Order Theory

Preorder

\begin{matrix} a ≤ a   (reflexive), \\ (a \leq b) \times (b \leq c) \to (a \leq c) (transitive) . \end{matrix}

Meet

\begin{matrix} x \land y \leq x, x \land y \leq y, \\ (a \leq x) \times (a \leq y) \to a \leq x \land y . \end{matrix}

Join

\begin{matrix} x \leq x \lor y, y \leq x \lor y, \\ (x \leq a) \times (y \leq a) \to x \lor y \leq a . \end{matrix}

Partial Order

(x \leq y) \times (y \leq x) → x = y   (antisymmetric) .

Order

Lattice

Comparison with Category Theory

References