# Boolean semilattices

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### Definition

A Boolean semilattice is a structure A = (A,∨,0, ∧,1,¬,·) such that

A is in the variety generated by complex algebras of semilattices.

Let S = (S,·) be a semilattice. The complex algebra of S is Cm(S) = (P(S),∪,Ø, ∩,S,−,·), 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\}.

Remark :  The complex algebra of any groupoid is a Boolean algebra with one binary operator.

<h3>Morphismsh3>

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).

<h3>Some resultsh3>

<h3>Examplesh3>

<h3>Propertiesh3> <table border = 2> 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 definable principal congruences Amalgamation property Strong amalgamation property Epimorphisms are surjective <h3>Finite membersh3> [Size 1]? :   1
[Size 2]? :   1
[Size 3]? :   0
[Size 4]? :   5
[Size 5]? :   0
[Size 6]? :   0
[Size 7]? :   0
[Size 8]? :
ge 96\$ out of 104

[Some members of BSlat]?

### Subclasses

[Variety generated by complex algebras of linear semilattices]?

### Superclasses

[Commutative Boolean semigroups]?

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