Semimodular Lattice
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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! 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…mehr

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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! 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. 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.