Find whether the following partially ordered sets are lattices, whether they have a zero and unit elements and whether they are complemented lattices:
(d) The set of all positive integers, partially ordered by $\leq$, where: $z_{1}$ $\leq$ $z_{2}$ iff $z_{2}$ is an integral multiple of $z_{1}.$
I do not know exactly what is the meaning of supremum and infimum in our case but I feel it is a lattice. I do not see that it has either zero or unit element.
This question is sooo difficult for my imagination, Could anyone help me in it?
thanks.
For all $n\ne1$, you have $1\le n$. For all primes numbers $p$, you have $k\le p \Rightarrow k=1$. So certainly $1$ is your zero element and thus any two elements must have an infinium (which is actually their $\gcd$). And the supremum is no more than their product (and is actually their ${\rm lcm}$).
There is no greatest element so this is not a complemented lattice.