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boolean_modules_over_a_relation_algebra [2010/07/29 15:23] (current)
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 +=====Boolean modules over a relation algebra=====
 +
 +Abbreviation: **BRMod**
 +====Definition====
 +A \emph{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 \emph{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 \emph{normal}: $f_r(0)=0$.
 +
 +==Morphisms==
 +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 $f_r$:
 +
 +$h(f_r(x))=f_r(h(x))$
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  | |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  | |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  |unbounded |
 +^[[Congruence distributive]]  |yes |
 +^[[Congruence modular]]  |yes |
 +^[[Congruence n-permutable]]  |yes, $n=2$ |
 +^[[Congruence regular]]  |yes |
 +^[[Congruence uniform]]  |yes |
 +^[[Congruence extension property]]  |yes |
 +^[[Definable principal congruences]]  |no |
 +^[[Equationally def. pr. cong.]]  |no |
 +^[[Amalgamation property]]  | |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &\\
 +f(3)= &\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +[[One-element algebras]]
 +
 +====Superclasses====
 +[[Boolean algebras with operators]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]
 +
 +
 +
 +
 +
 +