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Let $L$ be a finite lattice and $x \in L$ with covers $r_1,...,r_l$ in $L$.

One can define $row(x):= \min \{ y | y \leq r_1 \lor \cdots \lor r_l $ and $ y \nleq r_1 \lor \cdots \lor \overline{r_t} \lor \cdots \lor r_l $ for $t=1,...,l \}$, where an overlined symbol means that it is omitted.

In case $L$ is distributive this should just be the classical rowmotion map ( see https://arxiv.org/abs/1108.1172). Usually (actually in all references that I found) this map is defined as sending an order ideal $I$ of a poset $P$ to $row(I)$ being equal to the order ideal generated by the minimal elements in $P \setminus I$.

Question 1: Is there a reference that this is the classical rowmotion for distributive lattices? Id like to have a reference with a definition like this that does not refer to distributive lattices being isomorphic to order ideals of a poset.

Question 2: Can on characterise the elements $x$ in a lattice such that $row(x)$ is well defined and consists of a unique element?

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    $\begingroup$ I think the paper you linked to before (arxiv.org/abs/1712.10123v2) explains this, especially when discussing the work of Barnard (arxiv.org/abs/1610.05137) in Section 6. In fact you might be interested in the semi-distributive case, where a lot can be said. $\endgroup$
    – Sam Hopkins
    Jul 24, 2020 at 13:36
  • $\begingroup$ Regarding your last comment from the edit, you should definitely look at the semidistributive case. $\endgroup$
    – Sam Hopkins
    Jul 24, 2020 at 14:45
  • $\begingroup$ @SamHopkins Ok, thank you. I will try to read the article in detail now. My expierence with non-distributive lattices is very small. Rowmotion recently came up naturally in a homological algebra problem, and it seems the construction works if and only if the lattice is distributive. $\endgroup$
    – Mare
    Jul 24, 2020 at 14:49

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