The following preprint is available: F. Borceux - M. Grandis JORDAN-HOLDER, MODULARITY AND DISTRIBUTIVITY IN NON-COMMUTATIVE ALGEBRA Dip. Mat. Univ. Genova, Preprint 474 (Feb 2003), 34 p. Abstract. A study of lattices of subgroups or subrings adequate for non-commutative homological algebra can be pursued in a setting of *weakly exact* categories. These extend the Puppe-exact ones and the semiabelian ones, and are essentially based on a notion of *gamma-category* introduced by Burgin. In this context, subobjects form *w-modular w-lattices*, equipped with a normality relation. The free w-modular w-lattice generated by two chains with normality conditions is determined and proved to be *weakly distributive*, by a construction inspired by the well-known Birkoff theorem for free modular lattices. We show that this theorem is relevant for the study of double filtrations, much in the same way as the Birkoff theorem in the commutative case; similarly, it should be of use in the study of spectral sequences. Available in dvi and ps: http://www.dima.unige.it/~grandis/BGwe.dvi http://www.dima.unige.it/~grandis/BGwe.ps ________ The interested reader can also download the following preprint on w-exact categories: M. Grandis Weakly exact categories and their relations, December 2002. [Slightly revised version of: Weakly exact categories and their relations, Dip. Mat. Univ. Genova 20 (1987)]. Available in ps: http://www.dima.unige.it/~grandis/wEx.ps ________ Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/~grandis/