2.2. Symmetric monoidal preorders#
from IPython.display import Image
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.#
See also Exercise 2.5.
Image('raster/2023-10-28T17-43-10.png', metadata={'description': "7S answer"})
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Exercise 2.8.#
See also Exercise 2.8.
Image('raster/2023-10-28T17-43-36.png', metadata={'description': "7S answer"})
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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.#
See also Exercise 2.20.
Image('raster/2023-10-28T17-44-37.png', metadata={'description': "7S answer"})
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2.2.3. Applied examples#
Chemistry#
Exercise 2.21.#
See also Exercise 2.21.
Image('raster/2023-10-28T18-07-23.png', metadata={'description': "7S answer"})
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Manufacturing#
Informatics#
2.2.4. Abstract examples#
Exercise 2.29.#
See also Exercise 2.29.
Image('raster/2023-10-28T18-08-22.png', metadata={'description': "7S answer"})
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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:
We have, multiplying both sides of the first premise by \(y_2\):
Multiplying both sides of the second premise by \(x_1\):
And combining the last two equations with transitivity:
Image('raster/2023-10-28T18-10-26.png', metadata={'description': "7S answer"})
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Exercise 2.33.#
See also Exercise 2.33.
Image('raster/2023-10-28T18-10-52.png', metadata={'description': "7S answer"})
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Exercise 2.34.#
See also Exercise 2.34.
Image('raster/2023-10-28T18-12-02.png', metadata={'description': "7S answer"})
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See also Exercise 2.29; we could have constructed a NMY preorder from the max
operation as well.
Exercise 2.35.#
See also Exercise 2.35.
Image('raster/2023-10-28T18-12-44.png', metadata={'description': "7S answer"})
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Exercise 2.36.#
See also Exercise 2.36.
Image('raster/2023-10-28T18-13-36.png', metadata={'description': "7S answer"})
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Example 2.37#
See also Extended real number line (which adds both a +∞ and -∞).
Exercise 2.39.#
See also Exercise 2.39.
Image('raster/2023-10-28T18-14-01.png', metadata={'description': "7S answer"})
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Exercise 2.40.#
See also Exercise 2.40.
Image('raster/2023-10-28T18-15-01.png', metadata={'description': "7S answer"})
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2.2.5. Monoidal monotone maps#
Exercise 2.43#
See also Exercise 2.43.
Image('raster/2023-10-28T18-15-36.png', metadata={'description': "7S answer"})
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Exercise 2.44#
See also Exercise 2.44.
Image('raster/2023-10-28T18-15-59.png', metadata={'description': "7S answer"})
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Exercise 2.45#
See also Exercise 2.45.
Image('raster/2023-10-28T18-17-16.png', metadata={'description': "7S answer"})
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![../../_images/b39eca649b56daa2cb3a5a2154d5ae21f94f7855b3cbaa56d51220fcd63c2718.png](../../_images/b39eca649b56daa2cb3a5a2154d5ae21f94f7855b3cbaa56d51220fcd63c2718.png)