# Set Theory terminology: ' well-founded partial ordering'

I am currently dealing with a set theory problem where I am told to show that any well-founded partial ordering is also a well-ordering (in those words exactly, no other givens). I know I have the ability to solve this problem - the issue is, I am unsure exactly what a 'well-founded partial ordering' is. You see, by definition (at least, as I have learned them):

If $$R \subseteq A \times A$$ is a partial ordering, it is reflexive, transitive, and antisymmetric.

If $$R \subseteq A \times A$$ is well-founded, then $$\forall B \subseteq A(B \neq \emptyset \Rightarrow \exists x \in B \forall y \in B (\lnot yRx)).$$

(In other words, every nonempty subset of $$A$$ has a minimal element.)

The problem? For each $$B$$ and each minimal element $$x \in B$$, wouldn't this imply that $$\lnot xRx$$ for each of these minimal elements spanning across the various nonempty subsets of $$A$$? However, we stated that $$R$$ is a partial order, and hence reflexive (ergo $$xRx$$ for all $$x \in A$$). I see a contradiction here...

So is this entity known as a 'well-founded partial ordering' necessarily strict (read: irreflexive)? If so, why wouldn't the author have just stated that instead of calling it a 'well-founded partial ordering'?

Anyways, any clarification would be helpful from a set theory minded person. Also, not trying to be sarcastic or anything, I just don't see what good ambiguity brings to learning undergrad mathematics.

• According to wikipedia, "In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation." So given a partial order $R$ there is a strict order $S$ which is defined as $xSy$ iff $x\neq y$ and $xRy.$ This essentially means that your expression should end $\forall y\in B(y=x\lor \lnot yRx).$ en.wikipedia.org/wiki/Well-founded_relation – Thomas Andrews Oct 1 '19 at 17:24
• On the other hand, it is not true that any well-founded partial ordering is a well-ordering, unless you have an unusual definition of well-ordering. – Thomas Andrews Oct 1 '19 at 17:28

There are two equivalent ways of defining a partial order on a set $$A$$:
(1) a partial order is a subset $$R\subset A\times A$$ such that $$R$$ is reflexive, anti-symmetric (for every $$x,y \in A$$ if $$xRy$$ and $$yRx$$ then $$x=y$$) and transitive.
(2) a partial order is a subset $$R'\subset A\times A$$ such that $$R'$$ is anti-reflexive (for every $$x \in A$$ it is the case that $$x\not R'x$$) and transitive.
Indeed, if you have an $$R'$$, the corresponding $$R$$ is $$R' \cup \{(x,x):x\in A \}$$ and if you have an $$R$$ you just remove the couples of the form $$(x,x)$$ in order to get the corresponding $$R'$$.