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A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
A small multiple (sometimes called trellis chart, lattice chart, grid chart, or panel chart) is a series of similar graphs or charts using the same scale and axes, allowing them to be easily compared. It uses multiple views to show different partitions of a dataset.
A common type of lattice graph (known under different names, such as grid graph or square grid graph) is the graph whose vertices correspond to the points in the plane with integer coordinates, x-coordinates being in the range 1, ..., n, y-coordinates being in the range 1, ..., m, and two vertices being connected by an edge whenever the corresponding points are at distance 1.
A totally ordered set is a distributive lattice. 21. A metric lattice is modular. [6] 22. A modular lattice is semi-modular. [7] 23. A projective lattice is modular. [8] 24. A projective lattice is geometric. (def) 25. A geometric lattice is semi-modular. [9] 26. A semi-modular lattice is atomic. [10] [disputed – discuss] 27. An atomic ...
For example, a crystal, viewed as a lattice with a single kind of atom located at every lattice point (the simplest basis form), may also be viewed as a lattice with a basis of two atoms. In this case, a primitive unit cell is a unit cell having only one lattice point in the first way of describing the crystal in order to ensure the smallest ...
A lattice is distributive if and only if none of its sublattices is isomorphic to M 3 or N 5; a sublattice is a subset that is closed under the meet and join operations of the original lattice. Note that this is not the same as being a subset that is a lattice under the original order (but possibly with different join and meet operations).
For an example, the lattice of subgroups of the dihedral group of order 8 is not modular. The smallest non-modular lattice is the "pentagon" lattice N 5 consisting of five elements 0, 1, x, a, b such that 0 < x < b < 1, 0 < a < 1, and a is not comparable to x or to b. For this lattice, x ∨ (a ∧ b) = x ∨ 0 = x < b = 1 ∧ b = (x ∨ a) ∧ b
Various different characterizations exist. For example, the following is an equivalent law that avoids the use of choice functions [citation needed].For any set S of sets, we define the set S # to be the set of all subsets X of the complete lattice that have non-empty intersection with all members of S.