Ordered set which is not linear

[Silviu]

Hello! I was introduced in the Real Analysis class the notion of ordered set. I am not sure I understand the concept of linear and non-linear ordering. Can someone explain this to me? Thank you!

 

[fresh_42]

Hello! I was introduced in the Real Analysis class the notion of ordered set. I am not sure I understand the concept of linear and non-linear ordering. Can someone explain this to me? Thank you!

As I see it, is linear ordering a rarely and in my view misleading term for the common total ordering, which means, it is an order (reflexive, transitive, anti-symmetric) and any of two elements $x,y$ can be compared by either $x \prec y$ or $y \prec x$, e.g. the usual order of real numbers. It is not the case with the order by inclusion of subsets of a set. Here we can have $x \nprec y$ and $y \nprec x$.

 

[math771]

Sure, I'd be happy to explain it to you! In an ordered set, also known as a partially ordered set, there is a relation between the elements that determines their order. This relation can be linear or non-linear.

In a linear ordering, every pair of elements in the set can be compared to each other. This means that for any two elements, one is always greater than or equal to the other. Think of it like a number line, where every number has a clear position and can be compared to any other number. Examples of linearly ordered sets include the set of real numbers and the set of integers.

On the other hand, in a non-linear ordering, there may be elements that cannot be compared to each other. This can happen when the relation between elements is not transitive, meaning that if A is greater than B and B is greater than C, it does not necessarily mean that A is greater than C. An example of a non-linearly ordered set is the set of all shapes, where a triangle cannot be compared to a square in terms of "greater than" or "less than."

I hope this helps clarify the concept of linear and non-linear ordering in an ordered set. Let me know if you have any other questions!