Submission date: 8 May 2010.

Abstract:

We observe simple links between preorders, semi-groups, rings and categories (and between equivalence relations, groups, fields and groupoids), which are infinitely 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 category defined by a pure n-type in a small structure is the conjunction of definable categories. For a group G_A defined by an n-type over some arbitrary set A in a small and simple structure, we deduce that
1) if G_A is included in some definable set such that boundedly many translates of G_A cover X, then G_A is the conjunction of definable groups.
2) for any finite tuple g in G_A, there is a definable group containing g.

Mathematics Subject Classification: 03C45, 03C60, 20L05, 20M99.

Keywords and phrases: Small theory, simple theory, infinitely definable, semi-groups, rings and categories, field, ring, preodred, equivalence relation, category groupoid.

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