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A pre-$\lambda$-ring is a commutative ring $R$ with identity element $1$ and a set of mappings $\lambda^n : R \rightarrow R$, $n = 0,1,2,\ldots$ such that

i) $\lambda^0(x) = 1$ for all $x \in R$;

ii) $\lambda^1(x) = x$ for all $x \in R$;

iii) $\lambda^n(x+y) = \sum_{i+j=n} \lambda^i(x) \lambda^j(y)$.

Examples are, for instance, the topological $K$-groups $K(M)$ and $K_G(M)$, $G$ a compact Lie group (cf. [[K-theory|$K$-theory]]), and the complex representation ring $R(G)$ of a finite group $G$ (cf. [[Representation of a compact group]]). In all these cases the $\lambda^n$ are induced by taking exterior powers. For instance, for $M = \text{pt}$, $K(M) = \mathbf{Z}$ and the $\lambda$-structure is given by $\lambda^n(m) = \binom{m}{n}$ (binomial coefficients; the formula $\binom{m_1+m_2}{n} = \sum_{i+j=n} \binom{m_1}{i} \binom{m_2}{j}$ follows by the binomial expansion theorem from $(X+Y)^{m_1+m_2} = (X+Y)^{m_1} (X+Y)^{m_2}$.

Let $R$ be any commutative ring with unit element 1. Consider the set $\Lambda(R) = 1 + t R[[t]]$ of power series in $t$ over $R$ with constant term 1. Multiplication of power series turns $\Lambda(R)$ into an Abelian group. A pre-$\lambda$-ring structure on $\Lambda(R)$ defines a homomorphism of Abelian groups $\lambda_t : R \rightarrow \Lambda(R)$, $\lambda_t(x) = \lambda^0(x) + \lambda^1(x) t + \lambda^2(x) t^2 + \cdots$, and vice versa.

Let $\alpha(t) = 1 + a_1 t + a_2 t^2 + \cdots$, $\beta(t) = 1 + b_1 t + b_2 t^2 + \cdots$ be two elements of $\Lambda(R)$. Formally, write
$$
\alpha(t) = \prod_{i=1}^\infty \left({ 1 - \xi_i t }\right)
$$
$$
\beta(t) = \prod_{i=1}^\infty \left({ 1 - \eta_i t }\right)
$$
and consider the expressions
$$
\prod_{i,j=1}^\infty \left({ 1 - \xi_i \eta_j t }\right) = 1 + P_1 t + P_2 t^2 + \cdots \ ,
$$
$$
\prod_{i_1 < i_2 < \cdots < i_n}^\infty \left({ 1 - \xi_{i_1} \xi_{i_2} \cdots \xi_{i_n} t }\right) = 1 + L_{1,n} t + L_{2,n} t^2 + \cdots \ .
$$

The $P_i$ and $L_{i,n}$ are symmetric polynomial expressions in the $\xi$'s and $\eta$'s and hence can be written as universal polynomial expressions in the $a$'s and $b$'s. Now define a multiplication on $\Lambda(R)$ by
$$
\alpha(t) * \beta(t) = 1 + P_1(a,b) t + P_2(a,b) t^2 + \cdots
$$
($a = (a_1,a_2,\ldots)$, $b = (b_1,b_2,\ldots)$), and define operations (mappings) $\lambda^n : \Lambda(R) \rightarrow \Lambda(R)$ by
$$
\lambda^n \alpha(t) = 1 + L_{1,n}(a,b) t + L_{2,n}(a,b) t^2 + \cdots \ .
$$

The ring $\Lambda(R)$ with these operations is a pre-$\lambda$-ring. Given two pre-$\lambda$-rings $R_1$, $R_2$, a $\lambda$-ring homomorphism $\phi : R_1 \rightarrow R_2$ is a homomorphism of rings such that $\phi(\lambda^n(x)) = \lambda^n(\phi(x))$ for all $x \in R_1$, $n = 0,1,2,\ldots$.

A pre-$\lambda$-ring $R$ is a $\lambda$-ring if $\lambda_{-t} : R \rightarrow \Lambda(R)$, $\lambda_{-t}(x) = 1 - \lambda^1(x) t + \lambda^2(x) t^2 - \cdots$, is a homomorphism of pre-$\lambda$-rings. The ring $\Lambda(R)$ is always a $\lambda$-ring and so are the standard examples $K(M)$, $K_G(M)$, $R(G)$ of pre-$\lambda$-rings mentioned above.

On the other hand, consider a finite group $G$. A finite $G$-set is a finite set together with a [[group action]] of $G$. Using disjoint union and Cartesian products with diagonal action, the isomorphism classes of finite $G$-sets form a [[semi-ring]], $A^+(G)$. The associated [[Grothendieck ring]] $A(G)$ is called the ''Burnside ring''. On $A^+(G)$, define operations $\lambda^n : A^+(G) \rightarrow A^+(G)$ by taking $\lambda^n(S)$ to be the set of$n$-element subsets of $S$ with the natural induced $G$-action. This generalizes the $\lambda$-operations $\lambda^n$ on $\mathbf{N} \subset \mathbf{Z}$, $\lambda^n(m) = \binom{m}{n}$. Using iii), the $\lambda^n$ extend to $A(G)$, making the Burnside ring into a pre-$\lambda$-ring. As a rule this pre-$\lambda$-ring is not a $\lambda$-ring, [[#References|[a9]]].

Instead of pre-$\lambda$-ring and $\lambda$-ring one also finds, respectively, the phrases $\lambda$-ring and special $\lambda$-ring in the literature.

Let $R$ be a pre-$\lambda$-ring. One defines new operations $\psi^i : R \rightarrow R$ by the formula
$$
-t \frac{d}{dt} \log \lambda_{-t} (x) = \sum_{i=1}^\infty \psi^i(x) t^i \ .
$$

These operations are called the ''Adams operations'' on the pre-$\lambda$-ring $R$. They were introduced in the case $R = K(M)$ by J.F. Adams ([[#References|[a10]]]).

The Adams operations satisfy

iv) $\Psi^1(x) = x$;

v) $\Psi^n(x+y) = \Psi^n(x) + \Psi^n(y)$.

Let $R$ be a torsion-free pre-$\lambda$-ring; then $R$ is a $\lambda$-ring if and only if the Adams operations satisfy in addition

vi) $\Psi^i(1) = 1$;

vii) $\Psi^n(xy) = \Psi^n(x) \Psi^n(y)$;

viii) $\Psi^{ij}(x) = \Psi^i(\Psi^j(x))$.

A ring $R$ with operations $\Psi^i$ satisfying iv)–viii) is sometimes called a ''$\Psi$-ring''.

The ring $\Lambda(R)$ is isomorphic to the ring $W(R)$ of (big) Witt vectors (cf. (the editorial comments to) [[Witt vector|Witt vector]]):
$$
\bar E : W(R) \rightarrow \Lambda(R)
$$
$$
(a_1,a_2,\ldots) \mapsto \prod_{i=1}^\infty \left({ 1 - a_i t^i }\right)
$$

Under this isomorphism the Adams operations $\Psi^n$ on $\Lambda(R)$ correspond to the Frobenius operations $\mathbf{f}_n : W(R) \rightarrow W(R)$.

The $\lambda$-structures on the rings $\Lambda(R)$ define a functorial morphism of ring-valued functors $\lambda_{-t}(\cdot) : \Lambda(\cdot) \rightarrow \Lambda(\Lambda(\cdot)) $. Together with $\Lambda(R) \rightarrow R$, $1 + a_1 t + a_2 t^2 + \cdots \mapsto a_1$, this defines a [[co-triple]] structure on the functor $\Lambda$, and the $\lambda$-rings are precisely the [[co-algebra]]s of this co-triple.

Via the isomorphism $\bar E$ one finds "exponential homomorphisms" 
$$
E : W(R) \rightarrow W(W(R))
$$
$$
E' : W(R) \rightarrow \Lambda(W(R))
$$
which should be seen as (generalizing) the so-called [[Artin–Hasse exponential]] ([[#References|[a11]]], [[#References|[a12]]]).

Let $w_n(R) : W(R) \rightarrow R$ be the ring homomorphism
$$
w_n (a_1,a_2,\ldots) = \sum_{d|n} d a_d^{n/d} \ .
$$
Then the Artin–Hasse exponential $E(R) : W(R) \rightarrow W(W(R))$ is functorially characterized by
$$
w_n(W(R)) \circ E(R) = \mathbf{f}_n(R)
$$
where $\mathbf{f}_n$ is the Frobenius homomorphism.

Let $V : \lambda\textsf{-Ring} \rightarrow \textsf{Ring}$ be the forgetful functor. Then the functor $\Lambda : \textsf{Ring} \rightarrow \lambda\textsf{-Ring}$ is right adjoint (cf. [[Adjoint functor]]) to $V$:
$$
\textsf{Ring}(V(S),R) \equiv \lambda\textsf{-Ring}(S,\Lambda(R))
$$
(cf. [[#References|[a5]]], p. 20).

There are (besides the identity) three natural automorphisms of the Abelian group $\Lambda(R) = 1 + t R[[t]]$, given by the substitution $t \mapsto -t$, the "inversion" $\alpha(t) \mapsto \alpha(t)^{-1}$, and the combination of the two. Correspondingly there are four natural ways to introduce a ring structure on $\Lambda(R)$; the corresponding unit elements are $1-t$, $1+t$, $(1-t)^{-1}$, $(1+t)^{-1}$. All four occur in the literature. The most frequently occurring have $1-t$ or $1+t$ as their unit element — here, in the above, $1-t$ is the unit element —, and $(1+t)^{-1}$ seems to be the most rare case.

$\lambda$-rings were introduced by A. Grothendieck in an algebraic-geometric setting [[#References|[a2]]] and were first used in group representation theory by M.F. Atiyah and D.O. Tall ([[#References|[a1]]]).

In case $x \in R$ is one-dimensional, i.e. $\lambda^n(x) = 0$ for $n \ge 2$, the terminology derives from the case $R = K(M)$ or $R = R(G)$; one has $\Psi^n(x) = x^n$, whence the name ''power operations'' for the $\Psi^n$. On the $\Lambda(R)$ the operations $\Psi^n$ are directly defined by
$$
\Psi^n \left({ \prod_{i=1}^\infty (1 - \xi_i t) }\right) = \prod_{i=1}^\infty (1 - \xi_i^n t) \ .
$$

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> M.F. Atiyah, D.O. Tall, "Group representations, $\lambda$-rings and the $J$-homomorphism" ''Topology'' , '''8''' (1969) pp. 253–297 {{MR|244387}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Grothendieck, "La théorie des classes de Chern" ''Bull. Soc. Math. France'' , '''86''' (1958) pp. 137–154 {{MR|0116023}} {{ZBL|0091.33201}} </TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Grothendieck, "Classes de faisceaux et théorème de Riemann–Roch" , ''Sem. Géom. Algébrique'' , '''6''' , Springer (1972) pp. 20–77 {{MR|}} {{ZBL|0229.14008}} </TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) pp. 144ff {{MR|0506881}} {{MR|0463184}} {{ZBL|0454.14020}} </TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top"> D. Knutson, "$\lambda$-rings and the representation theory of the symmetric group" , Springer (1974) {{MR|0364425}} {{ZBL|0272.20008}} </TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top"> P. Berthelot, "Généralités sur les $\lambda$-anneaux" , ''Sem. Géom. Algébrique'' , '''6''' , Springer (1972) pp. 297–365</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top"> W. Fulton, S. Lang, "Riemann–Roch algebra" , Springer (1985) {{MR|0801033}} {{ZBL|0579.14011}} </TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top"> T. tom Dieck, "Transformation groups and representation theory" , Springer (1979) {{MR|}} {{ZBL|0445.57023}} </TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top"> C. Siebeneicher, "$\lambda$-Ringstrukturen auf dem Burnsidering der Permutationsdarstellungen einer endlichen Gruppe" ''Math. Z.'' , '''146''' (1976) pp. 223–238 {{MR|0390035}} {{ZBL|0306.20011}} </TD></TR>
<TR><TD valign="top">[a10]</TD> <TD valign="top"> J.F. Adams, "Vectorfields on spheres" ''Ann. of Math.'' , '''75''' (1962) pp. 603–632 {{DOI|10.2307/1970213}} {{ZBL|0112.38102}}</TD></TR>
<TR><TD valign="top">[a11]</TD> <TD valign="top"> M. Hazewinkel, "Twisted Lubin–Tate formal group laws, ramified Witt vectors and (ramified) Artin–Hasse exponentials" ''Trans. Amer. Math. Soc.'' , '''259''' (1980) pp. 47–63 {{MR|0561822}} {{ZBL|0437.13014}} </TD></TR>
<TR><TD valign="top">[a12]</TD> <TD valign="top"> E. Artin, H. Hasse, "Die beide Ergänzungssätze zum Reciprozitätsgesetz der $\ell^n$-ten Potenzreste im Körper der $\ell^n$-ten Einheitswurzeln" ''Abh. Math. Sem. Univ. Hamburg'' , '''6''' (1928) pp. 146–162</TD></TR>
<TR><TD valign="top">[a13]</TD> <TD valign="top"> G. Whaples, "Generalized local class field theory III: Second form of the existence theorem, structure of analytic groups" ''Duke Math. J.'' , '''21''' (1954) pp. 575–581 {{MR|73645}} {{ZBL|}} </TD></TR>
</table>

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