I would like to know if there is a name for the class of commutative (i.e., $\phi(x,y)=\phi(y,x)$) lattice-morphisms $\phi : L_1\times L_{1} \rightarrow L_2$ with the following property:
$\phi(x \sqcap y, x \sqcup y) = \phi(x, y)$.
Note that when $L_{1}$ is linearly ordered, the equality is automatically satisfied.
Are objects of this kind studied somewhere?
Thank you in advance!