The wikipedia article about partially ordered set describes three of the possible partial orders on the Cartesian product of two partially ordered sets, and mentions that similar constructions can be done on the Cartesian product of more than two sets. However, if there are more than two sets, the constructions can be combined with each other in various ways. If we restrict ourselves to the lexicographic order and the product order, how can we describe the various way in which these orders can be combined?
For three sets, we could define $(a_1, a_2, a_3) \leq(b_1,b_2,b_3)$ via:
- $a_1 < b_1$ or ($a_1=b_1$ and ($a_2 < b_2$ or ($a_2=b_2$ and $a_3 \leq b_3$))) (lexicographical order)
- $a_1 \leq b_1$ and $a_2 \leq b_2$ and $a_3 \leq b_3$ (product order)
- $a_1 < b_1$ or ($a_1=b_1$ and $a_2 \leq b_2$ and $a_3 \leq b_3$) (first lexicographical order, then product order)
- ($a_1 \leq b_1$ and $a_2 \leq b_2$ and ($a_1 < b_1$ or $a_2 < b_2$)) or ($a_1 = b_1$ and $a_2 = b_2$ and $a_3 \leq b_3$) (first product order, then lexicographical order)
- $a_1 \leq b_1$ and ($a_2 < b_2$ or ($a_2=b_2$ and $a_3 \leq b_3$)) (first product order, then ...)
- ($a_1 < b_1$ or ($a_1=b_1$ and $a_3 \leq b_3$)) and ($a_2 < b_2$ or ($a_2=b_2$ and $a_3 \leq b_3$)) (first product order, then ...)
I think these are all orders I want to consider for three sets (ignoring permutations), but how can I be sure without a systematic way to describe these orders. Is there a systematic way to describe the orders on the Cartesian product of a finite number of partial ordered sets?
Does the description gets easier, if we look at bounded lattices (or semilattices with identity element) instead of partially ordered sets?