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is there a name for the class of functions $f: L\times L \rightarrow L$, where $L$ is a lattice and $L\times L$ is the product lattice (ordered pointwise), with the following property:

$f(x,y)=f( x \vee y, x \wedge y)$ ?

The symbols $\vee$ and $\wedge$ denote the binary join and meet operations on lattices.

thank you in advance.

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    @BrianM.Scott : Thanks for the observations Brian. My question actually comes from some thoughts about the midpoint operation of the plane.2011-12-21

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