<p>I have come across many informal examples that provide a physical illustration for Brouwer's fixed point theorem (some due to Brouwer himself). </p>

<ul>
<li>A person walks from the bottom of a hill to the top. The next day the person walks from the top of the hill to the bottom. If both journeys start and stop at the same time, there exists a place where the person is at the same time on both days.</li>
<li>If a paper is crumpled and dropped onto another identical, flat sheet, then flattened, there exists a point that is directly above itself. </li>
<li>If a floor plan of a building is taken into the building, there is always a point that can be marked "You are here".</li>
<li>Etc, etc etc.</li>
</ul>

<p>I have seen similar illustrations for the hairy-ball theorem, and the ham-sandwich theorem. I am interested in Tarski's Fixed Point Theorem.</p>

<blockquote>
  <p><strong>Theorem.</strong> A monotone function on a complete lattice has a fixed point. </p>
</blockquote>

<p>Are there intuitive, physical-world examples that illustrate this theorem? </p>

CC BY-SA 3.0
2013-08-11T17:59:33.263Z
What is an "intuitive, physical-world" instance of a complete lattice?  Finite partial orders are arguably the most natural, but they lack the "surprise factor" of the continuous examples cited. For instance: suppose $n$ tennis players are ranked from $1$ to $n$ (with ties possible, so $1,2,2,2,5$ is a possible ranking of 5 players).  After the next tournament, no player drops in rank.  Then someone must retain their rank.
András Salamon
CC BY-SA 3.0
2013-08-11T19:05:33.300Z
Is is the lattice n elements with less-than or all possible rank functions? If no player drops rank, the function mapping current to next rankings is non-decreasing but not necessarily monotone.
Vijay D
CC BY-SA 3.0
2013-08-12T08:06:46.710Z
I was intending a "real-world" interpretation, with the usual order.  But this is not a good example; I don't think anyone would be surprised that there is a player that retains their rank.
András Salamon
CC BY-SA 3.0
2013-08-13T07:18:03.103Z
Actually I was saying that your example does not illustrate the fixed point theorem. An update in the ranking satisfies that x <= f(x) but a function f with this property need not be monotone (eg {3 -> 1, 2-> 2, 1 -> 1} is not monotone), so the existence of a fixed point does not follow from Tarski's theorem.
Vijay D
CC BY-SA 3.0
2013-08-13T12:34:11.807Z
You are right, my condition "no player drops in rank" is too simplistic.  I was trying to capture "monotone" using a simple English phrase, but this isn't it.  Something like "every pair of players retains their relative ranks" is closer, but yet more text is needed to deal with $x<y$ vs. $x=y$.
András Salamon
<teaching><lattice><order-theory>
Illustrative Examples of Tarski's Fixed Point Theorems
Vijay D
# Illustrative Examples of Tarski's Fixed Point Theorems

(Asked by: username_0 on 2013-08-10)

I have come across many informal examples that provide a physical illustration for Brouwer's fixed point theorem (some due to Brouwer himself).

* A person walks from the bottom of a hill to the top. The next day the person walks from the top of the hill to the bottom. If both journeys start and stop at the same time, there exists a place where the person is at the same time on both days.
* If a paper is crumpled and dropped onto another identical, flat sheet, then flattened, there exists a point that is directly above itself.
* If a floor plan of a building is taken into the building, there is always a point that can be marked "You are here".
* Etc, etc etc.

I have seen similar illustrations for the hairy-ball theorem, and the ham-sandwich theorem. I am interested in Tarski's Fixed Point Theorem.

> **Theorem.** A monotone function on a complete lattice has a fixed point.

Are there intuitive, physical-world examples that illustrate this theorem?

**Tags:** teaching, lattice, order-theory
> **Comments:**
> **username_1**: What is an "intuitive, physical-world" instance of a complete lattice?  Finite partial orders are arguably the most natural, but they lack the "surprise factor" of the continuous examples cited. For instance: suppose $n$ tennis players are ranked from $1$ to $n$ (with ties possible, so $1,2,2,2,5$ is a possible ranking of 5 players).  After the next tournament, no player drops in rank.  Then someone must retain their rank.  
> **username_0**: Is is the lattice n elements with less-than or all possible rank functions? If no player drops rank, the function mapping current to next rankings is non-decreasing but not necessarily monotone.  
> **username_1**: I was intending a "real-world" interpretation, with the usual order.  But this is not a good example; I don't think anyone would be surprised that there is a player that retains their rank.  
> **username_0**: Actually I was saying that your example does not illustrate the fixed point theorem. An update in the ranking satisfies that x <= f(x) but a function f with this property need not be monotone (eg {3 -> 1, 2-> 2, 1 -> 1} is not monotone), so the existence of a fixed point does not follow from Tarski's theorem.  
> **username_1**: You are right, my condition "no player drops in rank" is too simplistic.  I was trying to capture "monotone" using a simple English phrase, but this isn't it.  Something like "every pair of players retains their relative ranks" is closer, but yet more text is needed to deal with $x<y$ vs. $x=y$.  

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