# Why is the zero-th power of the empty set {ε}?

It has been asked before why $\emptyset^\star=\{\epsilon\}$. The answer boils down to $\emptyset^\star$ being defined as $$L^\star = \bigcup_{i=0}^\infty L^i,$$ where a word in $L^i$ is the concatenation of $i$ words from $L$. So, for the empty language this leaves empty sets for all $i>0$, "but $L^0 = \{ \epsilon \}$ since $\epsilon$ is the concatenation of zero words from $L$. It doesn't matter if $L$ is empty or not, since we are choosing zero words from $L$." (quoted from the answer.)

And that's what I'm curious about: is there a good reason to say that the concatenation of zero words is $\epsilon$ even for the empty set? Can you compare it to $0^0=1$ in arithmetics (where it's just more or less arbitrary) or is there a strong logical reason for this to be true?

• This is discussed in Yuval's answer there and comments on it; what more do you want? – Raphael Jul 2 '16 at 15:56
• You're right, I didn't see the second answer - sorry! – lukas.coenig Jul 2 '16 at 16:30

Strings with concatenation and the empty word form a monoid.

In monoids, the empty concatenation/sum/product yields the neutral element, i.e. $\varepsilon$/0/1.

An empty concatenation does not need any words to start with, so $\emptyset^0 = \{\varepsilon\}$.

Intuitively, if you specialize

$L^n$ is the set of all words obtained by $n$-fold concatenation of words in $L$

to

$L^0$ is the set of all words obtained by zero concatenations (of words in $L$)

you can just stop reading after "concatenations" and the contents of $L$ do not matter; we don't even need it to be non-empty.

• N.B. the only reason this argument does not apply with equal force to justify $0^0 = 1$ is that exponentiation is not a monoid. – zwol Jul 2 '16 at 21:29
• @zwol: I don't think you've quite thought that through. Neither $L^n$ nor $m^n$ is a monoid operation. Raphael is using $L^n$ to denote repeated application of a monoid operation (namely concatenation); if you use $m^n$ the analogous way (to denote repeated multiplication), then $0^0 = 1$ is actually correct. The reason this argument doesn't apply to $0^0$ in general is that $x^y$ doesn't always denote repeated multiplication; $0^{0.1}$ (for example) has no monoid-related interpretation, and $\lim\limits_{x\to 0}0^x \ne 1$. – ruakh Jul 2 '16 at 22:54