Necklace ring

In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983) to elucidate the multiplicative properties of necklace polynomials.

Definition

If A is a commutative ring then the necklace ring over A consists of all infinite sequences $(a_{1},a_{2},...)$ of elements of A. Addition in the necklace ring is given by pointwise addition of sequences. Multiplication is given by a sort of arithmetic convolution: the product of $(a_{1},a_{2},...)$ and $(b_{1},b_{2},...)$ has components

$\displaystyle c_{n}=\sum _{[i,j]=n}(i,j)a_{i}b_{j}$

where $[i,j]$ is the least common multiple of $i$ and $j$, and $(i,j)$ is their greatest common divisor.

This ring structure is isomorphic to the multiplication of formal power series written in "necklace coordinates": that is, identifying an integer sequence $(a_{1},a_{2},...)$ with the power series $\textstyle \prod _{n\geq 0}(1{-}t^{n})^{-a_{n}}$.

See also

• Witt vector

References

• Hazewinkel, Michiel (2009). "Witt vectors I". Handbook of Algebra. Vol. 6. Elsevier/North-Holland. pp. 319–472. arXiv:0804.3888. Bibcode:2008arXiv0804.3888H. ISBN 978-0-444-53257-2. MR 2553661.
• Metropolis, N.; Rota, Gian-Carlo (1983). "Witt vectors and the algebra of necklaces". Advances in Mathematics. 50 (2): 95–125. doi:10.1016/0001-8708(83)90035-X. MR 0723197.