Partial order, where every interval is a powerset

Question

We are studying a partial order P with the following specific properties:

1. P is countably infinite.
2. There are infinitely many isolated elements (not comparable to any other element).
3. Every chain is finite.
4. There is a grading g: P -> N_0 (natural numbers), such that
   • x < y implies g(x) < g(y)
   • if y covers x (y > x, and nothing between), then g(y) = g(x) + 1.
5. If x <= y, then the interval [x,y] is isomorphic to the powerset of g(y) - g(x) elements (and the subset-relation).

We can't find such partial orders in the literature, and so we wonder whether something is known here? Especially Property 5 seems relevant.

P comes from a logical-combinatorial situation. The motivation for this question is, whether the above properties can relate our investigations to the theory of partial orders (or whether there seem to be no deeper relations)?

Answer 1

Let $(P; \preceq)$ be the poset $\bigcup_{n \in \mathbb N} \{n\} \times X_n$, where $X_n = \{ x \subseteq \mathbb N \mid x \neq \emptyset \wedge \operatorname{card}(x) \le n \}$, such that for $(m,x), (n,y) \in P$ $$ (m,x) \preceq (n,y) : \iff m = n \wedge x \subseteq y. $$ Furthermore, let $$ g \colon P \to \mathbb N_{0}, (n,x) \mapsto \operatorname{card}(x). $$

1. $P$ is the union of countably many countable sets and hence countable. (In fact, $P$ is easily seen to be countable without any choice.)
2. For every $k \in \mathbb N$ the set $(1,\{k\})$ is an isolated point of $P$.
3. Clearly every chain in $P$ is finite.
4. holds trivially.
5. Identify, for any $(n,x) \preceq (n,y)$ the interval $[(n,x), (n,y)]_{\preceq}$ with $[x,y]_{\subseteq}$ and note that the latter is canonically isomorphic to $(\mathcal{P}(\operatorname{card}(y \setminus x)), \subseteq)$.