> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/240-order-theory/list-of-order-structures.md).

# List of order structures in mathematics

<https://en.wikipedia.org/wiki/List_of_order_structures_in_mathematics>

In mathematics, and more specifically in *order theory*, several different types of *ordered set* have been studied. They include:

* **Cyclic order**, orderings in which triples of elements are either clockwise or counterclockwise
* **Lattice**, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list.
* **Partially ordered set (poset)**, orderings in which some pairs are comparable and others might not be
* **Preorder**, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities)
* **Semiorder**, partial orders determined by comparison of numerical values, in which values that are too close to each other are incomparable; a subfamily of partial orders with certain restrictions
* **Total order**, orderings that specify, for every two distinct elements, which one is less than the other. that is, all elements are comparable.
* **Weak order**, generalization of total order allowing ties (represented either as equivalence or, in strict weak orders, as transitive incomparability)
* **Well-order**, total order in which every non-empty subset has an infinum
* **Well-quasi-ordering**, a class of preorders generalizing the well-orders

## Refs

[List of order theory topics](https://en.wikipedia.org/wiki/List_of_order_theory_topics) [Glossary of order theory](https://en.wikipedia.org/wiki/Glossary_of_order_theory) [Categories](https://en.wikipedia.org/wiki/Help:Category) [Mathematics-related lists](https://en.wikipedia.org/wiki/Category:Mathematics-related_lists) [Order theory](https://en.wikipedia.org/wiki/Category:Order_theory)