Solution

• Monotonicity: \(x_1 \leq y_1 \land x_2 \leq y_2 \implies x_1x_2 \leq y_1y_2\)

  • Suppose without loss of generality that \(x_1\leq x_2\)

  • then \(x_1x_2=x_1\) and \(y_1y_2 = y_1\) or \(y_2\)

  • In the first case, we know this is true because we assumed \(x_1 \leq y_1\)

  • In the second case, we know it from transitivity: \(x_1 \leq x_2\leq y_2\)

• Unitality: \(min(x,yes)=x\)

• Associativity: probably

• Symmetry: table is symmetric.