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Consider the ordering relation x | y ⊆ N × N over natural numbers N such that x|y iff there exists z ∈ N such that x · z = y. A set is called lattice if every finite subset has a least upper bound and greatest lower bound. It is called a complete lattice if every subset has a least upper bound and greatest lower bound.is it a complete lattice or lattice explain ?
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Quoted from wiki :

  • The non-negative integers, ordered by divisibility. The least element of this lattice is the number 1, since it divides any other number. Perhaps surprisingly, the greatest element is 0, because it can be divided by any other number. The supremum of finite sets is given by the least common multiple and the infimum by the greatest common divisor. For infinite sets, the supremum will always be 0 while the infimum can well be greater than 1. For example, the set of all even numbers has 2 as the greatest common divisor. If 0 is removed from this structure it remains a lattice but ceases to be complete.

As the given relation is defined over set of natural numbers which is an infinite set, hence it will exclude thereby violating it to be a complete lattice though it is a lattice..

Hence the given lattice is not complete lattice..Had it been over set of whole numbers it would have been complete..However if we consider the standard ISO 80000-2 , where 0 is included in natural number, then the lattice is considered to be complete..

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