Abstract examples

The opposite of a symmetric monoidal preorder \((X, \geq, I, \otimes)\) is still a symmetric monoidal preorder

Proof(1)

• Monotonicity: \(x_1 \geq y_1 \land x_2 \geq y_2 \implies x_1 \otimes x_2 \geq y_1 \otimes y_2\)

  • This holds because monotonicity holds in the original preorder (\(a\geq b \equiv b \leq a\)).

• Unitality, symmetry, and associativity are not affected by the preorder.

There is a symmetric monoidal structure on \((\mathbb{N}, \leq)\) where the monoidal unit is ?zero and the product is ?\(+\) (\(6+4=10\)). Monotonicity (?\((x_1,x_2)\leq(y_1,y_2) \implies x_1+x_2 \leq y_1+y_2\)) and other conditions hold.

• Recall the divisibility order \((\mathbb{N}, |)\)

• There is a symmetric monoidal structure on this preorder with unit ?\(1\) and product ?\(*\).

• Monotonicity (?\(x_1|y_1 \land x_2|y_2 \implies x_1*x_2 | y_1*y_2\)) and other conditions hold.

• Lawvere’s symmetric monoidal preorder, Cost.

• Let \([0,\infty]\) represent the non-negative real numbers with infinity. Also take the usual notion of \(\geq\).

• There is a monoidal structure for this preorder: \(\mathbf{Cost}:=([0,\infty],\geq,0,+)\)

  • The monoidal unit being zero means ?“you can get from a to a at no cost."

  • The product being + means ?“getting from a to c is at most the cost of a to b plus b to c"

  • The ‘at most’ above comes from the ?\(\geq\).

Linked by

• Feasibility relationships as Bool-profunctors: referenced

Show there is a symmetric monoidal structure on \((\mathbb{N}, \leq)\) where the monoidal product is \(6*4=24\). What should the monoidal unit be?

Solution(1)

• Let the monoidal product be the standard product for integers, with 1 as unit.

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

  • Unitality: \(1*x_1=x_1=x_1*1\)

  • Associativity: \(a*(b*c)=(a*b)*c\)

  • Symmetry: \(a*b=b*a\)

Recall the divisibility order \((\mathbb{N}, |)\). Someone proposes \((0,+)\) as the monoidal unit and product. Does this satisfy the conditions of a symmetric monoidal structure?

Solution(1)

Conditions 2-4 are satisfied, but not monotonicity: \(1|1 \land 2|4\) but not \(3 | 5\)

• Consider the preorder defined by the Hasse diagram \(\boxed{no \rightarrow maybe \rightarrow yes}\)

• Consider a potential monoidal structure with \(yes\) as the unit and \(min\) as the product.

• Fill out a reasonable definition of \(min\) and check that it satisfies the conditions.

Solution(1)

\(min\)	no	maybe	yes
no	no	no	no
maybe	no	maybe	maybe
yes	no	maybe	yes

• 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.

Linked by

• Exercise 2-61: referenced

Check that there is a symmetric monoidal structure on the power set of \(S\) ordered by subset relation. (The unit is \(S\) and product is \(\cap\))

Solution(1)

• Monotonicity: \(x_1 \subseteq y_1 \land x_2 \subseteq y_2 \implies x_1 \cap x_2 \subseteq y_1 \cap y_2\)

  • This is true: ?if something is in both \(x_1,x_2\), then it is in both \(y_1,y_2\), i.e. in their intersection.

• Unitality: \(x \cap S = x = S \cap x\), since ?\(\forall x \in P(S): x \subseteq S\).

• Associativity and symmetry come from associativity and symmetry of \(\cap\) operator.

• Let \(Prop^\mathbb{N}\) be the set of all mathematical statements one can make about a natural number.

• Examples:

  • n is a prime

  • n = 2

  • \(n \geq 11\)

• We say \(P \leq Q\) if for all \(n \in \mathbb{N}\), \(P(n) \implies Q(n)\)

• Define a monoidal unit and product on \((Prop^\mathbb{N}, \leq)\).

Solution(1)

• Let the unit be \(\lambda x. true\) and product be \(\land\)

• montonicity: \(P_1(x)\leq Q_1(x) \land P_2(x) \leq Q_2(x) \implies (P_1 \land P_2)(x) \leq (Q_1 \land Q_2)(x)\)

  • If the \(P\) properties hold for a given number, then each of the \(Q\) properties hold

• unitality, associativity, symmetry: same as \(\mathbf{Bool}\)

Consider \(\mathbf{Cost}^{op}\). What is it as a preorder? What is its unit and product?

Solution(1)

As a preorder, the domain is still \([0,\infty]\) and ordered by the natural \(\leq\) relation. The unit and product are unchanged by taking the opposite preorder, so they are still \(0, +\) respectively.