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boolean_monoids [2010/07/29 15:23] (current)
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 +=====Boolean monoids=====
 +Abbreviation: **BMon**
 +====Definition====
 +A \emph{Boolean monoid} is a structure $\mathbf{A}=\langle A,\vee,0,
 +\wedge,1,\neg,\cdot,e\rangle$ such that
 +
 +
 +$\langle A,\vee,0,
 +\wedge,1,\neg\rangle $ is a [[Boolean algebra]]
 +
 +
 +$\langle A,\cdot,e\rangle $ is a [[monoids]]
 +
 +
 +$\cdot$ is \emph{join-preserving} in each argument:  
 +$(x\vee y)\cdot z=(x\cdot z)\vee (y\cdot z) \mbox{ and } x\cdot (y\vee z)=(x\cdot y)\vee (x\cdot z)$
 +
 +
 +$\cdot$ is \emph{normal} in each argument:  $0\cdot x=0 \mbox{ and } x\cdot 0=0$
 +
 +
 +Remark:
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean monoids.
 +A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves $\cdot$, $e$:
 +
 +$h(x\cdot y)=h(x)\cdot h(y) \mbox{ and } h(e)=e$
 +
 +====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====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &1\\
 +f(3)= &0\\
 +f(4)= &9\\
 +f(5)= &0\\
 +f(6)= &0\\
 +f(7)= &0\\
 +f(8)= &258\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Sequential algebras]]
 +
 +====Superclasses====
 +[[Boolean algebras with operators]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]