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# Exercise 37 - Chapter 2

edited June 2018

Since $$\textbf{Cost}$$ is a symmetric monoidal preorder, Proposition 2.35 says that $$\textbf{Cost}^{op}$$ is too. $$\textbf{Cost} = ( [ 0, \infty ], \ge, 0, + )$$

1. What is $$\textbf{Cost}^{op}$$ as a preorder?
2. What is its monoidal unit?
3. What is its monoidal product?

• Options
1.
edited May 2018

Proposition 2.20 implies that $$\textbf{Cost}^{op} = ( [ 0, \infty ], \le, 0, + )$$ is a perfectly good symmetric monoidal preorder.

Using $$0$$ for the monoidal unit, and $$+$$ as the monoidal product seem fine.

Thinking of cost as a negative value $$[ -\infty, 0 ]$$ gives an alternate $$op$$ mapping that I will call $$neg$$.

An alternate Proposition : Preorder Negation Suppose $$\mathcal{X} = (X, \le)$$ is a preorder and $$\mathcal{X}^{neg} = (X^{neg}, \le)$$ is an opposite. That is to say, all the arrows are reversed but the meaning of order function is retained. If $$(X, ≤, I, \otimes)$$ is a symmetric monoidal preorder then so is its negation, $$(X^{neg}, \le, I, \otimes)$$. As an example, consider the following. $$\textbf{Cost}_{neg} = ( [ -\infty, 0 ], \ge, 0, + )$$

$$\textbf{X} = \textbf{Cost}^{op} = ( [ 0, \infty ], \le, 0, + )$$ $$\textbf{X}^{op} = \textbf{Cost} = ( [ 0, \infty ], \ge, 0, + )$$ Using the negative real numbers augmented with $$- \infty$$.

$$\textbf{X}^{neg} = \textbf{Cost}^{neg^{op}} = ( [ - \infty, 0 ], \ge, 0, + )$$ $$\textbf{X}^{neg^{op}} = \textbf{Cost}^{neg} = ( [ - \infty, 0 ], \le, 0, + )$$ Does the $$neg$$ already have a name?

Comment Source:Proposition 2.20 implies that $$\textbf{Cost}^{op} = ( [ 0, \infty ], \le, 0, + )$$ is a perfectly good symmetric monoidal preorder. Using \$$0 \$$ for the monoidal unit, and \$$+ \$$ as the monoidal product seem fine. Thinking of cost as a negative value \$$[ -\infty, 0 ] \$$ gives an alternate \$$op\$$ mapping that I will call \$$neg\$$. An alternate **Proposition : Preorder Negation** Suppose \$$\mathcal{X} = (X, \le) \$$ is a preorder and \$$\mathcal{X}^{neg} = (X^{neg}, \le) \$$ is an opposite. That is to say, all the arrows are reversed but the meaning of order function is retained. If \$$(X, ≤, I, \otimes) \$$ is a symmetric monoidal preorder then so is its negation, \$$(X^{neg}, \le, I, \otimes) \$$. As an example, consider the following. $$\textbf{Cost}_{neg} = ( [ -\infty, 0 ], \ge, 0, + )$$ ![Does this generalize?](https://docs.google.com/drawings/d/e/2PACX-1vSaZS_iF5odXxpSMFNiPtH58VMEAgYuLuXV5JMT4dOwc01ZplZW1rU0oZ95wLFTlhxXDU8l_nqPR2V4/pub?w=327&h=204) $$\textbf{X} = \textbf{Cost}^{op} = ( [ 0, \infty ], \le, 0, + )$$ $$\textbf{X}^{op} = \textbf{Cost} = ( [ 0, \infty ], \ge, 0, + )$$ Using the negative real numbers augmented with \$$- \infty \$$. $$\textbf{X}^{neg} = \textbf{Cost}^{neg^{op}} = ( [ - \infty, 0 ], \ge, 0, + )$$ $$\textbf{X}^{neg^{op}} = \textbf{Cost}^{neg} = ( [ - \infty, 0 ], \le, 0, + )$$ Does the \$$neg\$$ already have a name?
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2.

@FredrikEisele : why do you need the negative numbers? Also, the first question ask “what is this as a preorder?”; I understood this to require an interpretation of what $$Cost^{op}$$ is, what it can represent - and I cannot figure this out: to me it seems just the same thing as $$Cost$$ except that it tells you which things are cheaper rather than which ones are more expensive. Any hint? Thanks!

Comment Source:@FredrikEisele : why do you need the negative numbers? Also, the first question ask “what is this as a preorder?”; I understood this to require an interpretation of what \$$Cost^{op}\$$ is, what it can represent - and I cannot figure this out: to me it seems just the same thing as \$$Cost\$$ except that it tells you which things are cheaper rather than which ones are more expensive. Any hint? Thanks!
• Options
3.
edited May 2018

@ValterSorana You do not need negative numbers; I wanted to point out a concept related to an alternative opposite. The alternative invokes the 'cost' set as a negative value. I believe your interpretation of it as a simple change in perspective between 'cheaper' vs. 'more expensive' is a correct interpretation.

Comment Source:@ValterSorana You do not need negative numbers; I wanted to point out a concept related to an alternative opposite. The alternative invokes the 'cost' set as a negative value. I believe your interpretation of it as a simple change in perspective between 'cheaper' vs. 'more expensive' is a correct interpretation. 
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4.

Got it. Thanks @FredrickEisele !

Comment Source:Got it. Thanks @FredrickEisele !