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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>
+)]
+
+
+
+
+
+