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 GA(where the set A may 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.

Any preorder P on a set X has an associated preorder P′, P″, P‴, … The proerties of this sequence are studied. When X is finite the sequence is eventually periodic with period P = 1 or p = 1, the eventual constant preorder is full p = 2 the possible forms which the eventual alternating order can take are examined: first, the possible combinations of components are enumerated; second, the notion of ramification at a caste is used to show that X may in a heuristic sense be of unbounded complexity. If X is orderdense the periodicity starts at P′.