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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> | ||
+ | )] |
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