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In the branch of mathematics known as order theory, a semimodular lattice, is a lattice that satisfies the following condition: ;Semimodular law: ''a'' ∧ ''b'' <: ''a'' implies ''b'' <: ''a'' ∨ ''b''. The notation ''a'' <: ''b'' means that ''b'' covers ''a'', i.e. ''a'' < ''b'' and there is no element ''c'' such that ''a'' < ''c'' < ''b''. An atomistic (hence algebraic) semimodular bounded lattice is called a matroid lattice because such lattices are equivalent to (simple) matroids. An atomistic semimodular bounded lattice of finite length is called a geometric lattice and corresponds to a matroid of finite rank.〔These definitions follow Stern (1999). Some authors use the term ''geometric lattice'' for the more general matroid lattices. But most authors only deal with the finite case, in which both definitions are equivalent to semimodular and atomistic.〕 Semimodular lattices are also known as upper semimodular lattices; the dual notion is that of a lower semimodular lattice. A finite lattice is modular if and only if it is both upper and lower semimodular. A finite lattice, or more generally a lattice satisfying the ascending chain condition or the descending chain condition, is semimodular if and only if it is M-symmetric. Some authors refer to M-symmetric lattices as semimodular lattices.〔For instance Fofanova (2001).〕 ==Birkhoff's condition== A lattice is sometimes called weakly semimodular if it satisfies the following condition due to Garrett Birkhoff: ;Birkhoff's condition: If ''a'' ∧ ''b'' <: ''a'' and ''a'' ∧ ''b'' <: ''b'', :then ''a'' <: ''a'' ∨ ''b'' and ''b'' <: ''a'' ∨ ''b''. Every semimodular lattice is weakly semimodular. The converse is true for lattices of finite length, and more generally for upper continuous relatively atomic lattices. 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「Semimodular lattice」の詳細全文を読む スポンサード リンク
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