Consider the set of integers {1, 2, 3, 4, 6, 8, 12, 24} together with the two binary operations LCM (lowest common multi...

Question

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? A. group  B. ring  C. field  D. lattice

Accepted Answer

The structure is a lattice (Option D). The set {1, 2, 3, 4, 6, 8, 12, 24} is exactly the set of all positive divisors of 24. Ordered by divisibility, any two of these numbers have a greatest common divisor (GCD) inside the set and a lowest common multiple (LCM) inside the set, and both stay within the divisors of 24. Two operations giving a meet (GCD) and a join (LCM) that satisfy the lattice laws make this a lattice. It is not a group, ring or field: GCD and LCM have no inverses (you cannot 'undo' a GCD), so the group-style structures do not apply.