Abbreviation: BRMod

A Boolean module over a relation algebra $\mathbf{R}$ is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,f_r\ (r\in R)\rangle$ such that

$\langle A,\vee,0,\wedge,1,\neg\rangle$ is a Boolean algebra

$f_r$ is join-preserving: $f_r(x\vee y)=f_r(x)\vee f_r(y)$

$f_{r\vee s}(x)=f_r(x)\vee f_s(x)$

$f_r(f_s(x))=f_{r\circ s}(x)$

$f_{1'}$ is the identity map: $f_{1'}(x)=x$

$f_0(x)=0$

$f_{r^\smile}(\neg (f_r(x)))\le \neg x$

Remark: Assuming that $f_r$ is order-preserving, the last identity is equivalent to the condition that $f_{r^\smile}$ and $f_r$ are conjugate operators. It follows that $f_r$ is normal: $f_r(0)=0$.

Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean modules over a realtion algebra. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves all $f_r$:

$h(f_r(x))=f_r(h(x))$

Example 1:

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$

One-element algebras

Boolean algebras with operators