Question

If x * y = x + y + xy then (G, *) is _____________

(a) Monoid

(b) Abelian group

(c) Commutative semigroup

(d) Cyclic group

The question was posed to me at a job interview.

This is a very interesting question from Group Axioms topic in chapter Groups of Discrete Mathematics

Answer 1

Right choice is (c) Commutative semigroup

The explanation: Let x and y belongs to a group G.Here closure and associativity axiom holds simultaneously. Let e be an element in G such that x * e = x then x+e+xe=a => e(1+x)=0 => e = 0/(1+x) = 0. So, identity axiom does not exist but commutative property holds. Thus, (G,*) is a commutative semigroup.