Let $G$ be a finite acylic directed graph. The transitive closure of $G$ defines a

An*independent set* $\mathcal{I} \subseteq G$ is a set of pairwise non-adjacent vertices of $G$.

\(
\def\D{{\color{#004db2}{\mathcal{D}}}}
\def\U{{\color{#ffcd6f}{\mathcal{U}}}}
\def\IU{{\color{#ffcd6f}{\mathcal{I}}}}
\def\ID{{\color{#004db2}{\mathcal{I}}}}
\)
posetWhich we call $G$-order

.An

Definition 1.
A pair $(\D,\U)$ of independent sets of $G$ is called *orthogonal* if there is no edge in $G$ from an element of $\D$ to an element of $\U$. An orthogonal pair of independent sets $(\D,\U)$ is called *tight* if whenever any element of $\D$ is

Write*top* for **t**ight **o**rthogonal **p**air, and $\mathrm{top}(G)$ for the set of all tops of $G$.

increasedremoved and replaced by a larger element with respect to $G$-order

or any element of $\U$ is decreasedremoved and replaced by a smaller element with respect to $G$-order

, or a new element is added to either $\D$ or $\U$, then the result is no longer an orthogonal pair of independent sets.Write

Theorem 1. Let $\mathcal{I}$ be an independent set of a directed acyclic graph $G$. Then there exists a unique $(\ID,\U) \in \mathrm{top}(G)$ and a unique $(\D,\IU)\in \mathrm{top}(G)$.

Definition 2. *Rowmotion*
Rowmotion was introduced by Duchet; studied for the Boolean lattice and products of chains by Brouwer and Schrijver; and related to matroid theory by Deza and Fukuda. Cameron and Fon-der-Flaass considered rowmotion on the product of two and then three chains. Because the orbit structure of rowmotion on Boolean lattices is so wild, much of the effort in the references above is dedicated to understanding which orbit sizes are realizable.
Its study then apparently lay dormant for over a decade until Panyushev resurrected it in the form of a series of conjectures of the orbit structure of rowmotion on the root posets of Lie algebras. The focus then shifted to finding equivariant bijections to natural combinatorial objects, and Stanley completely characterized the orbit structure of rowmotion on the product of two chains combinatorially. Armstrong, Stump, and Thomas then went on to resolve Panyushev's conjectures using noncrossing partitions, while Striker and Williams unified and extended various results by relating rowmotion to *jeu-de-taquin* and made linguistic contributions to the theory.
Interest in rowmotion has blossomed as part of a new research direction which Jim Propp has dubbed "dynamical algebraic combinatorics."
sends one completion to the other:
\[\mathrm{row}(\ID,\U):=(\D,\IU).\]

Definition 3.
The *flip* of $(\D,\U) \in \mathrm{top}(G)$ at an element $g \in G$ is the tight orthogonal pair $\mathrm{flip}_g(\D,\U)$ defined as follows: if $g \not \in \D$ and $g \not \in \U$, the flip does nothing. Otherwise, preserve all elements of $\D$ that are not less than $g$ and all elements of $\U$ that are not greater than $g$ (and delete all other elements); after switching the set to which $g$ belongs, then

greedily completeIn reverse linear extension order for $\D$ and linear extension order for $\U$.

$\D$ and $\U$ to a tight orthogonal pair.Definition/Theorem 2. We *independence poset* on $\mathrm{top}(G)$ as the reflexive and transitive closure of the relations $(\D,\U) \lessdot (\D',\U')$ if there is some $g \in \U$ such that \[\mathrm{flip}_g(\D,\U)=(\D',\U').\]

defineThis really does define a poset, and these relations are exactly its covers.

the Example 1. Let $G$ be the compatibility graph of a poset, so that independent sets of $G$ are antichains. For $(\D,\U) \in \mathrm{top}(G)$, both $\D$ and $\U$ are antichains and $\U$ consists of the minimum elements of $G\setminus \D$. The independence poset on $\mathrm{top}(G)$ recovers the distributive lattice of antichains, and flips are toggles.

Theorem 3. Rowmotion can be computed in

slow motionThat is, $\mathrm{row} = \displaystyle\prod\limits_{g \in \ell} \mathrm{flip}_{g}$, where $\ell$ is a linear extension of $G$-order.

.Theorem 4. If $\mathrm{top}(G)$ is a lattice, then it is a

trim latticeTrim lattices are extremal left-modular lattices. They are analogues of distributive lattices without the graded hypothesis: a graded trim lattice is a distributive lattice, and every distributive lattice is trim.

.Theorem 5. If $\mathrm{top}(G)$ is a graded lattice, then it is a distributive lattice.

Example 2.
Click here to build your own examples in CoCalc.

George Markowsky, *The factorization and representation of lattices*, Transactions of the
American Mathematical Society **203** (1975), 185-200.

Jessica Striker and Nathan Williams,*Promotion and rowmotion*, European Journal of
Combinatorics **33** (2012), no. 8, 1919-1942.

Hugh Thomas,*An analogue of distributivity for ungraded lattices*, Order **23** (2006), no. 2,
249-269.

Jessica Striker and Nathan Williams,

Hugh Thomas,

Hugh Thomas and Nathan Williams, *Rowmotion in slow motion*, Proceedings of the London
Mathematical Society, **119** (2019), 1149-1178.

Hugh Thomas and Nathan Williams,*Independence Posets*, Journal of Combinatorics **10.3** (2019): 545-578.

Hugh Thomas and Nathan Williams,