Map of lattices

The factual accuracy of part of this article is disputed. The dispute is about "26. A semi-modular lattice is atomic.". Please help to ensure that disputed statements are reliably sourced. See the relevant discussion on the talk page. (May 2017) (Learn how and when to remove this message)

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 Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Rng Semiring Near-ring Commutative ring Domain Integral domain Field Division ring Lie ring Ring theory
Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map of lattices Lattice theory
Module-like Module Group with operators Vector space Linear algebra
Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra
v t e

1. A boolean algebra is a complemented distributive lattice. (def)

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

3. A boolean algebra is

4. A distributive orthocomplemented lattice is

orthomodular

.

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.^[3]

16. A modular complemented lattice is relatively complemented.^[4]

17. A boolean algebra is

. (1,15,16)

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

19. A heyting algebra is distributive.^[5]

20. A

is a distributive lattice.

21. A metric lattice is modular.^[6]

22. A modular lattice is semi-modular.^[7]

23. A projective lattice is modular.^[8]

24. A projective lattice is geometric. (def)

25. A geometric lattice is semi-modular.^[9]

26. A semi-modular lattice is atomic.

]

27. An

atomic

lattice is a lattice. (def)

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

29. A

. (def)

Notes

References