 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-i/5/b/i/solution

Suppose

y * y = y

. Choose

z

with

y * z = e

, using the given right inverse element property. Multiply the idempotent relation on the right by

z

. By associativity and the given right identity element property,

e = y * z = (y * y) * z = y * (y * z) = y * e = y .

Thus the only idempotent element of a semigroup is $e$ . This argument uses no cancellation and does not assume a left identity element.

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