Are the free monoids always infinite?

Question

It's the Wikipedia's definition of the free monoid: **In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. ** My question is: Is the free monoid always infinite? there always exist infinite words of the given alphabet(and it doesn't depend on the cardinality of the alphabet), isn't it?

Answer 1

It's always infinite unless the alphabet is empty.

If the alphabet is empty, then there is only the empty word : the free monoid on $\emptyset$ is the trivial monoid.

If there is at least one letter (say $a$) in the alphabet then there is an infinite number of words : you can at least form $a^n = aaa\cdots a$ ($n$ times).

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I'm not sure what you mean about the cardinality : if the alphabet is $S$, then the cardinality of the free monoid on $S$ is $|S|\cdot \aleph_0$.

So if $S$ is finite (and non-empty) this gives $\aleph_0$, and if $S$ is infinite this gives $|S|$.