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?

Probably this holds for all $x$ if and only if $L$ is distributive.