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In general, different bases B will generate different lattices. However, if the transition matrix T between the bases is in () - the general linear group of R (in simple terms this means that all the entries of T are in R and all the entries of are in R - which is equivalent to saying that the determinant of T is in - the unit group of elements ...
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).
This matrix belongs to the modular group (,). This equivalence of lattices can be thought of as underlying many of the properties of elliptic functions (especially the Weierstrass elliptic function ) and modular forms.
Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self-dual. Given a pair (m,n) of nonnegative integers, an even unimodular lattice of signature (m,n) exists if and only if m−n is divisible by 8, but an odd unimodular lattice of signature (m,n) always exists. In particular, even ...
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.
In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets.. Formally, a complete lattice L is said to be completely distributive if, for any doubly indexed family {x j,k | j in J, k in K j} of L, we have
Since N 5 is a sublattice of S 7, it follows that the M-symmetric lattices do not form a subvariety of the variety of lattices. M-symmetry is not a self-dual notion. A dual modular pair is a pair which is modular in the dual lattice, and a lattice is called dually M-symmetric or M * -symmetric if its dual is M-symmetric.
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