Matches in DBpedia 2014 for { ?s ?p 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. 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.. }
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- Modular_lattice abstract "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. 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.".