Example
The full transformation semigroup T3 consists of all functions from the set {1, 2, 3} to itself; there are 27 of these. Write (a b c) for the function which sends 1 to a, 2 to b, and 3 to c. Since T3 contains the identity map, (1 2 3), there is no need to adjoin an identity.
The egg-box diagram for T3 has three D-classes. They are also J-classes, because these relations coincide for a finite semigroup.
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In T3, two functions are L-related if and only if they have the same image. Such functions appear in the same column of the table above. Likewise, the functions f and g are R-related if and only if
- f(x) = f(y) ⇔ g(x) = g(y)
for x and y in {1, 2, 3}; such functions are in the same table row. Consequently, two functions are D-related if and only if their images are the same size.
The elements in bold are the idempotents. Any H-class containing one of these is a (maximal) subgroup. In particular, the third D-class is isomorphic to the symmetric group S3. There are also six subgroups of order 2, and three of order 1 (as well as subgroups of these subgroups). Six elements of T3 are not in any subgroup.
Read more about this topic: Green's Relations
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