Partial Order
Definition 1. A relation $\leqq$ on a set $A$ is called a partial order relation if and only if the relation $\leqq$ is reflexive and transitive on $A$ and antisymmetric on $A$, that is, if $a\leqq b$ and $b \leqq a$, then $a = b$. A partially ordered set is a pair $(A, \leqq)$, where $A$ is a set and $\leqq$ is a partial order relation on $A$.
Total order
Definition 2. A total order relation $\leqq$ on a set $A$ is a partial order relation such that for any pair of elements $a$ and $b$ in $A$, either $a\leqq b$ or $b \leqq a$. A totally ordered set is a pair $(A, \leqq)$ where $A$ is a set and $\leqq$ is a total order relation.