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 $\mathcal{A}_{\mathrm{max}} := \{A \in \mathcal{A}\colon \forall_{B\in\mathcal{A}}\#B\le\#A\}$. 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.

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    $\begingroup$ In her Ph.D. thesis at dspace.mit.edu/handle/1721.1/104603?show=full, Efrat Engel computes $P'$ for some posets $P$. Also, the surjectivity of the map $P\to P'$ follows from Exercise 3.72(b) of EC1. $\endgroup$ – Richard Stanley Nov 25 '19 at 20:10
  • $\begingroup$ @RichardStanley: Thanks for these great references! $\endgroup$ – Sam Hopkins Nov 25 '19 at 20:16
  • $\begingroup$ (I realized the question about fixed points was silly: unless $P$ is empty, of course $J(P)$ will be strictly bigger than $\mathcal{A}_{\mathrm{max}} \subseteq J(P)$ because $J(P)$ has the emptyset, but $\mathcal{A}_{\mathrm{max}}$ won't.) $\endgroup$ – Sam Hopkins Nov 26 '19 at 1:06

The references given by Richard Stanley answer most of my array of questions.

In particular, Koh showed that the map $P \mapsto P'$ is surjective in "On the lattice of maximum-sized antichains of a finite poset", and gave some more information about the fibers of this map in "Maximum-sized antichains in minimal posets" (both linked below).

Also, in Engel's MIT PhD thesis (available online), she computes $P'$ for various posets $P$, including the case of the product of three chains; she describes (in Section 3.3) the distributive lattice in question as the lattice of Semistandard Young Tableaux of rectangular shape $c^r$ with entries at most $n$ with coordinate-wise order, but it is easy to see that this is the same as $J(\underline{\mathbf{c}} \times \underline{\mathbf{r}}\times\underline{\mathbf{n-r}})$. (Note that she views the map in question as being of the form $J(P)\to J(P')$, but of course by the Fundamental Theorem of Finite Distributive Lattices, this is no different than $P\to P'$.)

Koh, K. M., Maximum-sized antichains in minimal posets, Algebra Univers. 20, 217-228 (1985). ZBL0602.06001.described some

Koh, K. M., On the lattice of maximum-sized antichains of a finite poset, Algebra Univers. 17, 73-86 (1983). ZBL0524.06003.


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