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Consider the set of integers $\{1,2,3,4,6,8,12,24\}$ together with the two binary operations LCM (lowest common multiple) and GCD (greatest common divisor). Which of the following algebraic structures does this represent?

  1. group
  2. ring
  3. field
  4. lattice
asked in Set Theory & Algebra by Veteran (99.2k points) | 318 views

1 Answer

+8 votes
Best answer

ans is lattice .

lattice

 

answered by Veteran (55.6k points)
selected by
How did you draw this lattice? How two elements are related?
I think this lattice has been drawn like this:

For any two elements a and b in this lattice, if a < b then b is the LCM of a and b and a is the GCD of a and b. For example, if 1 and 3 are connected then 3 is the LCM of 1 and 3 and 1 is the GCD of 1 and 3.

This is lattice (D24,|)  over partial order relation divisibility. Where D24 indicates positive integral divisors of 24.

LCM is given by LUB and HCF is given by GLB.

 

how can we have two operations in a lattice?


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