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Regarding the question about the strength of the ultrafilter lemma for distributive lattices, let me cite the relevant theorem from Herrlich's book:

Theorem 4.32. Equivalent are:

1. Every lattice has a maximal filter.
2. Every complete lattice has a maximal filter.
3. Every distributive lattice has a maximal filter.
4. Every closed lattice has a maximal filter.
5. AC.

In this theorem a closed lattice is a lattice that is isomorphic to the lattice of closed subsets of a nonempty topological space (Definition 4.28).

Herrlich, Horst
Axiom of choice
Lecture Notes in Mathematics, 1876.
Springer-Verlag, Berlin, 2006.

Herrlich credits the equivalence of Item 3 and Item 5 to

G. Klimowsky.
El Theorema de Zorn y la existencia de filtros e ideales maximales en los reticulados distributivos.
Rev. Union Mat. Argentina, 18:160-164, 1958.

Where Herrlich writes in terms of AC and in terms of the existence of a maximal filter, Klimowsky writes that Zorn's Lemma is equivalent to En todo reticulado distributivo con primer elemento, todo filtro está contenido en un ultrafiltro.

Regarding the question about the strength of the ultrafilter lemma for distributive lattices, let me cite the relevant theorem from Herrlich's book:

Theorem 4.32. Equivalent are:

1. Every lattice has a maximal filter.
2. Every complete lattice has a maximal filter.
3. Every distributive lattice has a maximal filter.
4. Every closed lattice has a maximal filter.
5. AC.

In this theorem a closed lattice is a lattice that is isomorphic to the lattice of closed subsets of a nonempty topological space (Definition 4.28).

Herrlich, Horst
Axiom of choice
Lecture Notes in Mathematics, 1876.
Springer-Verlag, Berlin, 2006.

Herrlich credits the equivalence of Item 3 and Item 5 to

G. Klimowsky.
El Theorema de Zorn y la existencia de filtros e ideales maximales en los reticulados distributivos.
Rev. Union Mat. Argentina, 18:160-164, 1958.

Where Herrlich writes in terms of AC and in terms of the existence of a maximal filter, Klimowsky writes that Zorn's Lemma is equivalent to 'En todo reticulado distributivo con primer elemento, todo filtro está contenido en un ultrafiltro'. (In every distributive lattice with first element, every filter is contained in an ultrafilter.)

Regarding the question about the strength of the ultrafilter lemma for distributive lattices, let me cite the relevant theorem from Herrlich's book:

Theorem 4.32. Equivalent are:

1. Every lattice has a maximal filter.
2. Every complete lattice has a maximal filter.
3. Every distributive lattice has a maximal filter.
4. Every closed lattice has a maximal filter.
5. AC.

In this theorem a closed lattice is a lattice that is isomorphic to the lattice of closed subsets of a nonempty topological space (Definition 4.28).

Herrlich, Horst
Axiom of choice
Lecture Notes in Mathematics, 1876.
Springer-Verlag, Berlin, 2006.

Herrlich credits the equivalence of Item 3 and Item 5 to

G. Klimowsky.
El Theorema de Zorn y la existencia de filtros e ideales maximales en los reticulados distributivos.
Rev. Union Mat. Argentina, 18:160-164, 1958.

Regarding the question about the strength of the ultrafilter lemma for distributive lattices, let me cite the relevant theorem from Herrlich's book:

Theorem 4.32. Equivalent are:

1. Every lattice has a maximal filter.
2. Every complete lattice has a maximal filter.
3. Every distributive lattice has a maximal filter.
4. Every closed lattice has a maximal filter.
5. AC.

In this theorem a closed lattice is a lattice that is isomorphic to the lattice of closed subsets of a nonempty topological space (Definition 4.28).

Herrlich, Horst
Axiom of choice
Lecture Notes in Mathematics, 1876.
Springer-Verlag, Berlin, 2006.

Herrlich credits the equivalence of Item 3 and Item 5 to

G. Klimowsky.
El Theorema de Zorn y la existencia de filtros e ideales maximales en los reticulados distributivos.
Rev. Union Mat. Argentina, 18:160-164, 1958.

Where Herrlich writes in terms of AC and in terms of the existence of a maximal filter, Klimowsky writes that Zorn's Lemma is equivalent to En todo reticulado distributivo con primer elemento, todo filtro está contenido en un ultrafiltro.

Regarding the question about the strength of the ultrafilter lemma for distributive lattices, let me cite the relevant theorem from Herrlich's book:

Theorem 4.32. Equivalent are:

1. Every lattice has a maximal filter.
2. Every complete lattice has a maximal filter.
3. Every distributive lattice has a maximal filter.
4. Every closed lattice has a maximal filter.
5. AC.

In this theorem a closed lattice is a lattice that is isomorphic to the lattice of closed subsets of a nonempty topological space (Definition 4.28).

Herrlich, Horst
Axiom of choice
Lecture Notes in Mathematics, 1876.
Springer-Verlag, Berlin, 2006.

Regarding the question about the strength of the ultrafilter lemma for distributive lattices, let me cite the relevant theorem from Herrlich's book:

Theorem 4.32. Equivalent are:

1. Every lattice has a maximal filter.
2. Every complete lattice has a maximal filter.
3. Every distributive lattice has a maximal filter.
4. Every closed lattice has a maximal filter.
5. AC.

In this theorem a closed lattice is a lattice that is isomorphic to the lattice of closed subsets of a nonempty topological space (Definition 4.28).

Herrlich, Horst
Axiom of choice
Lecture Notes in Mathematics, 1876.
Springer-Verlag, Berlin, 2006.

Herrlich credits the equivalence of Item 3 and Item 5 to

G. Klimowsky.
El Theorema de Zorn y la existencia de filtros e ideales maximales en los reticulados distributivos.
Rev. Union Mat. Argentina, 18:160-164, 1958.