Let $L$ be a finite distributive lattice. In case $L$ is Boolean one can define a multiplication by $a*b:=(a \cup b) \setminus (a \cap b)=(a \cup b) \cap (a \cap b)^c$, where $(-)^c$ denotes the complement. With this multiplication $L$ is a group, isomorphic to an elementary abelian 2-group.

Now the rowmotion (see for example https://arxiv.org/pdf/1712.10123.pdf) generalises the complement of a Boolean lattice and one might define for any distributive lattice the multiplication $a*b=(a \lor b) \land row(a \land b)$. One then obtains a commutative magma that is in general not unital or associative.

Question 1: Has this multiplication been studied before?

Question 2: One might define a functor $F$ from the category of distributive lattices with lattice homomorphism that respect rowmotion to the category of commutative magmas. Does this functor have nice properties, can one describe the image and those resulting magmas have some nice structure? In case $F(L_1) \cong F(L_2)$, do we have $L_1 \cong L_2$?

I just calculated some small examples but those magmas with barely any structure are very confusing. Or might there be a better modification of $a*b:=(a \cup b) \setminus (a \cap b)$ to get a multiplication with better structure?

In the Boolean case, it seems that we can recover the Boolean lattice in the elementary abelian 2-group as the set of maximal subroups including the whole group where one orders the generators of those subgroups in a natural way. For the Klein four group $G=<a,b>$ we have $G>a , G>b , a>ab, b>ab$ and get the Boolean lattice on a 2-set.

gradedposet $P$, it might be better to define $a \ast b = (a \vee b) \wedge \mathrm{Rvac}(a \wedge b)$, where $\mathrm{Rvac}$ is "row-vacuation" instead of rowmotion (see Section 5.1 of arxiv.org/abs/2006.01568 for the definition). This is because $\mathrm{Rvac}$ is an involution, like complementation on the Boolean lattice. If for instance $P=[a]x[b]$ is the product of two chains, then $\mathrm{Rvac}$ is 180 degree rotation+complementation. $\endgroup$ – Sam Hopkins Jul 23 '20 at 14:53