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The concept of alattice arises inorder theory, a branch of mathematics. TheHasse diagram below depicts the inclusion relationships among some important subclasses of lattices.
| Algebraic structures |
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1. Aboolean algebra is acomplementeddistributive lattice. (def)
2. A boolean algebra is aheyting algebra.[1]
3. A boolean algebra isorthocomplemented.[2]
4. A distributive orthocomplemented lattice isorthomodular.
5. A boolean algebra is orthomodular. (1,3,4)
6. An orthomodular lattice is orthocomplemented. (def)
7. An orthocomplemented lattice is complemented. (def)
8. A complemented lattice is bounded. (def)
9. Analgebraic lattice is complete. (def)
10. Acomplete lattice is bounded.
11. A heyting algebra is bounded. (def)
12. A bounded lattice is a lattice. (def)
13. A heyting algebra isresiduated.
14. A residuated lattice is a lattice. (def)
15. A distributive lattice is modular.[3]
16. A modular complemented lattice is relatively complemented.[4]
17. A boolean algebra isrelatively complemented. (1,15,16)
18. A relatively complemented lattice is a lattice. (def)
19. A heyting algebra is distributive.[5]
20. Atotally ordered set is a distributive lattice.
21. Ametric lattice ismodular.[6]
22. A modular lattice is semi-modular.[7]
23. Aprojective lattice is modular.[8]
24. A projective lattice is geometric. (def)
25. Ageometric lattice is semi-modular.[9]
26. A semi-modular lattice is atomic.[10][disputed –discuss]
27. Anatomic lattice is a lattice. (def)
28. A lattice is a semi-lattice. (def)
29. Asemi-lattice is apartially ordered set. (def)