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=Burnside ring=
''of a group $G$''

For a $G$-set $X$, that is, a set with a [[group action]], let $[X]$ denote the isomorphism class of $X$.  These classes form a [[semi-ring]] with addition given by [[disjoint union]] $[X] + [Y] = [X \sqcup Y]$ and multiplication given by [[Cartesian product]] $[X] \cdot [Y] = [X \times Y]$.  The zero element of this semiring is the (class of the) empty set and the multiplicative identity is the (class of the) one-point set.  The ''Burnside ring'' $B(G)$ is the [[Grothendieck ring]] of this semiring. 

====References====
* David D. Benson, ''Representations and Cohomology: Volume 1, Basic Representation Theory of Finite Groups and Associative Algebras'' Cambridge University Press (1998) ISBN 0-521-63653-1 {{ZBL|0908.20001}}

=Necklace algebra=
The algebra $N(R)$ over a ring $R$ with additive group $R^{\mathbf{N}} = \{ a = (a_1,a_2,\ldots) : a_i \in R \}$ and multiplication given by
$$
(a * b)_n = \sum_{i,j : \mathrm{lcm}(i,j) = m} \mathrm{hcf}(i,j) a_i b_j \ .
$$
The definition of multiplication generalises the Metropolis–Rota multiplication formula for the [[necklace polynomial]]s.

For a [[binomial ring]] $R$, the necklace algebra $N(R)$ is isomorphic to the universal [[lambda-ring]] $\Lambda(R)$ via
$$
\prod_{n=1}^\infty \left({ 1 - (-t)^n }\right)^{a_n} \longleftrightarrow ( a_n )  \ ,
$$
which may be regarded as an abstraction of the [[Artin–Hasse exponential]] map.  Here the exponentiation by elements $a \in R$ is defined by the [[binomial series]]
$$
(1 + tx)^a = 1 + \sum_{n=1}^\infty \binom{a}{n} x^n t^n  \ .
$$

====References====
* Yau, Donald ''Lambda-rings'' World Scientific (2010) ISBN 978-981-4299-09-1 {{ZBL|1198.13003}}

=Necklace polynomial=
A polynomial of the form  
$$
M_n(x) = \frac{1}{n} \sum_{d | n} \mu(d) x^{n/d} \ .
$$
Here $\mu$ is the [[Möbius function]].  When $x$ is a natural number, $M_n(x)$ counts the number of "necklaces": assignments of $n$ colours to $x$ beads under cyclic symmetry which are "primitive", that is, not the repetition of a proper subsequence of colour assignments.

Metropolis and Rota showed that
$$
M_n(xy) = \sum_{[i,j]=n} (i,j) M_i(x) M_j(y)
$$
where $[,]$ denotes [[least common multiple]] and $(,)$ [[highest common factor]].

====References====
* Yau, Donald ''Lambda-rings'' World Scientific (2010) ISBN 978-981-4299-09-1 {{ZBL|1198.13003}}