Necklace ring

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

If A is a commutative ring then the necklace ring over A consists of all infinite sequences ${\displaystyle (a_{1},a_{2},...)}$ 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 ${\displaystyle (a_{1},a_{2},...)}$ and ${\displaystyle (b_{1},b_{2},...)}$ has components

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

where ${\displaystyle [i,j]}$ is the least common multiple of ${\displaystyle i}$ and ${\displaystyle j}$, and ${\displaystyle (i,j)}$ 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 ${\displaystyle (a_{1},a_{2},...)}$ with the power series ${\displaystyle \textstyle \prod _{n\geq 0}(1{-}t^{n})^{-a_{n}}}$.

• Witt vector

• 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.