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Boolean semilattices
Abbreviation: BSlat
Definition
A 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 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
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 \hyperbaseurl{http://math.chapman.edu/structures/files/}
Subclasses
Superclasses
References
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