Division lattice

The non-negative integers partially ordered by divisibility.

The division lattice is an infinite complete bounded distributive lattice whose elements are the natural numbers ordered by divisibility. Its least element is 1, which divides all natural numbers, while its greatest element is 0, which is divisible by all natural numbers. The meet operation is greatest common divisor while the join operation is least common multiple.[1]

The prime numbers are precisely the atoms of the division lattice, namely those natural numbers divisible only by themselves and 1.[2]

For any square-free number n, its divisors form a Boolean algebra that is a sublattice of the division lattice. The elements of this sublattice are representable as the subsets of the set of prime factors of n. [3]

References

  1. ^ Davey, B. A.; Priestley, H. A. (2002), Introduction to Lattices and Order, Cambridge University Press, p. 37, ISBN 978-0-521-78451-1
  2. ^ Adhikari, M. R.; Adhikari, A. (2003), Groups, Rings And Modules With Applications, Universities Press, p. 13, ISBN 9788173714290
  3. ^ Halmos, Paul R. (2018), Lectures on Boolean Algebras, Dover Books on Mathematics, Courier Dover Publications, p. 7, ISBN 9780486834573