On enveloping type-definable structures

Cédric Milliet (1)

1 ICJ - Institut Camille Jordan

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 0-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 0-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 g in G_A, there is a definable group containing g.

Mots clés: Small theory; Simple theory; Type-definable group; Semi-group; Field; ring; Preorder; Equivalence relation; Category; Groupoid; Envelope

Domaines: Logique [math.LO]

Cédric Milliet. On enveloping type-definable structures. The Journal of Symbolic Logic, 2011, 76 (3), pp.1023-1034. ⟨10.2178/jsl/1309952532⟩. ⟨hal-00582389⟩