|
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition: ;Modular law: ''x'' ≤ ''b'' implies ''x'' ∨ (''a'' ∧ ''b'') = (''x'' ∨ ''a'') ∧ ''b'', where ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. For an intuition behind the modularity condition see 〔http://math.stackexchange.com/a/443947/40167〕 and below. Modular lattices arise naturally in algebra and in many other areas of mathematics. For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice. Every distributive lattice is modular. In a not necessarily modular lattice, there may still be elements ''b'' for which the modular law holds in connection with arbitrary elements ''a'' and ''x'' (≤ ''b''). Such an element is called a modular element. Even more generally, the modular law may hold for a fixed pair (''a'', ''b''). Such a pair is called a modular pair, and there are various generalizations of modularity related to this notion and to semimodularity. ==Introduction== The modular law can be seen as a restricted associative law that connects the two lattice operations similarly to the way in which the associative law λ(μ''x'') = (λμ)''x'' for vector spaces connects multiplication in the field and scalar multiplication. The restriction ''x'' ≤ ''b'' is clearly necessary, since it follows from ''x'' ∨ (''a'' ∧ ''b'') = (''x'' ∨ ''a'') ∧ ''b''. In other words, no lattice with more than one element satisfies the unrestricted consequent of the modular law. (To see this, just pick non-maximal ''b'' and let ''x'' be any element strictly greater than ''b''.) It is easy to see that ''x'' ≤ ''b'' implies ''x'' ∨ (''a'' ∧ ''b'') ≤ (''x'' ∨ ''a'') ∧ ''b'' in every lattice. Therefore the modular law can also be stated as ;Modular law (variant): ''x'' ≤ ''b'' implies ''x'' ∨ (''a'' ∧ ''b'') ≥ (''x'' ∨ ''a'') ∧ ''b''. By substituting ''x'' with ''x'' ∧ ''b'', the modular law can be expressed as an equation that is required to hold unconditionally, as follows: ;Modular identity: (''x'' ∧ ''b'') ∨ (''a'' ∧ ''b'') = (∧ ''b'') ∨ ''a'' ) ∧ ''b''. This shows that, using terminology from universal algebra, the modular lattices form a subvariety of the variety of lattices. Therefore all homomorphic images, sublattices and direct products of modular lattices are again modular. The smallest non-modular lattice is the "pentagon" lattice ''N''5 consisting of five elements 0,1,''x'',''a'',''b'' such that 0 < ''x'' < ''b'' < 1, 0 < ''a'' < 1, and ''a'' is not comparable to ''x'' or to ''b''. For this lattice ''x'' ∨ (''a'' ∧ ''b'') = ''x'' ∨ 0 = ''x'' < ''b'' = 1 ∧ ''b'' = (''x'' ∨ ''a'') ∧ ''b'' holds, contradicting the modular law. Every non-modular lattice contains a copy of ''N''5 as a sublattice. Modular lattices are sometimes called Dedekind lattices after Richard Dedekind, who discovered the modular identity in several motivating examples. 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「Modular lattice」の詳細全文を読む スポンサード リンク
|