Answer:
(M, +, ε)
Step-by-step Explanation
Let S be an arbitrary set. We can construct a free commutative monoid on S as follows:
Define M to be the set of all finite sequences of elements of S.
Define the binary operation on M as concatenation of sequences, denoted by +.
That is, for any two sequences
a = (a1, a2, ..., an) and b = (b1, b2, ..., bm), a + b is the sequence (a1, a2, ..., an, b1, b2, ..., bm).
Define the identity element of M as the empty sequence, denoted by ε.
That is, for any sequence a, ε + a = a + ε = a.
The resulting structure (M, +, ε) is a free commutative monoid on S.
To see why this is the case, note that every element of S can be identified with a sequence of length 1, and every element of M can be expressed as a concatenation of sequences of elements of S.
Moreover, the operation + is commutative, since concatenation of sequences is commutative, and the identity element ε acts as a neutral element for +.
Therefore, (M, +, ε) satisfies the axioms of a commutative monoid, and is a free object on the set S.