A $k$-ary function $f$ on a bounded distributive lattice $L$ is called compatible if for any congruence relation $\theta$ on $L$ and $(a_i, b_i)\in \theta$ for $i=1,\ldots,k$ we always have $(f(a_1,\ldots,a_k), f(b_1,\ldots,b_k))\in\theta$.
It's easy to see that all polynomials are compatible.
A lattice $L$ is called affine complete if conversely every compatible function on $L$ is a polynomial.
Given a cardinal $\kappa > 0$, is $[0,1]^\kappa$ an affine complete lattice?
The answer is yes, and a key ingredient is the following theorem due to George Grätzer:
A bounded distributive lattice is affine complete if and only i f it does not contain a proper interval that is a Boolean lattice in the ind uced order. (G. Grätzer, Boolean functions on distributive lattices, Universal Algebra and Applications, vol.9, Banach Center Publications, Warsaw, 97-104.)
First we show that $[0,1]$ is affine-complete: Given any interval $[a,b]\subseteq [0,1]$ with $a<b$, it's easy to see that $\frac{a+b}{2}$ doesn't have a completement, therefore $[0,1]$ doesn't contain any proper Boolean intervals.
Second, according to Theorem 4.1. of this note, products of affine complete lattices are affine complete.
