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boolean_semilattices [2010/07/29 15:18]
jipsen created
boolean_semilattices [2010/09/04 16:55] (current)
jipsen delete hyperbaseurl
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-f+=====Boolean semilattices=====
+
+Abbreviation: **BSlat**
+
+====Definition====
+A \emph{Boolean semilattice} is a structure $\mathbf{A}=\langle A,\vee,0, +\wedge,1,\neg,\cdot\rangle$ such that
+
+$\mathbf{A}$ is in the variety generated by complex algebras of semilattices
+
+Let $\mathbf{S}=\langle S,\cdot\rangle$ be a [[semilattice]]. The
+\emph{complex algebra} of $\mathbf{S}$ is
+$Cm(\mathbf{S})=\langle P(S),\cup,\emptyset,\cap,S,-,\cdot\rangle$,
+where $\langle P(S),\cup,\emptyset, +\cap,S,-\rangle$ is the Boolean algebra of subsets of $S$, and
+
+$X\cdot Y=\{x\cdot y\mid x\in X,\ y\in Y\}$.
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean semilattices.
+A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves $\cdot$:
+
+$h(x\cdot y)=h(x)\cdot h(y)$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+^[[Classtype]]  |variety |
+^[[Finitely axiomatizable]]  |open |
+^[[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]]  | |
+^[[Congruence extension property]]  |yes |
+^[[Definable principal congruences]]  | |
+^[[Equationally def. pr. cong.]]  | |
+^[[Amalgamation property]]  | |
+^[[Strong amalgamation property]]  | |
+^[[Epimorphisms are surjective]]  | |
+
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &0\\ +f(4)= &5\\ +f(5)= &0\\ +f(6)= &0\\ +f(7)= &0\\ +f(8)= &\ge 97\text{ out of }104\\ +\end{array}$
+
+[[Some members of BSlat]]
+
+====Subclasses====
+[[Variety generated by complex algebras of linear semilattices]]
+
+====Superclasses====
+[[Commutative Boolean semigroups]]
+
+
+====References====
+
+[(Ln19xx>
+)]\end{document}
+%</pre>