Order Dimension, Grids, and Products
Stefan Felsner, Torsten Mütze, Maximilian Wittmann · Order · 2025
Abstract 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 $$\dim (P\times F) \leqslant \dim (P) +1$$ dim ( P × F ) ⩽ dim ( P ) + 1 for every poset P . Whether $$\dim (P \times Q) \geqslant \dim (P) + \dim (Q) -2$$ dim ( P × Q ) ⩾ 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 $$\dim (P \times Q)$$ dim ( P × Q ) and $$\dim (P) + \dim (Q)$$ dim ( P ) + dim ( Q ) can be explained in terms of the covering properties of one of the factors.