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Supersolvable Lattice
In mathematics, a supersolvable lattice is a graded poset, graded Lattice (order), lattice that has a maximal total order#Chains, chain of elements, each of which obeys a certain modularity relationship. The definition encapsulates many of the nice properties of lattice of subgroups, lattices of subgroups of supersolvable groups. Motivation A finite group (mathematics), group G is said to be ''supersolvable'' if it admits a maximal subgroup series, chain (or ''series'') of subgroups so that each subgroup in the chain is normal in G. A normal subgroup has been known since the 1940s to be left and (dual) right Modular pair, modular as an element of the lattice of subgroups. Richard P. Stanley, Richard Stanley noticed in the 1970s that certain geometric lattices, such as the partition lattice, obeyed similar properties, and gave a lattice-theoretic abstraction. Definition A finite graded lattice L is supersolvable if it admits a maximal chain \mathbf of elements (called an M-chain o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Poset
In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) ''P'' equipped with a rank function ''ρ'' from ''P'' to the set N of all natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...s. ''ρ'' must satisfy the following two properties: * The rank function is compatible with the ordering, meaning that for all ''x'' and ''y'' in the order, if ''x'' < ''y'' then ''ρ''(''x'') < ''ρ''(''y''), and * The rank is consistent with the covering relation of the ordering, meaning that for all ''x'' and ''y'', if ''y'' covers ''x'' then ''ρ''(''y'') = ''ρ''(''x'') + 1. The value of the rank functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
