I am currently studying equivalents of the Axiom of choice such as the well ordering theorem and the Zorn's lemma. I understand partially and linearly ordered set is a primitive definition in both definitions.
So this are the definitons i have got so far:
A partial ordering on a non-empty set $S$ is a binary relation $R$ such that for all $ a, b , c \in S$ satisfies:
- $R$ is reflexive, i.e., $aRa$.
- $R$ is antisymmetric, i.e., if $aRb$ and $bRa$ then $ a = b $.
- $R$ is transitive, i.e., if $aRb$ and $bRc$ then $aRc$.
A partially ordered set, $(S,R)$ is a linear ordered set or a totally ordered set if any two elements in $S $ is comparable
A linearly ordered set $(S,\leq)$ is a well ordered set, if for every $A \subseteq S$, there exists an element $s_0$ where $s_0 < s \forall s \in T$.
From my understanding when we consider a partially ordered set or a linealy ordered set the relation it's defined under can be any binary relation such as addition, subset etc. However, when considering a well ordered set we only consider the partially ordered set (S,<), i.e. the set under the less than ordering.
This is also the case when we look at upper bound and maximal element. The partially ordered set we consider is the set defiend under the less than relation
Is this right?
Thank you in advance.
You appear to believe that $\le$ has some specific meaning - perhaps you're thinking that $a \le b$ if and only if $b - a$ is nonnegative, for example.
On the contrary, $\le$ is used as a symbol to indicate any relation that is intended to be interpreted as some kind of ordering. The meaning of $\le$ will be defined before it is used. It need not be a relation on $\mathbb{N}$ or $\mathbb{R}$; I could define the relation-class $\le$ on the class of all sets by $X \le Y$ if and only if $X$ injects into $Y$, for example. Or I could define $\le$ on the genuine set $\{\text{the ordinals less than $\alpha$}\}$ by $\beta \le \gamma$ iff $\beta$ is isomorphic to an initial segment of $\gamma$.