Littérature scientifique sur le sujet « Lattice-ordered abelian groups »

Abstract First we give a necessary and sufficient condition for an abelian lattice ordered group to admit an expansion to a Riesz space (or vector lattice). Then we construct a totally ordered abelian group with two non-isomorphic Riesz space structures, thus improving a previous paper where the example was a non-totally ordered lattice ordered abelian group. This answers a question raised by Conrad in 1975. We give also a partial solution to another problem considered in the same paper. Finally, we apply our results to MV-algebras and Riesz MV-algebras.

Abstract We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation.

It is a well-known fact that the notion of an archimedean order cannot be formalized in the first-order calculus. In [12] and [18], A. Robinson and E. Zakon characterized the elementary class generated by all the archimedean, totally-ordered abelian groups (o-groups) in the language 〈+,<〉, calling it the class of regularly ordered or generalized archimedean abelian groups. Since difference (−) and 0 are definable in that language, it is immediate that in the expanded language 〈 +, −, 0, < 〉 the definable expansion of the class of regular groups is also the elementary class generated by the archimedean ones. In the more general context of lattice-ordered groups (l-groups), the notion of being archimedean splits into two different notions: a strong one (being hyperarchimedean) and a weak one (being archimedean). Using the representation theorem of K. Keimel for hyperarchimedean l-groups, we extend in this paper the Robinson and Zakon characterization to the elementary class generated by the prime-projectable, hyperarchimedean l-groups. This characterization is also extended here to the elementary class generated by the prime-projectable and projectable archimedean l-groups (including all complete l-groups). Finally, transferring a result of A. Touraille on the model theory of Boolean algebras with distinguished ideals, we give the classification up to elementary equivalence of the characterized class.We recall that a lattice-ordered group, l-group for short, is a structure

2014 - 2015
This thesis is a contribution to the research program ‘toposes as bridges’ introduced in [12], which aims at developing the unifying potential of the notion of Grothendieck topos as a means for relating different mathematical theories to each other through topos-theoretic invariants. The general methodology outlined therein is applied here to study already existing categorical equivalences of particular interest arising in the field of many-valued logics and also to produce new ones. The original content of the disseration is contained in [22], [21] and [23]... [edited by Author]
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Dans le cadre de la théorie des topos, nous généralisons deux équivalences classiques qui s'inscrivent dans le domaine des MV-algèbres: l'équivalence de Mundici entre la catégorie des MV-algèbres et celle des groupes abéliennes réticulés (1-groupes) avec unité forte et l'équivalence de Di Nola-Lettieri entre la catégorie des MV-algèbres parfaites et celle des 1-groupes. Ces généralisations produisent deux équivalences de Morita; l'une entre la théorie MV des MV-algèbres et la théorie Lu des 1-groupes avec unité forte et l'autre entre la théorie P des MV-algèbres parfaites et la théorie L des 1-groupes. Les deux équivalences de Morita nous permet d'appliquer la technique 'topos comme ponts' pour transférer des propriétés et des résultats d'une théorie à l'autre, en obtenant des nouvelles connaissances sur ces théories. Parmi elles, nous mentionnons une correspondance biunivoque entre les extensions géométriques de MV et celles de Lu, une forme de complétude et de compacité de la théorie infinitaire Lu, trois niveaux différents de bi-interprétabilitité entre la théorie P et la théorie L et un théorème de représentation pour les objets finiment présentables de la variété de Chang comme produits finis de MV-algèbres parfaites. Nous montrons ensuite que l'équivalence de Morita, résultant de l'équivalence de Di Nola-Lettieri, est seulement l'une de la classe des équivalences de Morita que nous établissons entre les théories des MV-algèbres locales dam des variétés propres des MV-algèbres et les appropriées extensions de la théorie des 1-groupes. En outre, nous généralisons dans ce cadre les résultats de représentation obtenus dans le cas de la variété de Chang
In the thesis we generalize to a topos-theoretic setting two classical equivalences arising in the field of MV-algebras: Mundici's equivalence between the category of MV-algebras and the that of lattice-ordered abelian groups (1-groups, for short) with strong unit and Di Nola-Lettieri's equivalence between the category of perfect MV-algebras and that of 1-groups. These generalizations yield respectively a Morita-equivalence between the theory MV of MV-algebras and the theory Lu of 1-groups with strong unit and one between the theory P of perfect MV-algebras and the theory L of 1-groups. These Morita-equivalences allow us to apply the `bridge technique' whence to transfer properties and results from one theory to the other, obtaining new insights on the theories which are not visible by using classical techniques. Among these results, we mention a bijective correspondence between the geometric extensions of the theory MV and those of the theory Lu, a form of completeness and compactness for the infinitary theory Lu, three different levels of bi-interpretabilitity between the theory P and the theory L and a representation theorem for the finitely presentable objects of Chang's variety as finite products of perfect MV-algebras. We then show that the Morita-equivalence arising from Di Nola-Lettieri's equivalence is just one of a whole class of Morita¬equivalences that we establish between theories of local MV-algebras in proper varieties of MV-algebras and appropriate extensions of the theory of 1-groups. Furthermore, we generalize to this setting the representation results obtained in the case of Chang's variety

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.