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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>
		+	)]