1.1. More than the sum#

ERROR:

a solution can be found in Chapter 1

The author likely intends to refer to the appendix.

Exercise 1.1.

An order-preserving function, part of the Category of preordered sets. See also Order theory:

\[ f(x) = x + 1 \]

A function that is not order-preserving:

\[ f(x) = -x \]

A metric-preserving function:

\[ f(x) = x + 1 \]

A function that is not metric-preserving:

\[ f(x) = 2x \]

An addition-preserving function:

\[ f(x) = 2x \]

A function that is not addition-preserving:

\[ f(x) = x^2 \]

1.1.1. A first look at generative effects#

See Category of topological spaces.

1.1.2 Ordering systems#

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Exercise 1.7. Only the last (4.) is false.

See “Observation preserves order but not join” in Applied Category Theory book - Math SE for valuable commentary on section 1.1.2.

For non-mathematicians this section in particular fails to address the (simple) difference between a Binary relation and a Binary operation. Operations are convertible to relations (see Binary operations as ternary relations) but are in no way the same. As pointed out in a comment by John Baez on the Math SE question referenced above, what it means for each to preserve structure is rather different.

If you do end up reading through those Wikipedia articles, notice that the Relation (mathematics) article was started as a copy of the Binary relation article. Please continue to improve the former article, and be understanding of the situation if you decide to read much from it.

That is, \(\leq\) is a binary relation, and \(\vee\) is a binary operation. The map \(\Phi\) is an order-preserving function because if \(A \leq B\) it implies that \(\Phi(A) \leq \Phi(B)\). The order-preserving relationship (unlike the metric-preserving, addition-preserving, and join-preserving relationships) does not require the result before and after to be equal. It’s defined so you only need the result of an order comparison before function application to imply the result of an order comparison after function application (the Material conditional, not the Material biconditional). For the function to be join-preserving, we need \(\Phi(A) \vee \Phi(B)\) (applying the function before) to equal \(\Phi(A \vee B)\) (applying the function after).