\[ \boxed { \begin{array} {} & \text {preorders} & \\ & \text {reflexive:} \; x \leq x & \\ & \text {transitive:}\; x\leq y \;\&\; y\leq z \Rightarrow x\leq z & \\ \boxed { \begin{array} {} \text {partial order} \\ \text {skeletal:}\; x\leq y \;\&\; y\leq x \Rightarrow x=y \\ \text {example: power set } X\mathcal P \\ \end{array} } & \boxed { \begin{array} {} \text{intersection} \\ \text{discrete order: only $x=x$ allowed } \\ \end{array} } & \boxed { \begin{array} {} \text {equivalence relation} \\ \text {symmetric: } x\leq y \Rightarrow y\leq x \\ \text{example: congruence } m \equiv n \; (\text{mod } k) \\ \end{array} }\\ \end{array} } \]
