in nitely de nable groups and elds [1, Hrushovski]. Consequently, in an omega-stable structure, an in nitely de nable group is de nable, as is an in nitely de nable eld in a superstable structure. Pillay and Poizat proved that an in nitely ∅-de nable equivalence relation on a small structure is enveloped, provided that it be coarser than the equality of pure types [8]. Kim generalised Pillay and Poizat's result to arbitrary in nitely ∅-de nable equivalence relations on a small structure [3]. In [10], Wagner deduces from Kim's result that if a small structure loosely envelopes an in nitely ∅-de nable group of nite arity, it must envelope it. He asked whether an in nitely ∅-de nable group in a small structure should be enveloped [10, Problem 6.1.14]. We shall show 2000 Mathematics Subject Classi cation. 03C45, 03C60, 20L05, 20M99.