Differences

This shows you the differences between two versions of the page.

boolean_monoids [2010/07/29 15:23] (current)
Line 1: Line 1:
+=====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>
+)]