Generators for Boolean algebras #
In this file, we provide an alternative constructor for Boolean algebras.
A set of Boolean generators in a compactly generated complete lattice is a subset S
such that
- the elements of
S
are all atoms, and - the set
S
satisfies an atomicity condition: any compact element below the supremum of a subsets
of generators is equal to the supremum of a subset ofs
.
Main declarations #
IsCompactlyGenerated.BooleanGenerators
: the predicate described above.IsCompactlyGenerated.BooleanGenerators.complementedLattice_of_sSup_eq_top
: ifS
generates the entire lattice, then it is complemented.IsCompactlyGenerated.BooleanGenerators.distribLattice_of_sSup_eq_top
: ifS
generates the entire lattice, then it is distributive.IsCompactlyGenerated.BooleanGenerators.booleanAlgebra_of_sSup_eq_top
: ifS
generates the entire lattice, then it is a Boolean algebra.
An alternative constructor for Boolean algebras.
A set of Boolean generators in a compactly generated complete lattice is a subset S
such that
- the elements of
S
are all atoms, and - the set
S
satisfies an atomicity condition: any compact element below the supremum of a finite subsets
of generators is equal to the supremum of a subset ofs
.
If the supremum of S
is the whole lattice,
then the lattice is a Boolean algebra
(see IsCompactlyGenerated.BooleanGenerators.booleanAlgebra_of_sSup_eq_top
).
The elements in a collection of Boolean generators are all atoms.
- finitelyAtomistic (s : Finset α) (a : α) : ↑s ⊆ S → CompleteLattice.IsCompactElement a → a ≤ s.sup id → ∃ t ⊆ s, a = t.sup id
The elements in a collection of Boolean generators satisfy an atomicity condition: any compact element below the supremum of a finite subset
s
of generators is equal to the supremum of a subset ofs
.
Instances For
A lattice generated by Boolean generators is a distributive lattice.
Equations
- hS.distribLattice_of_sSup_eq_top h = { toLattice := CompleteLattice.toConditionallyCompleteLattice.toLattice, le_sup_inf := ⋯ }
Instances For
A compactly generated complete lattice generated by Boolean generators is a Boolean algebra.