Let $(P, \succeq)$ be a poset (you can assume $\succeq$ is not empty). I am interested in finding a poset $(X, \succeq^*)$ that embeds $(P, \succeq)$, and $\forall x,y \in X$ both of the following properties are satisfied:

1. If $x\succ^* y$, then $\exists z\in X$ such that $x \succ^* z$, and $z$ and $y$ are not greater than or equal to the same non-zero element in $X$.
2. If $S_{x,y}=\{z| x\succeq^* z \quad and \quad y \succeq^* z\}\neq \emptyset$, then $S_{x,y}$ has a greatest element.

If it's easier, you can replace the second property by requiring $x\wedge y$ to exist for every $x,y \in X$.

For example, a lattice constructed by the lower cones of the elements of $P$ (i.e., the canonical embedding in a Boolean algebra) satisfies the above. But is there another embedding that is perhaps smaller, in which the maximal elements of $(P, \succeq)$ continue to be maximal? Is it possible to construct the embedding in a single-shot (like the canonical Boolean embedding), rather than characterizing it through an algorithm whose outcome is not known at the outset?

I hope this question makes sense and thanks for your help!

Let $(P, \succeq)$ be a poset (you can assume $\succeq$ is not empty). I am interested in finding a poset $(X, \succeq^*)$ that embeds $(P, \succeq)$, and $\forall x,y \in X$ both of the following properties are satisfied:

1. If $x\succ^* y$, then $\exists z\in X$ such that $x \succ^* z$, and $z$ and $y$ are not greater than or equal to the same non-zero element in $X$.
2. If $S_{x,y}=\{z| x\succeq^* z \quad and \quad y \succeq^* z\}\neq \emptyset$, then $S_{x,y}$ has a greatest element.

If it's easier, you can strengthen the second property by requiring $x\wedge y$ to exist for every $x,y \in X$.

For example, a lattice constructed by the lower cones of the elements of $P$ (i.e., the canonical embedding in a Boolean algebra) satisfies the above. But is there another embedding that is perhaps smaller, in which the maximal elements of $(P, \succeq)$ continue to be maximal? Is it possible to construct the embedding in a single-shot (like the canonical Boolean embedding), rather than characterizing it through an algorithm whose outcome is not known at the outset?

I hope this question makes sense and thanks for your help!

Let $(P, \succeq)$ be a poset (you can assume $\succeq$ is not empty). I am interested in finding a poset $(X, \succeq^*)$ that embeds $(P, \succeq)$, and $\forall x,y \in X$ both of the following properties are satisfied:

1. If $x\succ^* y$, then $\exists z\in X$ such that $x \succ^* z$, and $z$ and $y$ do not cover the same non-zero element in $X$.
2. If $S_{x,y}=\{z| x\succeq^* z \quad and \quad y \succeq^* z\}\neq \emptyset$, then $S_{x,y}$ has a greatest element.

If it's easier, you can relax the second requirement by requiring $x\wedge y$ exists for every $x,y \in X$.

For example, a lattice constructed by the lower cones of the elements of $P$ (i.e., the canonical embedding in a Boolean algebra) satisfy the above. But is there another embedding that is perhaps smaller, in which the maximal elements of $(P, \succeq)$ continue to be maximal? Is it possible to construct the embedding in a single-shot (like the canonical Boolean embedding), rather than characterizing it through an algorithm whose outcome is not known at the outset?

I hope this question makes sense and thanks for your help!

Let $(P, \succeq)$ be a poset (you can assume $\succeq$ is not empty). I am interested in finding a poset $(X, \succeq^*)$ that embeds $(P, \succeq)$, and $\forall x,y \in X$ both of the following properties are satisfied:

1. If $x\succ^* y$, then $\exists z\in X$ such that $x \succ^* z$, and $z$ and $y$ are not greater than or equal to the same non-zero element in $X$.
2. If $S_{x,y}=\{z| x\succeq^* z \quad and \quad y \succeq^* z\}\neq \emptyset$, then $S_{x,y}$ has a greatest element.

If it's easier, you can replace the second property by requiring $x\wedge y$ to exist for every $x,y \in X$.

For example, a lattice constructed by the lower cones of the elements of $P$ (i.e., the canonical embedding in a Boolean algebra) satisfies the above. But is there another embedding that is perhaps smaller, in which the maximal elements of $(P, \succeq)$ continue to be maximal? Is it possible to construct the embedding in a single-shot (like the canonical Boolean embedding), rather than characterizing it through an algorithm whose outcome is not known at the outset?

I hope this question makes sense and thanks for your help!