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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).
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms ...
A lattice is positive definite if the norm of all nonzero elements is positive. The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (a i, a j), where the elements a i form a basis for the lattice. An integral lattice is unimodular if its determinant is 1 or −1.
The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2 n n!. As a matrix group it is given by the set of all n × n signed permutation matrices. This group is isomorphic to the semidirect product
The unimodular matrix used (possibly implicitly) in lattice reduction and in the Hermite normal form of matrices. The Kronecker product of two unimodular matrices is also unimodular. This follows since det ( A ⊗ B ) = ( det A ) q ( det B ) p , {\displaystyle \det(A\otimes B)=(\det A)^{q}(\det B)^{p},} where p and q are the dimensions of A and ...
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The hpc lattice (left) and the ccf lattice (right) The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of ...