On enveloping type-definable structures

We observe simple links between preorders, semi-groups, rings and categories (and between equivalence relations, groups, fields and groupoids), which are type-definable 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 ∅-type-definable group of finite arity in a small structure is the conjunction of definable groups. A ∅-type-definable category of finite arity in a small structure is the conjunction of definable categories. For an A-type-definable group GA of finite arity (where the set A maybe be infinite) in a small and simple structure, we deduce that 1) if GA is included in some definable set X such that boundedly many translates of GA cover X, then GA is the conjunction of definable groups. 2) for any finite tuple ḡ in GA, there is a definable group containing ḡ. In a universe M, a A-type-definable set, instead of being defined by a formula, is the conjunction of infinitely many formulae with parameters in some set A. A A-typedefinable structure in M is any structure whose domain, functions and relations are A-type-definable in M. Definition. Let S be a class of structures, and A an element of S which is typedefinable in M. We say that M loosely envelopes A with respect to S if A is contained in some definable structure belonging to S. We say that M envelopes A with respect to S if A is the conjunction of definable structures in S. In the sequel, the class S will consist either of groups, semi-groups, fields, rings, preorders, equivalence relations, categories or groupoids and will be obvious from the context. For instance, we shall say that a structure envelopes a type-definable group G to say that G is the conjunction of definable groups. Note that being enveloped is strictly stronger that being loosely enveloped. A stable structure is known to envelope type-definable groups and fields of finite arities [2, Hrushovski]. Consequently, in an omega-stable structure, a type-definable group of finite arity is definable, as is a type-definable field of finite arity in a superstable structure. Pillay and Poizat proved that a ∅-type-definable equivalence relation on a small structure is enveloped, provided that it be coarser than the equality of pure 1-types [9]. Kim generalised Pillay and Poizat’s result to arbitrary ∅-type-definable equivalence relations on a small structure [4]. In [11], Wagner deduces from Kim’s result that if a small structure loosely envelopes a ∅-type-definable group of finite 2000 Mathematics Subject Classification. 03C45, 03C60, 20L05, 20M99.