Answer to Question #162308 in Abstract Algebra for K

Question #162308
Suppose * is an associative and commutative binary operation of a set S. Show that H = { a is an element of S | a*a=a} is closed under the operation *. 
1
Expert's answer
2021-02-12T10:34:21-0500

Let * be an associative commutative binary operation on a set SS. Let us show that H={aS  aa=a}H=\{a\in S\ |\ a*a=a\} is closed under operation .*. Let a,bHa,b\in H. Then aa=aa*a=a and bb=bb*b=b . It follows from commutativity and associativity of * that(ab)(ab)=a(b(ab))=a((ba)b)=a((ab)b)=a(a(bb))=a(ab)=(aa)b=ab.(a*b)*(a*b)= a*(b*(a*b))= a*((b*a)*b)= a*((a*b)*b)= a*(a*(b*b))= a*(a*b)= (a*a)*b=a*b.

Therefore, abHa*b\in H, and HH is closed under operation .*.



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