What is the point of a partial order?

[CuriousBanker]

I have a feeling the question I am about to ask, I won't be able to ask it the way I am trying to...but I will try. I will break it up into two questions.

1) What are its real life applications? Much easier for me to get it when I can see a real life application
2) Why is it the way it is? I get that it is reflexive, transitive, and antisymmetric. Why is that an important combo though? Why did the person who create partial orders decide they should be antisymmetric rather than symmetric? Is there something about the combo of reflexive, transitive, and antisymmetric that is good?

If you are reading this and thinking "What the hell is this guy asking", then I am sorry, lol.

 

[MarneMath]

1) Programming. One common example would be scheduling task or organizing data from different time zones and the dates. *Hint think of conditional statements.

2) I really don't have a good answer for this, not really a set theory person. I imagine the answer has to do with the fact that it made sense. It seems like this properties would have to be true if you were going to attempt to generalize the total set. It seems natural to me.

 

[CuriousBanker]

Thanks for the reply! Not sure I quite get it though...Organizing data from different time zones. How is that reflexive? How is it antisymmetric? Isnt antisymmetric when if x is y, y is not x? Well if it is 9 am on the east coast it is 6 am west coast, and if it is 6 am west coast it is 9 am east coast...isn't that symmetric and therefore not a partial order?

 

[Stephen Tashi]

1) What are its real life applications? Much easier for me to get it when I can see a real life application

If you consider doing abstract mathematics "real life", the answer is straightforward. It often happens that one mathematical object is a sub-object of another.

For example, a group can be a subgroup of another group, a vector space can be a subspace of another vector space, a graph can be a subgraph of a graph. In may cases the fact that A is sub-object B requires that some set associated with A be a subset of a set associated with B. To this are added other conditions. For example, if A is a subgroup of B then the elements in A are a subset of the elements of B. However, the subset relation of A to B is not enough by itself to make A a subgroup of B. So the relation "A is a subgroup of B" is not the same relation as "A is a subset of B".

The question naturally arises whether the relation "A is a sub so-and-so of B" behaves somewhat like the relation "A is a subset of B". That leads to the question of how to abstract the important properties of the relation "A is a subset of B". Doing that leads you to the definition of a partial order.

There is no universal rule about whether the relation "A is a sub-object of B" is a partial order. It depends on technical details of how the subobject relation is defined, but for many important types of mathematical objects the relation "A is a sub-object of B" does have properties that imitate the properties of "A is a subset of B". So there are lots of important partial orders in theoretical mathematics.

 

[nomadreid]

Stephen Tashi's answer is spot on when looking at mathematical objects.

To bring it more towards the everyday, one can think of a hierarchy in a company. We "measure" each person by two factors: his/her salary, and his/her identity. That is, let's say that A ≤ B if and only if
either
A's salary is less than B's,
or
A is the same person as B
(exclusive OR, obviously).
Now we have a set which is reflexive, antisymmetric, and transitive. But if your set contains Sam and Alex who have the same salary but are different persons, this is not a linear order, as Sam and Alex are incomparable.
When you think about political correctness, similar situations are very common in "real life".

 

[lugita15]

I have a feeling the question I am about to ask, I won't be able to ask it the way I am trying to...but I will try. I will break it up into two questions.

1) What are its real life applications? Much easier for me to get it when I can see a real life application
2) Why is it the way it is? I get that it is reflexive, transitive, and antisymmetric. Why is that an important combo though? Why did the person who create partial orders decide they should be antisymmetric rather than symmetric? Is there something about the combo of reflexive, transitive, and antisymmetric that is good?

If you are reading this and thinking "What the hell is this guy asking", then I am sorry, lol.

CuriousBanker, it may be easier to understand what a partial order is if you know what a total order is. A total ordering of a set is a relation that has the same basic properties as "less than or equal to" for numbers:
1. A total order must be reflexive, because a number is always less than or equal to itself.
2. It's impossible for both a≤b and b≤a to be true if a≠b, so a total order must be antisymmetric. 3. Since a≤b and b≤c implies that a≤c, a total order must be transitive.
4. For any numbers a and b, either a≤b or b≤a, so a total order must have that same property, called being total.

Now it turns out that some relations satisfy conditions 1, 2, and 3, so they have almost all the properties of being a total order, except they don't satisfy number 4. So it's like a total order without the "total" part, which is why we call it a "partial" order.

Here's an example of a partial order: instead of "less than or equal to" for numbers, let's consider the relation "ancestor or equal to" for people. Since you're obviously equal to yourself, you are an ancestor or equal to yourself, so "ancestor or equal" is reflexive. If your grandfather is an ancestor or equal to you, you can't also be an ancestor or equal to your grandfather, so "ancestor or equal" is antisymmetric. If your grandfather is an ancestor or equal to you, and you are an ancestor or equal to your grandson, then your grandfather is an ancestor or equal to your grandson, so "ancestor or equal" is transitive.

But presumably you're not the ancestor or equal to your brother, and he's not the ancestor or equal to you, so condition 4 above is not satisfied, and thus "ancestor or equal" is not total, so it's a partial order.

Does that make sense?