# 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 (P(S),∪,Ø, ∩,S,−) is the Boolean algebra of subsets of S, and

X·Y = {x·y | x ∈ Xy ∈ Y}.

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

### Morphisms

Let A and B be Boolean semilattices. A morphism from A to B is a function h : AB that is a Boolean homomorphism and preserves ·: h(x·y) = h(xh(y).

### 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 definable principal congruences Amalgamation property Strong amalgamation property Epimorphisms are surjective

### Finite members

[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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