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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. == 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.〔Rutherford (1965), p.77.〕 3. A boolean algebra is orthocomplemented.〔Rutherford (1965), p.32-33.〕 4. A distributive orthocomplemented lattice is orthomodular.〔(PlanetMath: orthomodular lattice )〕 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.〔Rutherford (1965), p.22.〕 16. A modular complemented lattice is relatively complemented.〔Rutherford (1965), p.31.〕 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.〔Rutherford (1965), Th.25.1 p.74.〕 20. A totally ordered set is a distributive lattice. 21. A metric lattice is modular.〔Rutherford (1965), Th.8.1 p.22.〕 22. A modular lattice is semi-modular.〔Rutherford (1965), p.87.〕 23. A projective lattice is modular.〔Rutherford (1965), p.94.〕 24. A projective lattice is geometric. (def) 25. A geometric lattice is semi-modular.〔Rutherford (1965), Th.32.1 p.92.〕 26. A semi-modular lattice is atomic.〔Rutherford (1965), p.89.〕 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) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「map of lattices」の詳細全文を読む スポンサード リンク
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