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boolean_semilattices [2010/07/29 15:18]
jipsen created
boolean_semilattices [2010/09/04 16:55] (current)
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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>