Talk:Product order

Partial order or preorder?

The lead says that this is a product of two preorders, then says that the result is a partial order. However, the product of a preorder A with itself would only be a preorder, not partial order. Which one is it supposed to be? 46.162.87.99 (talk) 17:51, 21 November 2022 (UTC)

I guess, the product of two preorders is a preorder again, and in the special case that both were partial orders, the product is a partial order. However, I have no appropriate citation at hand. - Jochen Burghardt (talk) 16:10, 22 November 2022 (UTC)

Ordering relation vs ordered set

@Jochen Burghardt I don't understand your reversion of my edits. As it stands, the introduction states that two preordered sets yield a partial order. This is obviously wrong.

Furthermore, the product should yield the same structure as its inputs. If it is two ordered sets, the output should be an ordered set. If it is two orders, the output should be an order. 46.162.75.255 (talk) 14:18, 31 May 2023 (UTC)

My point is: (1) the product of two given orders is again an order, while (2) the product of two given ordered sets is again an ordered set. Your most recent edit resulted in claiming "...the product order ... is a ... set". I agree that the result is partial order, not just a preorder.

The problem is that there is no appropriate name for the operation (2), since "product order" can only mean (1). What about the following introductory sentences:

"In mathematics, given a partial order $\preceq$ and $\sqsubseteq$ on a set $A$ and $B$, respectively, the product order (also called the coordinatewise order or componentwise order) is a partial order $\leq$ on the Cartesian product $A\times B$. Given two pairs $\left(a_{1},b_{1}\right)$ and $\left(a_{2},b_{2}\right)$ in $A\times B,$ declare that $\left(a_{1},b_{1}\right)\leq \left(a_{2},b_{2}\right)$ if $a_{1}\preceq a_{2}$ and $b_{1}\sqsubseteq b_{2}.$"

(I changed "iff" to "if" per MOS:MATH#DEFSYMBOL.) Would that be OK for you? - Jochen Burghardt (talk) 16:07, 31 May 2023 (UTC)

@Jochen Burghardt That is okay. Later in the text, the terms "order" and "ordered set" are again used interchangeably. I assumed this was unproblematic, as the two structures are isomorphic. 46.162.75.255 (talk) 18:07, 31 May 2023 (UTC)