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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 at 20:31

6 Answers 6

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 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 at 21:44
@mweiss I meant the usual alphabetical order used in dictionaries, which can also be useful to introduce total order in mathematics. –  dgrasines517 Jun 17 at 21:48
Oh, I see what you mean now. –  mweiss Jun 17 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 at 16:05

The classic example of a non-mathematical total ordering is people's heights. Each person is either taller or shorter than another. And if Alice is taller than Bob, who is taller than Carol, then Alice must be taller than Carol as well. Although measuring their heights converts this example into a numerical comparison, it's not strictly "mathematical" in nature.

An example of a partial ordering might be a corporate hierarchy. A CEO commands multiple managers, each of whom supervises multiple employees. Although a manager is strictly superior to his own employees, he may or may not have any relation to the employees that other managers supervise.

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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 $\ddot\smile$

Edit: Added two more orders.

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Anything that can be modelled by a directed acyclic graph is an order if extended suitably (transitive closure). So relations like "is the boss of," "branches off from" in a tree, "descends from" in genealogy (but be sure to consider both mother and father to make for more interesting examples).

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