Well Orders

A well order (also referred as well-ordering or well-order relation) is a special type of total order where every non-empty subset has a least element. A set with a well-order relation is called a well-ordered set.

For example, the set of natural numbers $$\mathbb{N}$$ under the usual order relation $$\le$$ forms a well-ordered set.

By definition, any well-ordered set is totally ordered. However, the converse is not true - the set of integers $$\mathbb{Z},$$ which is totally ordered, is not well-ordered under the standard ordering (since $$\mathbb{Z}$$ itself and some its subsets do not have least elements). Although, any finite totally ordered set is well-ordered.

In a well-ordered set, every element (except a possible greatest element) has a unique successor. However, not every element of a well-ordered set needs to have a predecessor.

The well-ordering theorem (also known as Zermelo's theorem) states that every set may be well-ordered. If so, we can find an order on the set of integers $$\mathbb{Z}$$ which makes it well-ordered. For example, instead of the regular order relation $$\le,$$ we can define the following order:

${0 \preccurlyeq - 1 \preccurlyeq 1 \preccurlyeq - 2 \preccurlyeq 2 \preccurlyeq - 3 \preccurlyeq 3 \preccurlyeq \ldots}$

That's a well order relation on $$\mathbb{Z},$$ in which $$0$$ is the least element.

The well-ordering theorem is equivalent to the axiom of choice.

Let $$A$$ and $$B$$ be two partially ordered sets. If there is a function $$f : A \to B$$ such that, for every $$x, y \in A,$$

$x \le y \Rightarrow f\left( x \right) \le f\left( y \right),$

then the sets $$A$$ and $$B$$ are said to be order-isomorphic. Isomorphic sets are denoted as $$A \cong B.$$

Order isomorphism preserves well-ordering

See solved problems on Page 2.