Liftings of distributive lattices by locally matricial algebras with respect to the Id c functor

We study representations of distributive -lattices, considered as join-semilattices, by semilattices of finitely generated two-sided ideals of locally matricial algebras over a field k, aiming to find a functorial solution of the problem. We find simple examples of a finite subcategory of the category Ld of distributive -lattices and of a subcategory of L_d corresponding to a partially ordered class which cannot be lifted with respect to the Id_c functor. On the other hand, we prove that there is such a lifting of every diagram in L_d or of a subcategory L_d1 of L_d whose objects are all distributive -lattices and whose morphisms are -embeddings. This paper is dedicated to Walter Taylor. Received February 8, 2005; accepted in final form August 11, 2005. The work is a part of the research project MSM 0021620839 financed by MSMT and partly supported by INTAS project 03-51-4110, the grant GAUK 448/2004/B-MAT, and the post-doctoral grant GAČR 201/03/P140.