The following preprint is available. It is a sequel of my previous work on the homology of equilogical spaces, announced in November. M. Grandis Inequilogical spaces, directed homology and noncommutative geometry Dip. Mat. Univ. Genova, Preprint 494 (2004), 22 p. Abstract. We introduce a preordered version of D. Scott's equilogical spaces, called *inequilogical spaces*, as a possible setting for Directed Algebraic Topology. The new structure can also express 'formal quotients' of spaces, which are not topological spaces and are of interest in noncommutative geometry, with finer results than the ones obtained with equilogical spaces, in a previous paper. This setting is compared with other structures which have been recently used for Directed Algebraic Topology: spaces equipped with an order, or a local order, or distinguished paths or distinguished cubes. http://www.dima.unige.it/~grandis/pEql.pdf http://www.dima.unige.it/~grandis/pEql.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/