Let $P$ be a finite poset. Let $\mathcal{A}$ denote the set of antichains of $P$. Equip $\mathcal{A}$ with a partial order $\preceq$ whereby $X \preceq Y$ means for all $x \in X$ there exists $y \in Y$ such that $x \leq_P y$. Then it is easy to see that $(\mathcal{A},\preceq)$ is isomorphic to $J(P)$, the distributive lattice of order ideals of $P$ (just send an antichain to the order ideal it generates).

Let $\mathcal{A}_{\mathrm{max}} := \{A \in \mathcal{A}\colon \#A=\mathrm{max}(\{\#B\colon B \in \mathcal{A}\})\}$, i.e., $\mathcal{A}_{\mathrm{max}}$ is the set of antichains of $P$ of maximal cardinality. Then it is known that $\mathcal{A}_{\mathrm{max}}$ is a sublattice of $\mathcal{A}$ with respect to the partial order $\preceq$ (see the paper of Freese linked to below). In particular, $\mathcal{A}_{\mathrm{max}}$ is a distributive lattice, and hence by Birkhoff's Fundamental Theorem of Finite Distributive Lattices, has the form $\mathcal{A}_{\mathrm{max}}=J(P')$ for some other (unique up to isomorphism) finite poset $P'$, namely the induced subposet of join-irreducible elements.

The map $P \mapsto P'$ defines a self-map on the class of all finite posets. This map is far from injective: for instance any poset with a unique antichain of maximal cardinality (for instance $P=n\underline{\mathbf{1}}$ an antichain) is sent to the empty poset. On the other hand, for $P=\underline{\mathbf{n}}$ a chain, we have $P' = \underline{\mathbf{n-1}}$. Also, experimentally it seems that with $P = \underline{\mathbf{a}} \times \underline{\mathbf{b}}$ a product of chains, with $a \geq b$, we have $P' = \underline{\mathbf{a-b}} \times \underline{\mathbf{b}}$. (And it looks like similar behavior may propagate to product of three chains and beyond.)

Broad question: How can this map $P \mapsto P'$ be understood? Is there a "simpler" description than the one I have given? Are there other families of posets on which it exhibits interesting behavior? (EDIT: Other questions along these lines: Is the map surjective?, Is $P'$ always isomorphic to a subposet of $P$?, et cetera.)

Specific question: Is it true that this map has $\underline{\mathbf{a}} \times \underline{\mathbf{b}} \mapsto \underline{\mathbf{a-b}} \times \underline{\mathbf{b}}$? What about products of many chains?

Freese, Ralph, An application of Dilworth’s lattice of maximal antichains, Discrete Math. 7, 107-109 (1974). ZBL0271.05011.