Lifting retracted diagrams with respect to projectable functors |
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Authors: | Friedrich Wehrung |
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Institution: | (1) LMNO, CNRS UMR 6139, Département de Mathématiques, Université de Caen, 14032 Caen Cedex, France |
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Abstract: | We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large.
As an application, we prove various results of which the following is a prototype: If every diagram, indexed by a lattice, of finite Boolean 〈∨ ,0 〉-semilattices with 〈∨ ,0 〉-embeddings, can be lifted, with respect to the Conc functor, by a diagram of lattices, then so can every diagram, indexed by a lattice, of finite distributive 〈∨ ,0 〉-semilattices with 〈∨ ,0 〉-embeddings. If the premise of this statement held, this would solve in turn the (still open) problem whether every distributive algebraic
lattice is isomorphic to the congruence lattice of a lattice. We also outline potential applications of our method to other
functors, such as the
functor on von Neumann regular rings.
Received August 12, 2004; accepted in final form June 6, 2005. |
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Keywords: | Primary 18A30 18A25 18A20 18A35 Secondary 06A12 06D05 08B25 18A40 19A49 |
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