Let ({p, q}, *) be a semigroup where p * p = q. Show that: a. p * q = q * p, and b. q * q = q.

Question

Accepted Answer

p*q = q*p and q*q = q both hold in the semigroup. A semigroup just means the operation * is associative (we get to regroup, but cannot assume commutativity or an identity). The set has only two elements {p, q}, and we are told p*p = q. Part (a) follows by sliding the brackets: p*q is p*(p*p), which regroups to (p*p)*p = q*p. Part (b) uses closure - q*q must be either p or q - together with associativity to pin it down to q in both possible cases.