# Minimum function

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Needs to be linked to order theory stuff. Also:
• Characteristic property of the minimum: $\forall x,u,v\in S\big[x\eq\text{min}(u,v)\iff\big(x\preceq u \wedge x\preceq v \wedge (x\eq u \vee x\eq v)\big)\big]$
• Proved for [ilmath]\implies[/ilmath] direction.
• I was working on the [ilmath]\impliedby[/ilmath] direction I was about to (attempt) to prove this lemma (and then use it)
• Unproven lemma: $\big(x\preceq u\wedge x\preceq v)\implies x\preceq \text{min}(u,v)$ - via contrapositive
• Some useful lemma: $\forall x,u,v\in S\big[x\eq\text{min}(u,v)\implies\big(x\preceq u\wedge x\preceq v\big)\big]$ which I (believe) I used in defining the minimum of random variables.

This page is short because I wrote it just prior to bed Alec (talk) 07:53, 28 July 2018 (UTC)

Let [ilmath](S,\preceq)[/ilmath] be a "totally ordered poset" ("oset"?)
TODO: Look into this
- it must be totally ordered so [ilmath]\forall x,y\in S[/ilmath] exactly one of [ilmath]x\prec y[/ilmath], [ilmath]x\succ y[/ilmath] or [ilmath]x\eq y[/ilmath] holds ("trichotomy law" or something)

Then:

• $\text{min}:S^2\rightarrow S$ is a function defined by $\text{min}(u,v)\mapsto\left\{\begin{array}{lr}\pi_1(u,v) & \text{if }u\prec v\\ \pi_2(u,v) & \text{if }u\succ v \\ \alpha & \text{otherwise}\end{array}\right.$ where [ilmath]\pi_1[/ilmath] and [ilmath]\pi_2[/ilmath][Note 1] are the characteristic projections of a product, and
• [ilmath]\alpha:\eq \pi_{\alpha'}(u,v)[/ilmath] for [ilmath]\alpha'\eq 1[/ilmath] or [ilmath]\alpha'\eq 2[/ilmath] (it doesn't matter as in the case where [ilmath]\alpha[/ilmath] is used we have {[ilmath]u\eq v[/ilmath] by the "trichotomy law" mentioned above.
• We include [ilmath]\alpha[/ilmath] to make the cases in analysis more explicit (as this is a Category:First-year friendly page, meaning the readers' hand is held as he reads the steps involved) but also because there is a practical case for a kind of (computer) arithmetic where it is possible that we have [ilmath]\alpha\neq u[/ilmath] and [ilmath]\alpha\neq v[/ilmath] - It might be signed zero, I remember doing it, not why I was doing it unfortunately.

## Notes

1. Explicitly:
• [ilmath]\pi_1:(a,b)\mapsto a[/ilmath] and
• [ilmath]\pi_2:(a,b)\mapsto b[/ilmath]