Let $(\mathcal{A}, \cup, \cap)$ a lattice (with minimum and maximum elements $\bot$ and $\top$). Let $X\subset \mathcal{A}$ a generator set (a set of minimal cardinality that generate $\mathcal{A}$ i.e. such that $\mathcal{A}$ is the intersection of all sublattices containing $X$).

I consider $\mathbb{R}[\mathcal{A}]$ the real vector space with bases the elements of $\mathcal{A}$, then its elements if of like: $r_1\cdot A_1+\ldots r_n\cdot A_n$ where $r_i$ is a real number and $A_i\in \mathcal{A}$.

If $\mathcal{A}$ is a sublattice of the class $\mathcal{P}(X)$ of subsets of a set $X$, we have (see $[1]$) a exact sequence of real vector spaces:

$0\to I\to \mathbb{R}[\mathcal{A}] \xrightarrow{q} Simp(\mathcal{A})\to 0 $

where $Simp(\mathcal{A})\subset \textbf{R}^X$ is the $\mathbb{R}$-vector subspace of generated of simple functions with support on $\mathcal{A}$ i.e. of the functions of type $r_1\cdot \chi_{A_1}+\ldots r_n\cdot \chi_{A_n}$ where $r_i\in \mathbb{R}, A_i\in \mathcal{A}$ for $1\leq i\leq n$ and $n\in\mathbb{N}$, and $\chi_A(X)=\{0,1\},\ \chi_{A}^{-1}(\{1\})=A$. Where $q$ is the natural map, and $I$ is the real vector subspace generated by the equalities like $\bot\equiv 0$ and:

*) $A+B \equiv A\cup B + A\cap B$ for $A, B\in \mathcal{A}$ .

In fact from $[1]$ (p. 28) a linear map $f: \mathbb{R}[\mathcal{A}]\to V$ has a (linear) extension to $Sim[\mathcal{A}]$ iff it is modular i.e. if it null on the elements of $I$.

From $[2]$ (p.60, 61, 36 (prop. 2.1.2(x)) we have that $I$ is also generated by the equalities like:

**) $A_1+\ldots A_n \equiv B_1+\ldots B_m$ where $A_1,\ldots A_n,\ B_1,\ldots B_m\in \mathcal{A}$ (may be with some repeated elements) such that: $\bigcup_{1\leq i_1<\ldots < i_k\leq n} (A_{i_1}\cap \ldots \cap A_{i_k})= \bigcup_{1\leq i_1<\ldots < i_k\leq n} (B_{i_1}\cap \ldots \cap B_{i_k})$ for each $k\leq max (n, m)$ (If $k>n$ we pose equal to $0$ the first member, analogously if $k>m$).

Let $(*)$ and $I(**)$ the subspaces generated by $(*)$ or $(**)$ respectively, of course $I(*)\subset I(**)$, now if a linear map $f$ as above is null on $I(*)$ then it is modular and then (see $[1]$ or $[2]$) it is null also on $I(**)$, then $I(*)\supset I(**)$.

Now, ** I want to generalize this sets construction** to a general alttice $\mathcal{A}$, then I define:

**DEF)** $Simp(\mathcal{A}):=\mathbb{R}[\mathcal{A}]/I$

where $I$ is the (real) vectorial subspace generated by $(*)$ or equivalently by $(**)$, and indicate an element of $Simp(\mathcal{A})$ as $[v]$ where $v\in \mathbb{R}[\mathcal{A}]$.

**Observation 1):**

We define the product on $\mathcal{A}$ as the intersection (or "inf") we have a multiplication on $\mathbb{R}[\mathcal{A}]$ that make this a (associative) algebra. Explicitly $(r_1\cdot A_1+\ldots + r_m\cdot A_m)\cdot (s_1\cdot B_1+\ldots + s_n\cdot B_n)= \sum_{i, j} (r_i\cdot s_j)\cdot\ A_i\cap B_j$ .

We have that $I$ is and ideal of the algebra, in fact if we multiply $(**)$ for a elements $v$ of $\mathbb{R}[\mathcal{A}]$ we get a elements of $I$, for distributivity its enough see this for $v=E$ where $E\in \mathcal{A}$ (a base vector), and easly this multiplication preserve $(**)$.

We call the class $Idmp(Simp(\mathcal{A}))$ of idempotents of $Simp(\mathcal{A})$ its a boolean algebra and we have a lattice morphism $\mathcal{A}\to Idmp(Simp(\mathcal{A}))$. Now I wish define a order on $Simp(\mathcal{A})$ in natural way, and such that the inducted order on $Idmp(Simp(\mathcal{A}))$ is just the one defined above.

I observe that give the elements $A_1,\ldots..A_n\in \mathcal{A}$ I can define (in a standard way) a sequence $C_1,\ldots C_N$ of disjoint elements of $Idmp(Simp(\mathcal{A}))$ such that each $A_i$ is a (disjoint) union of some elements of this sequence. Then each element $u= r_1\cdot [A_1]+\ldots + r_m \cdot [A_m]$ of $Simp(\mathcal{A})$ is writable like $u= r'_1\cdot a_1+\ldots + r'_M \cdot a_M$ where $a_1,\ldots,a_M$ are disjoint elements of $Idmp(Simp(\mathcal{A}))$. In similar way is $u, v\in \mathbb{R}[\mathcal{A}]$ we can write $[u]=r_1\cdot a_1+\ldots r_n\cdot a_n$, $[v]=s_1\cdot a_1+\ldots s_n\cdot a_n$ where $a_1,\ldots a_n$ are disjoint elements of $Idmp(Simp(\mathcal{A}))$.

My idea is to define a order on $Idmp(Simp(\mathcal{A}))$ as $u\leq v$ if given a representation $[u]=r_1\cdot a_1+\ldots r_n\cdot a_n$, $[v]=s_1\cdot a_1+\ldots s_n\cdot a_n$ with $a_1,\ldots, a_n$ disjoint, we have that $r_i \leq s_i\ \leq i\leq n$, but is this well defined?

If this is well defined I can consider the free $\sigma$-boolean algebra (relative to the generator set $X$) as in [3] and call this the "Borel algebra" of $\mathcal{A}$.

** Conjecture 1)**
Is the natural map $\mathcal{A}\to Simp(\mathcal{A})$ injective?

** Conjecture 2)**
If $[u]= r_1\cdot a_1+\ldots + r_m\cdot a_m $, $[v]=s_1\cdot b_1+\ldots + s_n\cdot b_n$ with $a_1\ldots \cdot , a_m$ and $b_1,\ldots \cdot , b_n$ disjoint, is true that $[u]= [v]$ iff $n=m$, $A_i= B_i,\ r_i=s_i\ 1\leq i\leq n$?

** Conjecture 3)**
If I consider the lattice $\mathcal{B}:=Idmp(Simp(\mathcal{A}))$ and do the some construction , is true that
$Idmp(Simp(\mathcal{A}))\cong Idmp(Simp(\mathcal{B}))$ ?

*I ask: are my conjectures true?*

Biblio:

[1]: Measure and Integration: An Advanced Course. Heinz Konig, Springer 2009.

[2]: Theory of Charges: A Study of Finitely Additive Measures Bhaskara Rao, M. Bhaskara Rao. AP

[3] Boolean algebras. Roman Sikorski. Springer, 1960.