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 θ-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 θ-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 ḡ in G A , there is a definable group containing ḡ. (shrink)

We investigate some topological properties of the spaces of classes of bounded type-definable equivalence relations.

§1. Introduction. In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others.The class of simple theories includes stable theories, but also many more, such as the theory of the random graph. Moreover, many of the theories of particular (...) algebraic structures which have been studied recently turn out to be simple. The interest is basically that a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. Stable theories will be defined formally in the next section. An exhaustive study of them is carried out in [33]. Without trying to read Shelah's mind, we feel comfortable in saying that the importance of stability for Shelah lay partly in the fact that an unstable theory T has 2λ many models in any cardinal λ ≥ ω1 + |T|. (shrink)

Let T be a countable, small simple theory. In this paper, we prove that for such T, the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.

Assume that G is a definable group in a stable structure M. Newelski showed that the semigroup SG of complete types concentrated on G is an inverse limit of the ∞-definable semigroups SG,Δ. He also showed that it is strongly π-regular: for every p∈SG,Δ, there exists n∈N such that pn is in a subgroup of SG,Δ. We show that SG,Δ is in fact an intersection of definable semigroups, so SG is an inverse limit of definable semigroups, and that the latter (...) property is enjoyed by all ∞-definable semigroups in stable structures. (shrink)