# Explanation: Summary of Different Order Relations

(related to Chapter: Order Relations)

We have learned three different possibilities of order relations "$\prec$" on a set $V$, all using the same notation $(V,\prec)$. We now want to recap all definitions again and compare them with each other. The following table summarizes the key differences:

Partial order Total (Linear) Order Strict Total Order
Poset Chain Strictly ordered set.
=^. $(V,\preceq )$ $(V,\preceq )$ $(V,\prec)$
For $u,v,w\in V$:
* reflexive: $u\preceq u$
* antisymmetric: $u\preceq v$ and $v\preceq u$ implies $u=v$
* transitive: $u\preceq v$ and $v\preceq w$ implies $u\preceq w$ For $u,v,w\in V$:
* reflexive: $u\preceq u$
* antisymmetric: $u\preceq v$ and $v\preceq u$ implies $u=v$
* transitive: $u\preceq v$ and $v\preceq w$ implies $u\preceq w$
* connex: $u\preceq v$ for all $u,v\in V.$ For $u,v,w\in V$:
* irreflexive: $u\not \prec u$
* asymmetric: $u\prec v$ implies $v\not\prec u$
* transitive: $u\prec v$ and $v\prec w$ implies $u\prec w$
Possibilities:
* equal: $u=v$
* less or equal: $u\preceq v$
* less: $u\preceq v$ and $u\neq v$
* greater or equal: $u\succeq v$
* greater: $u\succeq v$ and $u\neq v$
* else incomparable Possibilities:
* equal: $u=v$
* less or equal: $u\preceq v$
* less: $u\preceq v$ and $u\neq v$
* greater or equal: $u\succeq v$
* greater: $u\succeq v$ and $u\neq v$ Possibilities:
* less: $u\prec v$
* greater: $u\succ v$
* else equal: $u=v$
Eg. $(\mathbb N,\le)$ Eg. $(\mathbb N,<)$

Please note that since a total order of a set $V$ is reflexive while a strict total order on it is not, a total order never can be strict total and vice versa. A comparison of both types of order relations leads to the so-called diagonal1 $D$ of $V$ defined by $D:=\{(x,x)\in \preceq\}$: * We can transform a given chain $(V,\preceq )$ to a strictly ordered set $(V,\prec)$ by the set difference of the total order "$\preceq$" and $D$, i.e. the strict order is defined by $\prec := \preceq\setminus D.$ * On the other hand, if we change a given strictly ordered set $(V,\prec)$ into a chain $(V,\preceq )$ by the union of its strict order "$\prec$" and $D$, i.e. the total order is defined by $\preceq:=\prec\cup D.$

Github: ### References

#### Bibliography

1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition

#### Footnotes

1. The word "diagonal" is chosen reflecting that the pairs elements $(x,x)\in \preceq$ are exactly the diagonal elements of the matrix representation of the reflexive relation "$\preceq$".