# Words of the same length in a language

Let $$L\subseteq\Sigma^*$$ be a language, where $$\Sigma$$ is a set, and let $$n\in\mathbb N$$.

I am wondering if there is some good terminology for

$$L\cap\Sigma^n$$.

Of course I could say "the set of words in $$L$$ of length $$n$$" but maybe there is some special terminology like "level set of length", "grade", "homogeneous section" or I don't know what.... ?

• Nope. No special terminology (to my knowledge) apart from "the set of words from $L$ with length $n$" – nir shahar Feb 25 at 23:53
• The closest thing I know is the concept of a slice in parameterized complexity, which is explained e.g. here. One could argue that using the normal coding length as a parameter the set $L \cap \Sigma^n$ is the $n$th slice of $L$. – Watercrystal Feb 26 at 9:55
• Yes, some of the names that you mention are used. $\Sigma^*$ is the free monoid over the $\Sigma$. A free monoid is a graded monoid by the monoid homomorphism $d:\Sigma\to\mathbb{N}$, defined as $d(\Sigma^n)=\{n\}$. The elements of $\Sigma^n$ are called the degree $n$ elements. – plop Feb 26 at 15:43
• @Watercrystal "slices" sounds good, if it's not used a lot maybe it should be used more. I'll accept it as an answer. – Bjørn Kjos-Hanssen Feb 26 at 19:59
• That's a name that people use, yes. It would always need to be defined before using, but well. – plop Feb 26 at 20:26