boolean_semilattices - MathStructures

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=====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>

Trace: • cancellative_partial_monoids • transitive_relations • idempotent_semirings_with_identity_and_zero • bck-lattices • varieties • idempotent_semirings_with_zero • semilattices_with_zero • finite_commutative_binars • graphs • solvable_groups