## 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 $\cdot$:

$h(x\cdot y)=h(x)\cdot h(y)$

Example 1:

### Properties

Classtype variety open no unbounded yes yes yes, $n=2$ yes yes

### Finite members

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &0 f(4)= &5 f(5)= &0 f(6)= &0 f(7)= &0 f(8)= &\ge 97\text{ out of }104 \end{array}$

### References

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