Formal power series with cyclically ordered exponents
by
Gerard Leloup
University of LeMans, France

We say that a set C is cyclically ordered if it is equipped with a 3-ary relation (·, ·, ·) on C that satisfies (1), (2), (3) below.
(1) (·, ·, ·) enjoys for allx, y, z, (x, y, z) ===> x =/= y =/= z =/= x
(2) (·, ·, ·) enjoys for allx, y, z, (x, y, z) ===> (y, z, x)
(3) for all x in C, (x , ·, ·) is antisymetric and transitive. Hence it induces a total order on C \{ x }, we will note <= _x the associate order on C, with first element x, C_x this ordered set and min_x the minimum with the order <= _x.
Let C be an Abelian group and a cyclic order (·, ·, ·), we say that C is a cyclically ordered group if (·, ·, ·) is compatible, i.e. for allx, y, z, u, (x, y, z) ===> (x+u, y+u, z+u)
Any totally ordered set is a cyclically ordered one once equipped with the ternary relation (x, y, z) iff x < y < z or y < z < x or z < x < y. With the same relation, any totally ordered group is a cyclically ordered one. Such a cyclically ordered group is called a linear cyclically ordered one.
Let k be a commutative ring (or field), and C be a cyclically ordered group.
We generalize the usual definitions of k[[C]], the ring of formal power series, and k[C], the ring of polynomials, in such a way that they coincide with classical one when C is a linear cyclically ordered group. And applying Rieger's theorem, we show that there exists a totally ordered group G and z in G₊, z cofinal in G₊, and subrings R of k((G)) such that k[C] =~ k[G]/(1-X^z), k[[C]] =~ R/(1-X^z). We can prove that k[C] is integral if and only if k is integral and C is torsion free, and that k[[C]] is a field if and only if k is a field and C a linear cyclically ordered group. Then, we define the cyclic valuation on k[[C]], as the mapping v from C ×k[[C]] onto C \cup { \infty}, such that for all a in C and \sigma in k[[C]], v(a, \sigma) is the first element of the support of \sigma in the ordered set C_a with first element a, and point out some of its properties.
Next, we define cyclically valued groups. An Abelian group G is a
cyclically valued one if there exists a cyclically ordered set C and a mapping v from C ×G onto C \cup { \infty} such that for any a in C, (G, v(a, ·)) is a valued group, with the order <= _a on C; v(a, ·) will be called the a-valuation. For any \sigma in G, the support of \sigma will be the subset of C of elements a such that there exists b in C with v(b, \sigma)=a. \sigma is a monomial if its support contains exactly one element. Using such notions, we give two first order properties :
(1) the cyclic valuation on G is defined by the supports
(2) the monomials v-generate G,
and prove that :
(a) the subgroups of Hahn products of Abelian groups indexed by C are exactly the cyclically valued groups satisfying (1) and (2), and such that the supports of all the elements are well ordered,
(b) the Hahn products of Abelian group indexed by C are exactly the cyclically valued group (G, v) satisfying (1) and (2), and such that (G, v(a, ·)) is spherically complete for some a in C and is closed under section.
If we assume C is a cyclically ordered group, and extend the definition of cyclically valued group to cyclically valued ring, there exist some properties analogous to the properties of the cyclically valued groups, with formal power series rings with cyclically ordered exponents instead of Hahn products. If all the supports are finite, then such cyclically valued rings are isomorphic to polynomial rings. The valued fields of equal characteristic and perfect residue fields are subfields of cyclically valued one.
The usual valuations on fields are characterized by valuation rings. In the same way, we characterize the cyclically valued rings by the multiplicative subgroup of inversible monomials, where a subset is the analog of the positive cone of the subgroup of the inversible monomials, and by the set of elements of 0-valuation equal to 0.
Remark that if C is a linear cyclically ordered group, (that is a totally ordered group), and if all the supports are well ordered, then we can define a valuation in the usual sense on the ring. We will call it the infinite valuation denoted v(\infty, ·). We shall define the infinite valuation in a more general case, such that it satisfies the rule v(\infty, \sigma\tau)=v(\infty, \sigma)+v(\infty, \tau) if and only if C is his a linear cyclically ordered group. Now let T be a totally ordered group and k a field (or a ring), \nu the usual valuation on k[[T]]. The v(a, ·), a in T exist : for \sigma in k[[T]], v(a, \sigma) is the first element of the intersection of the support of \sigma with the final segment [a, \infty] (set v(a, \sigma)=\infty if the intersection is empty). v(a, ·) satisfies the ultrametric inequality, but v(a, \sigma)=\infty is not equivalent to \sigma = 0. If we consider T with the order <= _a deduced from the cyclic order, then v(a, \sigma)=\infty is equivalent to \sigma = 0. So the formal power series with exponents in a totally ordered group are a particular case of the formal power series with cyclically ordered exponents, and \nu = v(\infty, ·).

Date received: April 9, 1999