It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.4K
- Chat 504
- Study Groups 21
- Petri Nets 9
- Epidemiology 4
- Leaf Modeling 2
- Review Sections 9
- MIT 2020: Programming with Categories 51
- MIT 2020: Lectures 20
- MIT 2020: Exercises 25
- Baez ACT 2019: Online Course 339
- Baez ACT 2019: Lectures 79
- Baez ACT 2019: Exercises 149
- Baez ACT 2019: Chat 50
- UCR ACT Seminar 4
- General 75
- Azimuth Code Project 111
- Statistical methods 4
- Drafts 10
- Math Syntax Demos 15
- Wiki - Latest Changes 3
- Strategy 113
- Azimuth Project 1.1K
- - Spam 1
- News and Information 148
- Azimuth Blog 149
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 718

Options

Given preorders \(A\) and \(B\), let their **product** \(A \times B\) be the cartesian product of sets \(A\) and \(B\) made into a preorder as follows:

$$ (a,b) \le (a',b') \textrm{ if and only if } a \le a' \textrm{ and } b \le b' .$$ Draw the product of the two preorders drawn below:

## Comments

`<center> ![Product](https://docs.google.com/drawings/d/e/2PACX-1vSDPTMFV6q0bYV4ee-Khkg5_1f_5Kfv3dBj7Pc2NhqiT3b7PyGQexE87KbIqRfAdUf8N0iyM9hT9PE6/pub?w=282&h=280) </center>`

This is pretty fascinating to me. You can imagine acquiring the product by extruding \(A\) from \(B\). But it also works just as well by imagining extruding \(B\) from \(A\)!

It's like we're replacing every element in one preorder with a copy of the other preorder, and distributing edges of the original preorder through the copies.

Product constructions aren't particularly

newto me, but... There's something incredibly pleasant about a geometric explanation.`This is pretty fascinating to me. You can imagine acquiring the product by extruding \\(A\\) from \\(B\\). But it also works just as well by imagining extruding \\(B\\) from \\(A\\)! ![](https://i.imgur.com/iP02W2C.png) It's like we're replacing every element in one preorder with a copy of the other preorder, and distributing edges of the original preorder through the copies. Product constructions aren't particularly _new_ to me, but... There's something incredibly pleasant about a geometric explanation.`

Here's another image showing what I mean by "replacing every element". The outer edge "distributes" through the larger cells onto every analogous pair.

`Here's another image showing what I mean by "replacing every element". The outer edge "distributes" through the larger cells onto every analogous pair. ![](https://i.imgur.com/j9oGOC0.png)`

Nice illustrations @JonathanCastello . https://forum.azimuthproject.org/discussion/comment/17639/#Comment_17639 in particular shows commutivity of paths.

`Nice illustrations @JonathanCastello . https://forum.azimuthproject.org/discussion/comment/17639/#Comment_17639 in particular shows commutivity of paths.`