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. (def)

2. A boolean algebra is a heyting algebra.[1]

3. A boolean algebra is orthocomplemented.[2]

4. A distributive orthocomplemented lattice is orthomodular.[3]

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. An algebraic lattice is complete. (def)

10. A complete lattice is bounded.

11. A heyting algebra is bounded. (def)

12. A bounded lattice is a lattice. (def)

13. A heyting algebra is residuated.

14. A residuated lattice is a lattice. (def)

15. A distributive lattice is modular.[4]

16. A modular complemented lattice is relatively complemented.[5]

17. A boolean algebra is relatively complemented. (1,15,16)

18. A relatively complemented lattice is a lattice. (def)

19. A heyting algebra is distributive.[6]

20. A totally ordered set is a distributive lattice.

21. A metric lattice is modular.[7]

22. A modular lattice is semi-modular.[8]

23. A projective lattice is modular.[9]

24. A projective lattice is geometric. (def)

25. A geometric lattice is semi-modular.[10]

26. A semi-modular lattice is atomic.[11]

27. An atomic lattice is a lattice. (def)

28. A lattice is a semi-lattice. (def)

29. A semi-lattice is a partially ordered set. (def)

Notes

  1. ↑ Rutherford (1965), p.77.
  2. ↑ Rutherford (1965), p.32-33.
  3. ↑ PlanetMath: orthomodular lattice
  4. ↑ Rutherford (1965), p.22.
  5. ↑ Rutherford (1965), p.31.
  6. ↑ Rutherford (1965), Th.25.1 p.74.
  7. ↑ Rutherford (1965), Th.8.1 p.22.
  8. ↑ Rutherford (1965), p.87.
  9. ↑ Rutherford (1965), p.94.
  10. ↑ Rutherford (1965), Th.32.1 p.92.
  11. ↑ Rutherford (1965), p.89.

References

  • Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd. 
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