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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!

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what do you mean by 'commutative' in the question? –  Ittay Weiss Jan 8 '13 at 9:42
    
Ittay: i edited the question specifying what I mean by commutative. –  user55551 Jan 8 '13 at 9:53
    
Am I correct to assume that $L_1\times L_1$ is endowed with the lexicographical order? –  Willie Wong Jan 8 '13 at 10:05
    
No, it's the ordinary product on Lattices: $(a,b)\leq (c,d)$ iff $a\leq c$ AND $b\leq d$ –  user55551 Jan 8 '13 at 11:19
    
So if $L_1$ is actually a linear order, these are the functions that are symmetric about the diagonal? –  Brian M. Scott Jan 8 '13 at 14:27

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