Question

If (S,*) is a.semigroupand x € s show that (S,∆) is a semigroup if a∆b =a*x*b

Answer 1

Let $(S,*)$ be a semigroup and $x\in S$. Let us show that $(S,∆)$ is a semigroup if $a∆b =a*x*b.$

Since $x\in S$, then $a,b\in S$ imply that $a*x*b\in S,$ and hence $a∆b\in S.$ It follows that the operation $∆$ is defined on the set $S.$

Taking into account that

$(a∆b)∆c=(a*x*b)∆c=(a*x*b)*x*c\\=a*x*(b*x*c)=a*x*(b∆c)=a∆(b∆c),$

we conclude that the operation $∆$ is associative, and hence $(S,∆)$ is a semigroup.