Let $\mathcal L$ be a bounded modular lattice. Suppose also that $\mathcal L$ is complete and upper-continuous (i.e. for any directed subset $\{x\_i:i\in I\}$ of $\mathcal L$ and any $x\in \mathcal L$, 
$x\wedge\bigvee\_{i\in I}x\_i=\bigvee\_{i\in I}(x\wedge x\_i)$).


Is there any known sufficient condition for $\mathcal L$ to admit an order-preserving function $f\colon\mathcal L\to [0,1]$, such that:


1. $f(x)=0$ if and only if $x=0$;
2. if $a\leq b$ in $\mathcal L$, $f(a)=f(b)$ implies $a=b$;
3. $f$ commutes with (possibly infinite) joins of directed families.


Is there any known structure theorem for lattices admitting such a map?


Of course, the obvious case is when $\mathcal L$ is an atom, in which case $\mathcal L\cong\{0<1\}$. I am much more interested in situations where the image of $f$ is a dense subset of $[0,1]$.




---


**Motivating Example.** Let $D$ be an Archimedean valuation domain, and denote by $v:D^\*\to \mathbb R\_{\geq0}$ the valuation on $D$. It is possible to define a function $L\_v:Mod(D)\to \mathbb R\_{\geq0}\cup\{\infty\}$ via the following steps:


1. given an ideal $I$ of $R$, let $v(I):=\inf\{v(d):d\in I\}$;
2. given a finitely generated right $D$-module $F$, let $\{x\_1,\dots,x\_n\}$ be a generating set with the minimum possible cardinality. Define inductively
	* $F\_0=0$;
	* $F\_1=x\_1D$;
	* if $F\_k$ is defined for some $k<n$, then let $F\_{k+1}=F\_k+x\_{k+1}D$;


in particular $F\_n=F$. Let also $B\_j=Ann(x\_j+F\_{j-1})=\{d\in D: x\_jd\in F\_{j-1}\}$. We let 
$$L\_v(F)=v(B\_1\cdots B\_n)\,;$$
 3. for a given right $D$-module $M$, we let $L\_v(M)=\sup\{L\_v(F):F\leq M\text{ finitely generated}\}$.


One can prove that $L\_v$ is additive on short exact sequences (i.e. $L\_v(B)=L\_v(A)+L\_v(C)$ if $0\to A\to B\to C\to 0$ is exact) and it is upper-continuous (i.e. $L\_v(M)=sup\_iL\_v(M\_i)$ when $M=\bigcup\_iM\_i$ with $\{M\_i\}$ a directed set of submodules of $M$). 


Using these properties, one can see that 
$$Ker(L\_v):=\{M\in Mod(D):L\_v(M)=0\}$$
is a hereditary torsion class, and so we can define a new category
$$\mathfrak A:=Mod(D)/Ker(L\_v)$$
as the Gabriel quotient of $Mod(D)$ over $Ker(L\_v)$. Let $Q\_v\colon Mod(D)\to \frak A$ be the quotient functor. The function $L\_v$ induces a new function 
$$\bar{L\_v}\colon \mathfrak A\to \mathbb R\_{\geq0}\cup \{\infty\}$$
just letting $\bar{L\_v}(X)$ be $L\_v(M)$ for any $M\in Mod(D)$ such that $Q\_v(M)=X$. This new function has the same additivity and upper-continuity properties of $L\_v$ and, furthermore, it has the property that $\bar L\_v(X)=0$ iff $X=0$.


All the objects $X\in \mathfrak A$ such that $\bar {L\_v}(X)<\infty$ do have a lattice of subobjects with the properties described in the question (the wanted function to $[0,1]$ is just a rescaling of the values of $\bar L\_v$ on subobjects).



Functions from a lattice to $[0,1]$
Mathematical economists have been interested in special cases of the following general question, which they call "existence of utility representations."


$\quad$ Given $(X, \precsim)$ an ordered set, when does there exist a strictly increasing function $f: X \to \mathbb R$?


A classical result is that for $\precsim$ a complete preorder (total, transitive; not necessarily antisymmetric), it is necessary and sufficient that $X$ has a countable order-dense subset, and furthermore $f$ can be taken continuous with respect to the interval topology on $X$ and standard topology on $\mathbb R$. The first statement of this was Debreu (1954) (N.B. the proof in that paper has a nontrivial error.)


Economists tend to be interested in how structural axioms on the preorder translate to canonical functional forms for $f$ (and how to interpret these axioms), but it looks like others have continued this line of research in other directions.


Here's a recent paper that looks like it gives a general answer to your question (e.g. Theorem 3.1).


$\quad$ [Continuous Utility Representation Theorems in Arbitrary Concrete Categories](https://link.springer.com/article/10.1007/s10485-007-9097-0) 


$\quad$ Bosi, G. & Herden, G. Appl Categor Struct (2008) 16: 629. doi:10.1007/s10485-007-9097-0


I don't know if the characterization will be useful for your purposes, and don't know the non-economics literature well, but it seems to exist.


