title: Map of lattices
text: # 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.
  
  3. A boolean algebra is orthocomplemented.
  
  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.
  
  16. A modular complemented lattice is relatively complemented.
  
  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.
  
  20. A totally ordered set is a distributive lattice.
  
  21. A metric lattice is modular.
  
  22. A modular lattice is semi-modular.
  
  23. A projective lattice is modular.
  
  24. A projective lattice is geometric. (def)
  
  25. A geometric lattice is semi-modular.
  
  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 semi-lattice is a partially ordered set. (def)
  
  