## Well Ordered Set

Hello, infinite-set reader. Set theory is back.

As I said in the previous post, this post will discuss about the well ordered set. Happy reading.

In a poset, not necessarily the poset has the smallest element. So it is with the subsets. This gives rise to a new understanding which is a special occurrence of a poset.

**Definition :**

A poset is said to be well ordered set if for any non-empty subset of the poset, it has the smallest element.

**Example :**

(i) Poset does not have the smallest element. In the subset there are also those that do not have the smallest element. For the example , simply ordered subset , , and none of them has the smallest element. So the poset is not a well ordered set.

(ii) Poset is well ordered set.

**Theorem :**

Every well ordered set is simply ordered set.

**Proof :** (more…)

## Simply/Linearly/Totally Ordered Set

Hello, infinite-set reader. Set theory is back.

Today we will discuss about simply ordered set. Sometimes also called linearly or totally ordered set. To read this post, you should read this post first. You need the definition of an initial segment to understand the theorem we are going to discuss. Happy reading.

In a poset is not necessarily between one element with another can be compared. For example is a poset but between and is incommensurable.

**Definition :**

Partial order in the set it says simple (linear, total) order in if for any apply :

Then any poset it says simply (linearly, totally) ordered set if is a simple (linear, total) order in .

Example :

(i) is simply ordered set.

(ii) Poset of the set of all natural numbers it’s also simply ordered set.

**Theorem :**

Given any simply ordered set . Then the set of all initial segments of is simply ordered by .

**Proof :** (more…)

## Initial Segment

Hello, infinite-set reader. Set theory is back.

In this post will be introduced about the initial segment. Definition of initial segment will be used in the process of proving Zorn’s Lemma. So you have to make a special mark for this post. It will be important for the future.

**Definition :**

Given any poset . For any defined with :

Furthermore is called the initial segment of defined by . If is not empty set then is called \emph{proper initial segment} of .

**Example :**

Given poset then , , and . (more…)

## Bounded Set

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

Before reading this post, you should read the previous post. The previous post concerns the lower and upper bounds of a set. The previous definition became the basic of this post. Happy reading.

If is a subset of any poset that has a lower bound then is said to be bounded from below. If it has a upper bound then is said to be bounded from above. And if it has the upper bound and the lower bound then is said to be bounded.

For example is defined as the set of all lower bound of subsets of any poset . If the largest element of the set is then single existance. Next is denoted by that is the largest lower bound of. With the same understanding, the smallest upper bound of A is denoted by . Notice that the and are in . So the existence and not guaranteed in . To confirm this statement, please refer to the following example.

**Example :**

Given is subsets of poset . We get but .

Now, it will then be given the theorem of a poset relating to the smallest upper and lower bounds.

**Theorem :**

(i) Poset has the smallest element if and only if the empty set has the smallest upper bound in .

(ii) Poset has the largest element if and only if the empty set has the largest lower bound in .

**Proof :** (more…)

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