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Points critiques de couples de variétés d'algèbres

Abstract : The set of all congruences of a given algebra, ordered by inclusion, is an algebraic lattice (Birkhoff), its compact elements are the finitely generated congruences; they form a semilattice. A semilattice is liftable in a variety V if it is isomorphic to the semilattice of all compact congruences of an algebra in V. The works of Wehrung on CLP, and of Ploščica, illustrate that even for an easy to describe variety of algebras, like the variety of all lattices, or the finitely generated varieties, characterizing the liftable semilattices is hard. The critical point of two varieties V and W is the smallest cardinal of a semilattice liftable in V but not in W.

We introduce a tool, with categorical flavor, that gives a link between lifting diagrams of semilattices and lifting semilattices in a given variety. Given finitely generated varieties of lattices V and W such that W does not lift all semilattices liftable in V, we prove that the critical point of V and W is either finite or some aleph of finite index. We give two finitely generated varieties of modular lattices with critical point aleph 1, which disproves a conjecture of Tůma and Wehrung.

Using the theory of Von Neumann regular rings together with the dimension monoid of a lattice, we prove that the critical point of varieties of lattices generated by subspace lattice of vector spaces of the same finite dimension on finite fields is at least aleph 2. We prove the equality for dimensions 2 and 3.
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Submitted on : Wednesday, December 10, 2008 - 9:04:25 AM
Last modification on : Monday, April 27, 2020 - 4:14:03 PM
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  • HAL Id : tel-00345793, version 1

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Pierre Gillibert. Points critiques de couples de variétés d'algèbres. Mathématiques [math]. Université de Caen, 2008. Français. ⟨tel-00345793⟩

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