Abstract: Let be the set of positive integers. As
a semigroup under the binary operation given by the greatest common divisor of a
and does not have a minimal generating
set. For each the subsemigroup (under the binary operation given
by the greatest common divisor) has a unique minimal generating set T,
and The asymptotics of this result are
considered as Analogues of the above results are
considered for the binary operation given by least common multiple and for
minimal generating sets in some algebraic categories other than semigroups. This
note could find classroom/homework use as enrichment material in a variety of
undergraduate courses, especially courses on abstract algebra or elementary
number theory.
Keywords and phrases: minimal generating set, semigroup, greatest common divisor, least common multiple, natural numbers, prime number, cyclic group, vector space, unique factorization domain, polynomial ring.
