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For a poset $(X,R)$, where $R$ is a partial order on $X$,

  • let $\operatorname{Inf}(R)$ be the set of all $A\subseteq X$ which have an infimum in $(X,R)$.
  • let $\operatorname{Sup}(R)$ be the set of all $A\subseteq X$ which have a supremum in $(X,R)$.

Suppose $X$ is a set. Is there any better way to charaterize the set $$\Xi=\{(\operatorname{Inf}(R),\operatorname{Sup}(R))\mid R\text{ is a partial order on }X\}$$? (better means more set theoretic, less order-theoretic)

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  • $\begingroup$ user47958. Writing $(X,R)$ for a partially ordered set is not a standard practice. I do not recall ever seeing a mathematician use $R$ for a partial ordering relation. Usually one writes $\leq$ or maybe even $\preceq$ for a partially ordering. $\endgroup$ Apr 8 '15 at 22:31
  • $\begingroup$ yes, but it seems ugly and ambiguous to write $\operatorname{Inf}(\le)$ and $\le$ rarely is a variable as above. $\endgroup$ Apr 9 '15 at 2:40
  • $\begingroup$ If you don't like writing $\textrm{Inf}(\leq)$, then you could write $\textrm{Inf}(X,\leq)$ instead. And writing $\textrm{Inf}(\leq)$ is not any more ambiguous than writing $\textrm{Inf}(R)$. $\endgroup$ Apr 9 '15 at 3:16
  • $\begingroup$ I have seen $R$ used as a partial order, but also I have used $\leq$ as a variable. $\endgroup$ Apr 9 '15 at 11:46
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I do not believe that there is a nice way to characterize $\Xi$. Furthermore, I do not see any reason to separate sets with least upper bounds or greatest lower bounds from the underlying partial ordering. In this answer, I will give cases of where the partial ordering $\leq$ is completely determined by $\mathcal{A}$ and $\mathcal{B}$. In particular, since $\mathcal{A}$ and $\mathcal{B}$ sometimes completely determine the partial order $X$, we do not obtain a simpler structure when we observe the sets $\mathcal{A}$ and $\mathcal{B}$ as opposed to the partial ordering $\leq$.

Suppose that $X$ is a poset. Let $\preceq^{\circ}$ be the ordering on $X$ where $x\preceq^{\circ}y$ if and only if $A\cup\{x\}\in\mathcal{A}$ implies $A\cup\{y\}\in\mathcal{A}$. Notice how $\preceq^{\circ}$ only depends on $\mathcal{A}$ and not the partial ordering. It is clear that $\preceq^{\circ}$ is a preordering on $X$. Furthermore, if $X$ is a join-semilattice, then $x\preceq^{\circ}y$ whenever $x\leq y$. Define $\preceq$ to be the relation where $x\preceq y$ if and only if $x\preceq^{\circ}y$ and $x\preceq_{\circ}y$. If $X$ is a lattice, then $x\preceq y$ whenever $x\leq y$. In fact, as the following proposition shows, there are some lattices such that $x\preceq y$ if and only if $x\leq y$.

$\mathbf{Proposition}$. Suppose $X$ is a Heyting algebra such that if $x<y$, then there is some non-empty set $R\subseteq\{z\in X|x\leq z\leq y\}$ that does not have a least upper bound. Then $x\leq y$ if and only if $x\preceq^{\circ}y$.

$\mathbf{Proof}$. We only need to show that if $x\not\leq y$, then $\neg(x\preceq^{\circ}y)$. To prove this result, we will need a simple fact about Heyting algebras. In a Heyting algebra, if $\bigvee R$ exists and $a\in X$, then $\bigvee_{r\in R}(a\wedge r)$ exists and $a\wedge\bigvee R=\bigvee_{r\in R}(a\wedge r)$. Suppose that $x\not\leq y$. Then $y\wedge x<x$, so there is some non-empty set $R\subseteq\{z\in X|y\wedge x\leq z\leq x\}$ with no least upper bound. Then $\bigvee(R\cup\{x\})=x$, so $R\cup\{x\}\in\mathcal{A}$. However, $R\cup\{y\}\not\in\mathcal{A}$. Otherwise, if $R\cup\{y\}\in\mathcal{A}$, then $\bigvee_{a\in R\cup\{y\}}(a\wedge x)$ exists. However, it is clear that if $\bigvee_{a\in R\cup\{y\}}(a\wedge x)$ exists, then $\bigvee_{a\in R}(a\wedge x)=\bigvee R$ exists, a contradiction. Therefore we conclude that $\neg(x\preceq^{\circ}y)$.

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