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 common points between stable structures and weakly small structures and define a structureMto befineif the Cantor-Bendixson rank of the topological space${S_\varphi }\left} \right)$is an ordinal for every finite subsetAofMand every formula$\varphi \left$wherexis of arity 1. By definition, a theory isfineif all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subsetAof a fine groupG, the traces on the algebraic closure$acl\left$ofAof definable subgroups ofGover$acl\left$which are boolean combinations of (...) instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroups of finite index, nor Artin-Schreier extensions. (shrink)