Map of lattices

Concept in mathematics

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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.

[画像:Diagram showing implications of common lattice properties]

Diagram showing implications of common lattice properties

Proofs of the relationships in the map

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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 orthocomplemented.^[2]

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 relatively complemented. (1,15,16)

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

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

20. A totally ordered set 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.^[10]^{[disputed – discuss ]}

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

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^ Rutherford (1965), p.77.
^ Rutherford (1965), p.32-33.
^ 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

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

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