Lattice Definition and 510 Threads

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 two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.
Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.

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  1. E

    Exploring the Densest Space-Filling Polyhedra Lattice

    What is the densiest space-filling polyhedra lattice?My first guess would be the cubic one, but i am not sure.
  2. S

    Life on the Lattice: Matthew Nobes' QCD Technical Blog

    There is a new site called "Life on the Lattice" by Matthew Nobes which is what you might call a technical blog about Lattice QCD. He gets into some detail, but he is a clear guide, and it's not beyond the knowledge level of some I've seen post here. Give it a try, it's at...
  3. L

    Can an Equation Solver Handle Lattice Functions with Join and Meet Operators?

    Hi, I need some direction help. The problem I am facing is related to solutions of equations , where equations are montone functions over complete lattice , operators being meet and join and the solution set should range over elements of lattice. Normally data flow analysis problems in...
  4. L

    Toda lattice and Korteweg de Vries relation

    Does anybody know where I can find a good reference which describes in a simple way the relations between the Toda lattice and the Korteweg de Vries equation and in particular the former as the finite-dimensional equivalent of the latter?
  5. K

    Is Space-Time a Lattice? Exploring the Discreteness of the Universe

    It seems unlikely that space time is a lattice because it contradicts the fundamental assumptions and theories in physics. If space-time is a lattice its dimensions are likely to be on the order of the Plank time and distance scales. Opinions?
  6. S

    Can Renormalization Simplify the 2 x 2 Ising Lattice Problem?

    Hey all, I have a problem I'm working on. A 2 x 2 ising lattice, \ H = K_1\sum_{nn}\sigma_i\sigma_j \ + \ K_2\sum_{nnn}\sigma_i\sigma_j \ + \ K_3\sum_{sg}\sigma_i\sigma_j\sigma_k\sigma_l Were to find H as an explicit function the sigma's...
  7. W

    Is there any software to simulate lattice vibration?

    I really want to see the lattice wave.Who knows where could I get the software simulating it? Thank you very much!
  8. phoenixthoth

    Complete Lattice: Is Maximal Element Whole Lattice?

    by lattice, i mean definition 2 at http://en2.wikipedia.org/wiki/Complete+lattice . this is probably ill-posed but here goes nothing... is there a nonempty lattice such that the maximal element of the lattice is the whole lattice? so what i mean by a maximal element is an element b...
  9. B

    Proving Q is a Lattice but Not a (sigma)-Lattice

    I'm reading "A Course in Advanced Calculus" by Robert Borden, and one of the problems begins as follows: "Prove that the field Q is a lattice, but not a (sigma)-lattice, under the usual order" (pg.25) Q is of course the rational numbers. However, Q doesn't seem to be a lattice, since...
  10. G

    QFT on a Lattice: Researching Lattice Field Theory

    Hi everyone! I would like to post a new thread, related to my research work: QFT on a lattice, i.e. on computers! Is anyone interested?
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