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
Group-like Semigroup / Monoid Racks and quandles Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Rng Semiring Near-ring Commutative ring Domain Integral domain Field Ring theory
Lattice-like Lattice Semilattice Map of lattices Lattice theory
Module-like Module Group with operators Vector space
Algebra-like Algebra Associative Non-associative Lie algebra Graded Bialgebra

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

References