2.2. Symmetric monoidal preorders#

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2.2.1. Definition and first examples#

Definition 2.2.#

Define monoidal preorder#

See also Monoidal category - Monoidal preorders for an alternative definition of non-symmetric monoidal preorders. The full definition of a Monoidal category is not introduced (even roughly) until section 4.4.3.

You can remember the term “monoidal product” for the ⊗ operator because in general the “most important” example of a monoid in this context is going to be the real numbers with multiplication as the ⊗ operator. The usual formula for matrix multiplication (and by extension, for dot products) is based on this monoid; see Section 2.5.3. However, the symbol ⊗ (\otimes or “circled times”) is primarily associated with the Tensor product (see The Tensor Product, Demystified) and is the original source of this word. A final option is to associate this word to the “Cartesian product” when you are working with Set-categories (regular categories) (see Example 4.49).

In the short term we’re going to have to accept “+” as a “monoidal product” however. The article Monoid tries to avoid anything but the language “binary operator” in its definition.

Definitional variation#

Notational variation#

Example 2.4.#

Exercise 2.5.#

Exercise 2.8.#

Example 2.9.#

2.2.2. Introducing wiring diagrams#

Compare to String diagram and string diagram in nLab.

See also the concept of a “valid” argument in Chapter 3 Other logical notions ‣ Part I Key notions of logic ‣ forall x: Calgary.

Exercise 2.20.#

2.2.3. Applied examples#

Chemistry#

Exercise 2.21.#

Manufacturing#

Informatics#

2.2.4. Abstract examples#

Exercise 2.29.#

Exercise 2.31.#

Notice this question only asks for a monoidal structure, not a symmetric monoidal preorder structure. Since it seems to have both, we’ll show both here.

The monoidal unit should be 1 (satisfies unitality). The natural numbers under addition satisfy the associativity condition, so this symmetric monoidal preorder will as well. See Natural number - Algebraic properties satisfied by the natural numbers. The natural numbers also satisfy the commutativity condition (symmetry condition in the text).

The following is based (roughly) on Propositional calculus § Natural deduction system and Propositional calculus § Proofs in propositional calculus. To see the monotonicity condition is satisfied we’ll use the normal rules of algebra on the natural (positive) numbers, with these two premises:

\[\begin{split} \begin{align} \\ x_1 & \leq y_1 \\ x_2 & \leq y_2 \\ \end{align} \end{split}\]

We have, multiplying both sides of the first premise by \(y_2\):

\[\begin{split} \begin{equation} x_1 \ast y_2 \leq y_1 \ast y_2 \\ \end{equation} \end{split}\]

Multiplying both sides of the second premise by \(x_1\):

\[\begin{split} \begin{equation} \\ x_1 \ast x_2 \leq x_1 \ast y_2 \\ \end{equation} \end{split}\]

And combining the last two equations with transitivity:

\[ x_1 \ast x_2 \leq x_1 \ast y_2 \leq y_1 \ast y_2 \]

Image('raster/2023-10-28T18-10-26.png', metadata={'description': "7S answer"})

2.2. Symmetric monoidal preorders

Contents

2.2. Symmetric monoidal preorders#

2.2.1. Definition and first examples#

Definition 2.2.#

Define monoidal preorder#

Definitional variation#

Notational variation#

Example 2.4.#

Exercise 2.5.#

Exercise 2.8.#

Example 2.9.#

2.2.2. Introducing wiring diagrams#

Exercise 2.20.#

2.2.3. Applied examples#

Chemistry#

Exercise 2.21.#

Manufacturing#

Informatics#

2.2.4. Abstract examples#

Exercise 2.29.#

Exercise 2.31.#

Exercise 2.33.#

Exercise 2.34.#

Exercise 2.35.#

Exercise 2.36.#

Example 2.37#

Exercise 2.39.#

Exercise 2.40.#

2.2.5. Monoidal monotone maps#

Exercise 2.43#

Exercise 2.44#

Exercise 2.45#