A lattice is a partially ordered set in which every pair of elements has a unique supremum (least upper bound) and a unique infimum (greatest lower bound). In simpler terms, a lattice is a set with a certain structure that allows us to compare its elements in a particular way.
Now, let’s consider the set D36, which is the set of divisors of 36. The elements of D36 are {1, 2, 3, 4, 6, 9, 12, 18, 36}. We can order these elements by divisibility, so that x ≤ y if and only if x divides y. For example, 4 ≤ 12 because 4 divides 12.
To prove that D36 forms a lattice, we need to show that every pair of elements in D36 has a unique supremum and a unique infimum.
Let’s begin by drawing a Hasse diagram for D36:
In this diagram, the elements are arranged vertically, and the line connecting two elements indicates that the lower one divides the upper one. For example, 18 and 12 are connected by a line because 12 is divisible by 18.
Now, let’s consider a pair of elements in D36, say 4 and 6. The infimum of 4 and 6 is 2, since 2 is the largest number that divides both 4 and 6. The supremum of 4 and 6 is 12, since 12 is the smallest number that both 4 and 6 divide into. This can be seen from the Hasse diagram, as 2 is directly below 4 and 6, and 12 is directly above them.
Similarly, we can consider any other pair of elements in D36, and we will find that it has a unique infimum and a unique supremum. Therefore, D36 forms a lattice.