*Jérôme Lang*

In many real-world collective decision problems, the set of alternatives is a Cartesian product of finite value domains for each of a given set of variables. The prohibitive size of such domains makes it practically impossible to represent preference relations explicitly. Now, AI has been developing languages for representing preferences on such domains in a succinct way, exploiting structural properties such as conditional preferential independence. Here we reconsider voting and aggregation rules in the case where voters' preferences have a common preferential independence structure, and address the decompossition a voting rule or an aggregation function following a linear order over variables.