A symmetric monoidal structure on a preorder \((X, \leq)\)
Two additional constituents:
A monoidal unit \(I \in X\)
A monoidal product \(X \times X \xrightarrow{\otimes} X\)
Satisfying the following properties:
Monotonicity: \(\forall x_1,x_2,y_1,y_2 \in X: x_1 \leq y_1 \land x_2 \leq y_2 \implies x_1 \otimes x_2 \leq y_1 \otimes y_2\)
Unitality: \(\forall x \in X: I \otimes x = x = x \otimes I\)
Associativity: \(\forall x,y,z \in X: (x \otimes y) \otimes z = x \otimes (y\otimes z)\)
Symmetry: \(\forall x,y \in X: x \otimes y = y \otimes x\)