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Poset – Partially Ordered Set – Part 2

Hallo, infinite-set readers. Set theory is back.

We return to the previous post that any partial order P is denoted by \leq and \left(x,y\right)\in\leq often denoted by x\leq y. Today we will discuss about the partially ordered set. If you forget the previous post, you can go to this link. Happy reading.

Definition :

The pair \left(S,\leq\right) is said to be partially ordered set if \leq is the partial order in the set S.

If \left(S,\leq\right) is a partially ordered set then it is said that S partial order by \leq. Furthermore to further abbreviate the mention, partially ordered set \left(S,\leq\right) will be called a poset. Suppose given form :

    \[ \left\{ \ldots,\left(\ldots;a;b;\ldots;c;\ldots\right),\left(\ldots;m;n;\ldots\right),\ldots\right\} \]

then the intent of the above form is a partially ordered set \left(S,\leq\right) with :

    \[ S=\left\{ \ldots,a,b,\ldots,c,\ldots,m,n,\ldots\right\} \]

and \leq is partial order with x\leq y if x is on the left y under the condition x,y are in the same area / in the same brackets.

Example : (more…)

Poset – Partially Ordered Set – Part 1

Hallo, infinite-set readers. Set theory is back. Already reading the about relation? Today we will use that chapter. I hope you already know about relation.

In this post I will actually discuss about Zorn’s Lemma. But before that, it will be discussed first about the partially ordered set, simply ordered set, and well ordered set. For this post will be given first about the opening of the partially ordered set. Happy reading.

Definition :

Given any set S. A subset P\subseteq S\times S is called \emph{partial order} in set S if for any x,y,z\in S apply :

(i) \left(x,x\right)\in P (reflexive)

(ii) if \left(x,y\right)\in P and \left(y,x\right)\in P then x=y (antisymmetric)

(iii) if \left(x,y\right)\in P and \left(y,z\right)\in P then \left(x,z\right)\in P (transitive)

Example :

(a) Given S=\left\{ a,b,c\right\} then P=\left\{\left(a,a\right),\left(a,b\right),\left(b,b\right),\left(c,c\right)\right\} is partial order in S.

Proof :

(i) Reflexive

Therefore for any x\in S applies \left(x,x\right)\in P then P is proven reflexive.

(ii) Antisymmetric

Look at the P above. Therefore if x\neq y applies \left(x,y\right)\in P\Rightarrow\left(y,x\right)\notin P then for any x,y\in S statement \left(x,y\right)\in P\wedge\left(y,x\right)\notin P\Rightarrow x=y is true. Proven P is antisymmetric.

(iii) Transitive

Look at the P above. For x\neq z if known \left(x,y\right)\in P and \left(y,z\right)\in P then apply one of the following two things, i.e x=y or y=z. Consequently, for any x,y\in S if \left(x,y\right)\in P and \left(y,z\right)\in P then apply \left(x,z\right)\in P. Proved P is transitive.

So, P is partial order in S.

Example : (more…)