# Non-Mathematical Examples of Orders

Different properties of different types of orders including partial, total, scattered and well-orders are a part of any graduate/undergraduate set theory course. I am looking for interesting "non-mathematical" examples of these orders in physics, biology, sociology, politics, etc.

Question: What are some interesting "non-mathematical" examples of orders and other mathematical objects of a same nature? Any useful reference is welcome.

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What is a "scattered order"? – mbork Jun 19 '14 at 20:31

Preference relations in economics. e.g. I prefer Big Macs to Whoppers (denoted $\mathrm{Big Mac} \succ \mathrm{Whopper}$) and that induces a lattice on the space $\{\mathrm{BigMac},\mathrm{Whopper},\mathrm{Baconator}\}$. This idea has applications in choice theory, matching, and game/auction theory.

The preference-ordering framework is notably used to prove Arrow's Impossibility Theorem.

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Baconator? How have I lived this long and never heard of this? I pass a Wendy's almost daily. – JoeTaxpayer Jun 18 '14 at 0:41

Lexicographical (alphabetical) order is an example of total order, and it can be helpful to introduce a total order in $\mathbb{R}^n$

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The question was asking for examples in fields outside of mathematics, such as the natural sciences or social sciences. A lexicographic order on $\mathbb{R}^n$ doesn't really seem to fit the question. – mweiss Jun 17 '14 at 21:44
@mweiss I meant the usual alphabetical order used in dictionaries, which can also be useful to introduce total order in mathematics. – DGRasines Jun 17 '14 at 21:48
Oh, I see what you mean now. – mweiss Jun 17 '14 at 21:52

Wikipedia names genealogical descendancy as a partial order.

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+1. The transitive closure of parent-child relation. – Jyrki Lahtonen Jun 17 '14 at 16:05

Some examples:

• Pieces of clothing, shoes, etc., for example you put socks before shoes.
• Building towers using toy blocks, e.g. you need to use some blocks to form a base, then put some elements in the middle and so on.
• Any tree-like structure induces an ordering, for example genealogical tree, or even an ordering of leaves of some real tree if there's no inosculation.
• Subjects to learn, i.e. how more advanced subjects list other as prerequisites.
• Cooking, for example to bake a cake you need the dough and some topping which are prepared separately (each using its own steps) and joined later. Some recipes are really complicated!
• Any class of things ordered by its features, e.g. cars (speed, acceleration), buildings (height), mobile devices (screen size, processor, camera). Yet, this is not very non-mathematical.
• As for orders without lower bound, I would recommend human stupidity (and with its multitude of flavors, it might be even a partial order).
• As for total orders, you could try students by the time they can sit straight without making any noise.
• For more exotic orderings, one could use the Dershowitz-Manna ordering to get nice results:
• let's assume that we partially order chess pieces by relative strength: $$\text{Pawn} < \text{Knight},\text{Bishop} < \text{Rook} < \text{Queen}$$
• we can order some two (multi)sets $A$ and $B$ of chess pieces $A < B$ by $$\text{if A includes more pieces of some kind, then B contains } \\\text{more pieces of some other kind that dominates the first.}$$

I hope this helps $\ddot\smile$