This came about when I was studying the connection between matroids and strong greedoids, but it has broken through into a subject I am not particularly familiar with: submodular functions on lattices.

Let $L$ be a finite lattice (in the sense of combinatorics).

Let $\mathbb{N}=\left\{ 0,1,2,\ldots\right\} $. A function $f:L\to \mathbb{N}$ is said to be

*submodular*if it satisfies $f\left( a\right) +f\left( b\right) \geq f\left( a\wedge b\right) +f\left( a\vee b\right) $ for all $a,b\in L$.*isotone*if it satisfies $f\left( a\right) \leq f\left( b\right) $ whenever $a,b\in L$ satisfy $a\leq b$.*1-continuous*if it satisfies $f\left( b\right) -f\left( a\right) \in\left\{ 0,1\right\} $ whenever $a,b\in L$ satisfy $a\lessdot b$ (that is, $a<b$ but there exists no $c\in L$ satisfying $a<c<b$).

Note that the first two of these notions are standard, while the third is mine.

Now, assume that $L$ is the Boolean lattice $2^{E}$ of a finite set $E$ (so that the order relation $\leq$ on $L$ is the relation $\subseteq$ on $2^{E}$). Thus, the 1-continuous isotone submodular functions $f:L\to\mathbb{N}$ satisfying $f\left( \varnothing\right) =0$ are precisely the rank functions of matroids on the ground set $E$. If we drop the "1-continuous", then we obtain the rank functions of polymatroids instead. Note that "$a\lessdot b$" is equivalent to "$a \subseteq b$ and $\left|b \setminus a \right| = 1$" for any $a, b \in L = 2^E$.

Let $M$ be a sublattice of $L$, by which I mean a subset of $L$ that is a
lattice when equipped with the partial order it inherits from $L$ and that has
the same $0$, $1$, $\wedge$ and $\vee$ as $L$. (This may or may not be some
people's definition of a sublattice.) Let $g:M\to\mathbb{N}$ be a
function. An *extension* of $g$ to $L$ will mean a function
$f:L\to\mathbb{N}$ such that $f\mid_{M}=g$.

Theorem 1.If $g$ is an isotone submodular function on $M$, then there exists an isotone submodular extension of $g$ to $L$.

This theorem is (a particular case of) Lemma 5.1 in Donald M. Topkis,
*Minimizing a Submodular Function on a Lattice*, Operations Research **26**,
No. 2 (Mar. - Apr., 1978), pp. 305--321.
The proof defines the extension $f:L\to\mathbb{N}$ of $g$ to $L$ by
setting
\begin{align}
f\left( y\right) =\min\left\{ g\left( x\right) \ \mid\ x\in L\text{
satisfying }x\geq y\right\}
\end{align}
for all $y\in L$. It is easy to check that this extension $f$ is indeed isotone and
submodular. Note that $L$ could be any finite lattice here, not necessarily a Boolean one.

My question is: how well do other properties extend from $M$ to $L$ ? The specific question I am most interested in is:

Question 2.If $g$ is a 1-continuous isotone submodular function on $M$, then does there exist a 1-continuous isotone submodular extension of $g$ to $L$ ?

(Here, $L$ is still supposed to be Boolean.) A positive answer to Question 2 (specifically, in the case when $M$ is the lattice of order ideals of a certain poset structure on $E$) would yield a neat (if not quite one-to-one) correspondence between matroids and strong greedoids that I believe could help understand the latter. (I could elaborate if there is interest.) Note that Topkis's above construction of $f$ does not produce a 1-continuous $f$ even if it is applied to a 1-continuous $g$.

Having asked Question 2, we can vary the assumptions and get curious:

Question 3.If Question 2 has a positive answer, how far does it generalize? For example, can we replace the Boolean lattice $L$ by an arbitrary geometric lattice? ranked distributive lattice? ranked lattice?

Note that we cannot get rid of the "ranked" condition completely; e.g., the rank function on the long chain of the pentagon lattice $N_{5}$ is a 1-continuous isotone submodular function, but has no 1-continuous extension to the whole $N_{5}$.

Curiosity also suggests a different question:

Question 4.If $g$ is merely submodular (but not necessarily 1-continuous or isotone), then does there exist a submodular extension of $g$ to $L$ ?