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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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Such functions are obviously symmetric, and I suspect constant on each chain, so could be "factored" into a one-variable function from the chains of $L$ to $L$. – bgins Dec 18 '11 at 21:46
@bgins: They’re not in general constant on chains. Symmetric difference, union, and intersection are all operations of this kind in $\langle\wp(X),\subseteq\rangle$, and ordinary multiplication is one in $\mathbb{Z}^+$ ordered by divisibility. In the plane, with $(x,y)\le(u,v)$ iff $x\le u$ and $y\le v$, the operation of finding the midpoint of two points has this property. – Brian M. Scott Dec 20 '11 at 18:05
@BrianM.Scott : Thanks for the observations Brian. My question actually comes from some thoughts about the midpoint operation of the plane. – MM83 Dec 21 '11 at 19:56

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