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:

D36 Lattice By Learn Loner

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.

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