<p>I have a question about a term in a paper about formal power series. It was never defined but the author used <span class="math-container">$K \langle A \rangle$</span>, what set is <span class="math-container">$K \langle A \rangle$</span>? Here <span class="math-container">$K$</span> is a field (commutative or not) and <span class="math-container">$A$</span> is an alphabet. I know that K<span class="math-container">$\langle \langle A \rangle \rangle$</span> is the set of all formal power series from <span class="math-container">$A^*$</span> to <span class="math-container">$K$</span>.</p>
<p>Here the paper: <a href="https://www.sciencedirect.com/science/article/abs/pii/S0020019097001622" rel="nofollow noreferrer">https://www.sciencedirect.com/science/article/abs/pii/S0020019097001622</a></p>

CC BY-SA 4.0
2022-10-21T22:11:43.280Z
$K\langle A \rangle$ is (probably) the noncommutative polynomial ring whose set of variables is A. I believe this is the same notation as used in Berstel & Reutenauer's book "Noncommutative Rational Series with Applications"
Joshua Grochow
CC BY-SA 4.0
2022-10-21T22:55:21.843Z
In all texts on formal power series I have seen, $K\langle A \rangle$ is the subset of $K\langle\langle A \rangle\rangle$ with "finite support", meaning only a finite number of coefficients are non-zero. As @JoshuaGrochow says, they are usually called polynomials.
Martin Berger
CC BY-SA 4.0
2022-10-22T21:09:36.860Z
The book "Automata-Theoretic Aspects of Formal Power Series" by Arto Salomaa and Matti Soittola might be a good resource: https://link.springer.com/book/10.1007/978-1-4612-6264-0
Abuzer Yakaryilmaz
<automata-theory><definitions>
Definition in some old paper about formal power series (automata theory)
Tim
# Definition in some old paper about formal power series (automata theory)

(Asked by: username_0 on 2022-10-21)

I have a question about a term in a paper about formal power series. It was never defined but the author used $K \langle A \rangle$, what set is $K \langle A \rangle$? Here $K$ is a field (commutative or not) and $A$ is an alphabet. I know that K$\langle \langle A \rangle \rangle$ is the set of all formal power series from $A^\*$ to $K$.

Here the paper: <https://www.sciencedirect.com/science/article/abs/pii/S0020019097001622>

**Tags:** automata-theory, definitions
> **Comments:**
> **username_1**: $K\langle A \rangle$ is (probably) the noncommutative polynomial ring whose set of variables is A. I believe this is the same notation as used in Berstel & Reutenauer's book "Noncommutative Rational Series with Applications"  
> **username_2**: In all texts on formal power series I have seen, $K\langle A \rangle$ is the subset of $K\langle\langle A \rangle\rangle$ with "finite support", meaning only a finite number of coefficients are non-zero. As @JoshuaGrochow says, they are usually called polynomials.  
> **username_3**: The book "Automata-Theoretic Aspects of Formal Power Series" by Arto Salomaa and Matti Soittola might be a good resource: https://link.springer.com/book/10.1007/978-1-4612-6264-0  

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## 0 answers

