# 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

Algebraic structures |
---|

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

- Rutherford (1965), p.77.
- Rutherford (1965), p.32-33.
- PlanetMath: orthomodular lattice Archived 2007-10-11 at the Wayback Machine
- Rutherford (1965), p.22.
- Rutherford (1965), p.31.
- Rutherford (1965), Th.25.1 p.74.
- Rutherford (1965), Th.8.1 p.22.
- Rutherford (1965), p.87.
- Rutherford (1965), p.94.
- Rutherford (1965), Th.32.1 p.92.
- Rutherford (1965), p.89.

## References

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