I think the matching should be in ONE -to-MANY manners.
(a) Groups ----------------> (p) Associativity ,(q) Identity ,(s) Left inverse
(b) Semigroups -----------> (p) Associativity
(c) Monoids ----------------> (p) Associativity ,(q) Identity
(d) Abelian groups -------> (p) Associativity ,(q) Identity ,(s) Left inverse,(r) Commutativity