Question

what is partial ordered set?

Answer 1

Basically, a set that has a partial ordering imposed on it.

It's similar to a totally ordered set, but not every pair of elements from the set has to be able to be related.

Answer 2

can you please give me an example ?

Answer 3

A great example of partially ordered set (posets) is subset inclusion. {1, 2} is a subset of
{1, 2, 3}, and so it {2, 3}. However, {1, 2} is not a subset of {2, 3} and vice versa. So {1, 2} and {2, 3} are not related, but they are both related to {1, 2, 3}

Answer 4

okkk sot {1,2} & {2,3} are posets?

Answer 5

@KingGeorge please help me

Answer 6

are {1,2} & {2,3} posets of {1,2,3}?

Answer 7

{{1, 2}, {2, 3}, {1,2, 3}} would be the poset. You have a partially ordered set containing other sets, that are ordered by inclusion.

Answer 8

If you want a formal definition of a partial order:

It's a binary relation "\(≤\)" over a set \(P\) which is reflexive, antisymmetric, and transitive, i.e., for all \(a, b, c \in P\), we have that:
\(a ≤ a\) (reflexivity);
if \(a ≤ b\) and \(b ≤ a\) then \(a = b\) (antisymmetry);
if \(a ≤ b\) and \(b ≤ c\) then \(a ≤ c\) (transitivity).

Answer 9

if a≤b and b≤a then a=b (antisymmetry); then what is maximal and minimal ?

Answer 10

A maximal element \(a\) is an element in your set such that there is no element \(b\) so that \(a\leq b\).

A minimal element is an element \(c\) such that there is no element \(d\) such that \(d\leq c\).

It's important to note that a poset can have many maximal or minimal elements.

Answer 11

@KingGeorge you are the true king of thrones thanks a tonne for explaining poset so well

Answer 12

You're welcome.

Answer 13

@KingGeorge what is inclusion?

Answer 14

Sorry, I had to go get food.

Set inclusion is more commonly referred to as subsets.

Answer 15

no problem, thanks a tonne for the reply