> For the complete documentation index, see [llms.txt](https://adams-personal-organization.gitbook.io/compendium-of-predicates/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://adams-personal-organization.gitbook.io/compendium-of-predicates/definitions/monoidal-preorder.md).

# Monoidal Preorder

<details>

<summary><span class="math">\textbf{MonoidalPreorder} \; X \; R \; \otimes \; I</span></summary>

***

$$\textbf{Preorder} ; X ; R$$

$$\textbf{Monoid} ; X ; \otimes ; I$$

$$\forall (x\_1, x\_2, y\_1, y\_2 \in X : x\_1 \leq x\_2 \wedge y\_1 \leq y\_2 : x\_1 \otimes x\_2 \leq y\_1 \otimes y\_2)$$

***

```
pred MonoidalPreorder(X: set univ, R: univ->univ, op: univ->univ->univ, I: univ) {
  Preorder[X,R]
  Monoid[X,op,I]
  all x1,x2,y1,y2 | x1->x2 in R and y1->y2 in R implies op[x1,x2] -> op[y1,y2] in R
}
```

</details>