First-order Theories of r-projectable ℓ-groups and Priestley Products

F.Lucas

(A large part of the following belongs to a joint work with D.Gluschankof)

Boolean products where used in the study of first order properties of projectable ℓ-groups We introduce the Priestley products as a generalisation of both the Boolean products, and of the

Hahn products on a root system.

We also introduce the r-projectability property as a generalisation of the projectability property. It is more suitable in the case of Hahn products on a root system, when this root system is neither a chain nor an antichain.

Using these generalisations we obtain results in two different directions.

– A first order classification of a large class of r-projectable divisible ℓ-groups, (containing special valued ones), in terms of a lattice which encodes the maximal supports of the elements of the ℓ-group when carefully embedded in a Hahn product .

– A transfert result of the elementary equivalence for Priestley products on a given Priesley space, and for Priestley powers of totally ordered groups on different Priestley spaces, in terms of the elementary equivalence of two structures involved: on the one hand the totally ordered groups, on the other hand lattices related to the Priestley spaces.

D.Gluschankof, F.Lucas, Hyper-regular lattice-ordered groups, J.S.L. 58 (1993), 1342-1358.

D.Gluschankof, F.Lucas, The representation of ℓ-groups over Priestley spaces. Séminaire Structures Algébriques Ordonnées n49.

F.Lucas , First order theories of subgroups of divisible Hahn products. Séminaire Structures Algébriques Ordonnées n56.

V.Weispfenning, Model theory of abelian l-groups, in Lattice-ordered groups. A.M.W.Glass and W.Charles Holland eds. Kluwer Academic Press. pp41 − 79.

Address: Equipe Algèbre et géométrie (UPRESA 6093), Université d’Angers, 2 bd Lavoisier 49045 Angers Cedex

E-mail: Lucas@univ-angers.fr