If (S,*) is a.semigroupand x € s show that (S,∆) is a semigroup if a∆b =a*x*b
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Let be a semigroup and . Let us show that is a semigroup if
Since , then imply that and hence It follows that the operation is defined on the set
Taking into account that
we conclude that the operation is associative, and hence is a semigroup.