Question

Let S = [1,2,4,5,10,20, 25, 50, 100). Then show that 5 forms a lattice under divisibility.

Draw the Hasse diagram also.

Answer 1

Let us draw the Hasse diagram of lattices, $(L_1,<)$ and $(L_2,<)$ where $L_1 = \{1, 2, 3, 4, 6, 12\}$ and $L_2 = \{2, 3, 6, 12, 24\}$ and a < b if and only if a divides b.

Note that a Hasse diagram is a graphical rendering of a partially ordered set displayed via the cover relation of the partially ordered set with an implied upward orientation. A point is drawn for each element of the poset, and line segments are drawn between these points according to the following two rules:

1. If $x<y$  in the poset, then the point corresponding to $x$ appears lower in the drawing than the point corresponding to $y$.

2. The line segment between the points corresponding to any two elements  $x$ and  $y$ of the poset is included in the drawing iff  $x$ covers $y$  or  $y$  covers $x$.

In our case, $x<y$ if and only if $x|y.$ Therefore, the Hasse diagrams are the following: