"Not clear what this means?"


by Eric Bruylant Aug 8 2016

Show solution Note that our description of Hasse diagrams made use of the covers relation $~$\\prec$~$\. The covers relation, however, is not a helpful in many posets\. Consider the poset $~$\\langle \\mathbb R, \\leq \\rangle$~$ of the real numbers ordered by the standard comparison $~$\\leq$~$\. We have $~$0 < 1$~$, but how would we convey that with a Hasse diagram? The problem is that $~$0$~$ has no covers, even though it is not a maximal element in $~$\\mathbb R$~$\. In fact, for any $~$x \\in \\mathbb R$~$ such that $~$x > 0$~$, we can find a $~$y \\in \\mathbb R$~$ such that $~$0 < y < x$~$\. This "infinite density" of $~$\\mathbb R$~$ makes it impossible to depict using a Hasse diagram\.

Not clear what this means?