Show that the elements of the lattice (N,≤), where N is the set of positive intergers and a≤b if and only if a divides b, satisfy the distributive property. Since in distributive lattice , atmost one complement exist for each element. So , if we get 2 complements for an element then we can say given lattice is not distributive. So, to prove above statement , I assumed 2 complements 'b' and 'c' for an element 'a' now I am unable to check these complements exist or not for a given relation "x divides y" because we can't say about upper bound of the lattice because given set is set of positive integers. So , how to prove it. Please help!
2026-03-25 07:40:22.1774424422
Distributive Lattice
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The meet of two elements is the greatest common denominator
(gcd), the join is the least common multiple (lcm).
Show distributivity directly
a gcd (b lcm c) = (a gcd b) lcm (a gcd c).
Use the prime factorisation theorem to obtain
expressions for the gcf and lcm to prove this.
Clearly this lattice does not provide complements.
Thus you cannot use the unique complement property.