Map of lattices


The concept of a lattice arises in order theory, a branch of mathematics. The Hasse diagram below depicts the inclusion relationships among some important subclasses of lattices.

Proofs of the relationships in the map

1. A boolean algebra is a complemented distributive lattice.
2. A boolean algebra is a heyting algebra.
3. A boolean algebra is orthocomplemented.
4. A distributive orthocomplemented lattice is orthomodular.
5. A boolean algebra is orthomodular.
6. An orthomodular lattice is orthocomplemented.
7. An orthocomplemented lattice is complemented.
8. A complemented lattice is bounded.
9. An algebraic lattice is complete.
10. A complete lattice is bounded.
11. A heyting algebra is bounded.
12. A bounded lattice is a lattice.
13. A heyting algebra is residuated.
14. A residuated lattice is a lattice.
15. A distributive lattice is modular.
16. A modular complemented lattice is relatively complemented.
17. A boolean algebra is relatively complemented.
18. A relatively complemented lattice is a lattice.
19. A heyting algebra is distributive.
20. A totally ordered set is a distributive lattice.
21. A metric lattice is modular.
22. A modular lattice is semi-modular.
23. A projective lattice is modular.
24. A projective lattice is geometric.
25. A geometric lattice is semi-modular.
26. A semi-modular lattice is atomic.
27. An atomic lattice is a lattice.
28. A lattice is a semi-lattice.
29. A semi-lattice is a partially ordered set.