In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor N r from the category of special λ-rings into the category of special λ-rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant's paper (J. Algebra. 253 (2002); no.1, 167-188) to the case of graded Lie (super)algebras with a group action by applying the Euler-Poincaré principle.

Let R be a finite-dimensional torsion-free special l-ring. In this paper we generalize the results in Dress and Siebeneicher (Adv. in Math. 70 (1988) 89; 78 (1989) 1) by constructing R-analogue # O R ðGÞ of the Burnside ring of profinite groups # OðGÞ: In particular, we remark that the (Grothendieck) Lie-module denominator identity of free Lie algebras in Oh (Necklace rings and logarithmic functions, preprint, KIAS, 2003) is closely related to the canonical isomorphism between # O R ðGÞ and Grothendieck's ring of formal power series with coefficients in R and constant term 1. r

Let G be a profinite group and q an indeterminate. In this paper, we introduce and study a q-analog of the Möbius function and the cyclotomic identity arising from the lattice of open subgroups of G. When q is any integer, we show that they have close connections with the functors W q G , Nr q G , and Nr q G introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 257 (2007) 151-191]. In particular, we interpret the multiplicative property of the inverse of the table of marks and the Möbius function of G as a composition property of certain functors. Classification of W q G , Nr q G , and Nr q G up to strict natural isomorphism as q varies over the set of integers and its application will be dealt with, too.

Abstract. In this paper, we construct a q-deformation of the Witt-Burnside ring of a profinite group over a commutative ring, where q ranges over the set of integers. When q = 1, it coincides with the Witt-Burnside ring introduced by A. Dress and C. Siebeneicher (Adv. Math. 70 (1988), 87-132). To achieve our goal we first show that there exists a q-deformation of the necklace ring of a profinite group over a commutative ring. As in the classical case, i.e., the case q = 1, q-deformed Witt-Burnside rings and necklace rings always come equipped with inductions and restrictions. We also study their properties. As a byproduct, we prove a conjecture due to Lenart (J. Algebra. 199 (1998), 703-732). Finally, we classify W q G up to strict natural isomorphism in case where G is an abelian profinite group.

For every profinite group G, we construct two covariant functors ∆ G and AP G which are equivalent to the functor W G introduced in [A. Dress, C. Siebeneicher, The Burnside ring of profinite groups and the Witt vectors construction, Adv. Math. 70 (1988) 87-132]. We call ∆ G the generalized Burnside-Grothendieck ring functor and AP G the aperiodic ring functor (associated with G). In case G is abelian, we also construct another functor Ap G from the category of commutative rings with identity to itself as a generalization of the functor Ap introduced in [K. Varadarajan, K. Wehrhahn, Aperiodic rings, necklace rings, and Witt vectors, Adv. Math. 81 (1990) 1-29]. Finally, it is shown that there exist q-analogues of these functors (i.e., W G , ∆ G , AP G , and Ap G ) in case G is the profinite completion of the multiplicative infinite cyclic groupĈ.  2005 Elsevier Inc. All rights reserved.