Index >> Publications

- Pierre Gillibert; Gabriele Ranieri,
*"Julia Robinson Number"*, en préparation, 17 pages. - Wolfram Bentz; Pierre Gillibert; and Luís Sequeira,
*"Finite Abelian algebras are fully dualizable"*, préprint, 19 pages. HAL, PS, PDF. - Pierre Gillibert,
*"Finite Abelian algebras are dualizable"*, préprint, 13 pages. HAL, PS, PDF. - Pierre Gillibert,
*"Bounded congruence-preserving extensions of congruence-bounded lattices"*, préprint, 14 pages. PS, PDF. - Pierre Gillibert,
*"Categories of partial algebras for critical points between varieties of algebras"*, Algebra Universalis**71**(2014), no. 4, 299-357.HAL, PS, PDF. - Pierre Gillibert,
*"The possible values of critical points between strongly congruence-proper varieties of algebras"*, Advances in Mathematics**257**(2014), 546-566. HAL, PS, PDF. - Pierre Gillibert,
*"The finiteness problem for automaton semigroups is undecidable"*, International Journal of Algebra and Computation**24**, no. 1 (2014), 1-9. HAL, PS, PDF. - Pierre Gillibert; Miroslav Ploščica,
*"Congruence FD-maximal varieties of algebras"*, International Journal of Algebra and Computation**22**(2012). HAL, PS, PDF. - Pierre Gillibert,
*"The possible values of critical points between varieties of lattices"*, Journal of Algebra**362**(2012), 30-55. HAL, PS, PDF. - Pierre Gillibert; Friedrich Wehrung,
*"From objects to diagrams for ranges of functors"*, Springer Lecture Notes in Mathematics, Vol.**2029**, x+158 pages. HAL, PS, PDF. - Pierre Gillibert; Friedrich Wehrung,
*"An infinite combinatorial statement with a poset parameter"*. Combinatorica**31**, no. 2 (2011), 183-200. HAL, PS, PDF. - Pierre Gillibert,
*"Critical points between varieties generated by subspace lattices of vector spaces"*. Journal of Pure and Applied Algebra**214**(2010), 1306-1318. HAL, PS, PDF. - Pierre Gillibert,
*"Critical points of pairs of varieties of algebras"*. International Journal of Algebra and Computation**19**, no. 1 (2009), 1-40. HAL, PS, PDF.

PhD thesis, directed by Friedrich Wehrung, defended December, 8th 2008

Pierre Gillibert. *"Points critiques de couples de variétés d'algèbres".*

HAL, PS, PDF.

- Martin Goldstern, Professor, University of Technology in Wien
- Patrick Dehornoy, Professor, Université de Caen
- Maurice Pouzet, Professor, Adjunct à University of Calgary
- Miroslav Ploščica, Professor, Slovak Academy of Sciences
- Friedrich Wehrung, Directeur de recherches au CNRS, Université de Caen (advisor)

**Referees:**

- James B. Nation, Professor, University of Hawaii
- Martin Goldstern, Professor, University of Technology in Wien

**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 ℵ_{1}, which disproves a conjecture of Tuma 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 ℵ_{2}. We prove the equality for dimensions 2 and 3.