Interested in the dimension of the face lattices of d-dimensional grids, we are led to the study of fences and their products. This yields a characterization of generalized fences: F is a generalized fence if and only if \(\dim (P\times F) \leqslant \dim (P) +1\) for every poset P. Whether \(\dim (P \times Q) \geqslant \dim (P) + \dim (Q) -2\) for all posets P and Q is a long-standing question in dimension theory. Having understood products where one factor is a fence we further investigate the dimension of products where the factors are crowns and other 3-dimensional posets. We reconsider the covering property defined by Reuter for Ferrers relations and show that in many cases the gap between \(\dim (P \times Q)\) and \(\dim (P) + \dim (Q)\) can be explained in terms of the covering properties of one of the factors.
