# Map on class of all finite posets coming from maximal sized antichains

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.

• 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. – Richard Stanley Nov 25 '19 at 20:10
• @RichardStanley: Thanks for these great references! – Sam Hopkins Nov 25 '19 at 20:16
• (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.) – Sam Hopkins Nov 26 '19 at 1:06

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 $$P\to P'$$.)