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Binary Operations : Monoids

Let S be a set with an associative binary operation but with no identity. Choose an element 1 not belonging to S, write M = {1} or S, and define an operation on M by using the operation of S and 1s=s=s1 for all s belonging to S. Show that M is a monoid.

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Proof. Let M={1}or S. It suffices to show that M is associative and has an identity. By the definition of the operation on M, we know that element 1 is in fact the identity since for ...

Solution Summary

Associativity and identity is used to provide proof of a monoid.