Match the pairs in the following. (a) Groups (b) Semigroups (c) Monoids (d) Abelian groups with (p) Associativity (q) Identity (r) Commutativity (s) Left inverse

GATE 1990 · Discrete Mathematics · Group Theory · easy

Answer: a-s (Groups - Left inverse), b-p (Semigroups - Associativity), c-q (Monoids - Identity), d-r (Abelian groups - Commutativity)

1. Identify the defining property added at each level: (b) Semigroups need only associativity -> (p); (c) Monoids add an identity -> (q); the distinguishing axiom for (a) Groups beyond a monoid is the existence of inverses -> left inverse (s)
2. Match the abelian group: (d) Abelian groups are groups whose operation is also commutative, so they match commutativity -> (r); the remaining property associativity is the semigroup's, identity the monoid's