I found that GTM 27 (General Topology by John Kelley), GTM 73 (Algrbra by Hungerford), Topology by Munkres gave three different formulas of Zorn's Lemma. Are they equivalent?
All three books state Zorn's Lemma as: A partially ordered set has a maximal element, if every chain (linearly ordered subset) in it has an upper bound. But their definitions of terms appeared in this lemma are slightly different.
(1) GTM 27's definition (K-definition): A partially ordering relationship $<$ is a transitive relationship. $X$ is partially ordered by $<$. $x$ is an upper bound of $A\subseteq X$, if $y\in A\implies (y<x)\vee (y=x)$. $x\in A$ is a maximal element of $A$ if $y\in A\implies y<x\vee x\not<y$. A linearly oredering relationship $<$ is a partially ordering relationship subject to (a) $x<y\wedge y<x\implies y=x$ (b) $x\ne y$, $x, y\in\{a\in X|\ \exists b\in X,\ a<b \vee b<a\}\implies x<y\vee y<x$.
(2) GTM 73's definition (H-definition): A partially ordering relationship $\le$ on $X$ is a transitive relationship, and in addition (a) $x\le x$ $\forall x\in X$ (b) $x\le y\wedge y\le x\implies x=y$. $x$ is an upper bound of $A\subseteq X$, if $y\in A\implies y\le x$. $x\in A$ is a maximal element of $A$ if $y\in A\implies y\le x\vee x\not\le y$. A linearly oredering relationship $\le$ is a partially ordering relationship subject to $x,y\in X\implies x\le y\vee y\le x$.
(3) Munkres's definition (M-definition): A partially ordering relationship $<$ on $X$ is a transitive relationship, and in addition $x\not<x$ $\forall x\in X$. $x$ is an upper bound of $A\subseteq X$, if $y\in A\implies (y<x)\vee (y=x)$. $x\in A$ is a maximal element of $A$ if $\nexists y\in A\ x<y$. A linearly oredering relationship $\le$ is a partially ordering relationship subject to $x,y\in X,\ x\ne y\implies x< y\vee y< x$.
The K-Zorn's Lemma obviously implies another two formulas of Zorn's Lemma. Can we prove M$\implies$ K, H$\implies$ K, and M$\iff$ H?
Best Answer
Well, the confusion comes from different notions called partial orders by different materials.
Hungerford's partial order is a usual partial order, and Munkres' partial order is a strict partial order. (Ignore the totality or connected property on the liked page.) Kelley's "partial order" is just a transitive relation.
It is known that you can always construct a strict partial order from a usual partial order, and vice versa. For example, if $<$ is a strict partial order, then $\le$ defined by $$x\le y \iff (x<y)\lor (x=y)$$ is a usual partial order. Such-and-such manipulation shows these two notions are in some sense equivalent, and it will give the equivalence of the latter two (that is, Hungerford's and Munkres')
What about Kelley's one? Well, let us turn the transitive relation $<$ to a preorder by taking $$x\le y \iff (x<y)\lor (x=y).$$ Clearly, $\le$ is reflective and transitive. Now consider the separative quotient $X/\sim$, where $x\sim y$ iff $x\le y$ and $y\le x$. Then $(X/\sim ,\le)$ becomes a usual partial order, and you can see that Munkres' Zorn's lemma applied to $(X/\sim ,\le)$ shows Kelley's Zorn's lemma.