The notation for monoidal product and unit may vary depending on context. \(I, \otimes\) are defaults but it may be best to use \((0,+),(1,*),(true, \land)\) (etc.)
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: \(\forall x,y \in X: x \otimes y = y \otimes x\)
A weak monoidal structure on a preorder \((X, \leq)\)
Definition is identical to a symmetric monoidal structure, replacing all \(=\) signs with \(\cong\) signs.
A monoid is a set \(M\) with a monoid unit \(e \in M\) and associative monoid multiplication \(M \times M \xrightarrow{\star} M\) such that \(m \star e=m=e \star m\)
Every set \(S\) determines a discrete preorder: \(\mathbf{Disc}_S\)
It is easy to check if \((M,e,\star)\) is a commutative monoid then \((\mathbf{Disc}_M, =, e, \star)\) is a symmetric monoidal preorder.
Let \(H\) be the set of all poker hands, ordered by \(h \leq h'\) if \(h'\) beats or ties hand \(h\).
One can propose a monoidal product by assigning \(h_1 \otimes h_2\) to be “the best hand one can form out of the ten cards in \(h_1 \bigcup h_2\)"
This proposal will fail monotonicity with the following example:
\(h_1 := \{2\heartsuit, 3\heartsuit,10 \spadesuit,J\spadesuit,Q\spadesuit\} \leq i_1 := \{4\spadesuit,4\spadesuit,6\heartsuit,6\diamondsuit,10\diamondsuit\}\)
\(h_2 := \{2\diamondsuit,3\diamondsuit,4\diamondsuit,K\spadesuit,A\spadesuit\} \leq i_2 := \{5\spadesuit,5\heartsuit,7\heartsuit,J\diamondsuit,Q\diamondsuit\}\)
\(h_1 \otimes h_2=\{10\spadesuit,J\spadesuit,Q\spadesuit,K\spadesuit,A\spadesuit\} \not \leq i_2 \otimes i_2 = \{5\spadesuit, 5\heartsuit,6\heartsuit,6\diamondsuit,Q\diamondsuit\}\)
Consider the reals ordered by our normal \(\leq\) relation. Do \((1,*)\) as unit and product for a symmetric monoidal structure?
No, monotonicity fails: \(x_1\leq y_1 \land x_2 \leq y_2 \not \implies x_1x_2 \leq y_1y_2\) (Counterexample: \(x_1=x_2=-1, y_1=y_2=0\))
Check if \((M,e,\star)\) is a commutative monoid then \((\mathbf{Disc}_M, =, e, \star)\) is a symmetric monoidal preorder, as described in this example.
Monotonicity is the only tricky one, and is addressed due to the triviality of the discrete preorder.
We can replace \(x \leq y\) with \(x \leq x\) because it is a discrete preorder.
\(x_1 \leq x_1 \land x_2 \leq x_2 \implies x_1 \otimes x_2 \leq x_1 \otimes x_2\)
\(True \land True \implies True\) is vacuously true due to reflexivity of preorder.
Unitality/associativity comes from unitality/associativity of monoid
Symmetry comes from commutitivity of monoid.