General Interpolation by Polynomial Functions of Distributive Lattices

International audienceFor a distributive lattice $L$, we consider the problem of interpolating functions $f\colon D\to L$ defined on a finite set $D\subseteq L^n$, by means of lattice polynomial functions of $L$. Two instances of this problem have already been solved. In the case when $L$ is a distributive lattice with least and greatest elements $0$ and $1$, Goodstein proved that a function $f\colon\{0,1\}^{n}\to L$ can be interpolated by a lattice polynomial function $p\colon L^{n}\to L$ if and only if $f$ is monotone; in this case, the interpolating polynomial $p$ was shown to be unique.The interpolation problem was also considered in the more general setting where $L$ is a distributive lattice, not necessarily bounded, and where$D\subseteq L^{n}$ is allowed to range over cuboids $D=\left\{ a_{1},b_{1}\right\} \times\cdots\times\left\{ a_{n},b_{n}\right\} $ with $a_{i},b_{i}\in L$ and $a_{i}