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

- All Categories 2.4K
- Chat 505
- 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 719

Options

Show that a skeletal dagger preorder is just a discrete preorder, and hence just a set.

A **dagger preorder** is a preorder obeying the **symmetry** axiom:

$$ x \le y \Leftrightarrow y \le x $$ Also recall the skeletal preorders (Remark 1.26) and discrete preorders (Example 1.27):

**Remark 1.26** (Partial orders are skeletal preorders).

A preorder is a **partial order** if we additionally have that
3. \( x \cong y \) implies \( x = y \).

**Example 1.27** (Discrete preorders).

Every set \(X\) can be made into a **discrete** preorder.
This means that the only order relations on \(X\) are of the form \( x \le x \); if \( x \neq y \) then neither \( x \le y \) or \( y \le x \) hold.

## Comments

Recall that a

skeletal dagger preorderis a preorder such that:Combining these properties(*), we have that \(x \le y \Rightarrow x = y\), which is the defining property of a discrete preorder.

(*) Since \(x \le y\) and \(y \le x\) are equivalent by (2), we can remove \(y \le x\) from the precondition on (1). Formally, this involves substitution followed by idempotence of \(\wedge\).

`Recall that a _skeletal dagger preorder_ is a preorder such that: 1. \\(x \le y \wedge y \le x \Rightarrow x = y\\), and 2. \\(x \le y \Leftrightarrow y \le x\\). Combining these properties(\*), we have that \\(x \le y \Rightarrow x = y\\), which is the defining property of a discrete preorder. (\*) Since \\(x \le y\\) and \\(y \le x\\) are equivalent by (2), we can remove \\(y \le x\\) from the precondition on (1). Formally, this involves substitution followed by idempotence of \\(\wedge\\).`