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In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumptions of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way. ==Definition== Recall that a lattice is a partially ordered set in which any two elements and have a least upper bound and a greatest lower bound . In a lattice, or more generally a partially ordered set, an element covers another element (written as or ) if and there is no third element between and . A lattice or partially ordered set is graded when it can be given a rank function mapping its elements to integers, such that whenever and in particular whenever even . In a lattice having a bottom element, one may assume without loss of generality that its rank is zero. The atoms of such a lattice are the elements with rank one, and the lattice is atomistic if every element is the least upper bound of some set of atoms. A graded lattice is semimodular if, for every and , its rank function obeys the identity〔, Theorem 15, p. 40. More precisely, Birkhoff's definition reads "We shall call P (upper) semimodular when it satisfies: If ''a''≠''b'' both cover ''c'', then there exists a ''d''∈''P'' which covers both ''a'' and ''b''" (p.39). Theorem 15 states: "A graded lattice of finite length is semimodular if and only if ''r''(''x'')+''r''(''y'')≥''r''(''x''∧''y'')+''r''(''x''∨''y'')".〕 : A matroid lattice is a lattice that is both atomistic and semimodular,〔.〕〔.〕 and a geometric lattice is a matroid lattice with finitely many elements.〔, p. 51.〕 Some authors consider only finite matroid lattices, and use the terms "geometric lattice" and "matroid lattice" interchangeably for both.〔.〕 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「geometric lattice」の詳細全文を読む スポンサード リンク
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