Boolean semilattices

Abbreviation: BSlat

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 ∈ X, y ∈ 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 : A→B that is a Boolean homomorphism and preserves ·: h(x·y) = h(x)·h(y).

Some results

Examples

Properties

Finite members

[Some members of BSlat]?

Subclasses

Superclasses