Order Dimension, Grids, and Products

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 $$\dim (P\times F) \leqslant \dim (P) +1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>×</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> <mml:mo>⩽</mml:mo> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> for every poset P . Whether $$\dim (P \times Q) \geqslant \dim (P) + \dim (Q) -2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>×</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> <mml:mo>⩾</mml:mo> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> 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 $$\dim (P \times Q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>×</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$\dim (P) + \dim (Q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> can be explained in terms of the covering properties of one of the factors.