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Consider the ordering relation $x\mid y \subseteq N \times N$ over natural numbers $N$ such that $x \mid y$ if there exists $z \in N$ 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. Then,

  1. $\mid$ is an equivalence relation.
  2. Every subset of $N$ has an upper bound under $|$.
  3. $\mid$ is a total order.
  4. $(N, \mid)$ is a complete lattice.
  5. $(N, \mid)$ is a lattice but not a complete lattice.
asked in Set Theory & Algebra by Veteran (28.1k points)   | 148 views

1 Answer

+1 vote
i think ans will be E)

as every subset  of this will not have LUB and GLB .
answered by Boss (9.2k points)  
edited by
Yes, it is a lattice , but how Complete ?

what is Least upper bound if  Subset is {x | x>=50}

I think for this Subset there is not LUB i.e. LUB exists for every finite subset but not any Infinite subset..
yeah you are right , i guess . for every subset LUB and GLB is not possible .
What does it mean by X.Z=Y?
Here prime numbers are not related to each other..how it will be a lattice?

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