We define and study a notion of ring of formal power series with exponents in a cyclically ordered group. Such a ring is a quotient of various subrings of classical formal power series rings. It carries a two variable valuation function. In the particular case where the cyclically ordered group is actually totally ordered, our notion of formal power series is equivalent to the classical one in ...

We investigate finite-state systems with costs. Departing from classical theory, in this paper the cost of an action does not only depend on the state of the system, but also on the time when it is executed. We first characterize the terminating behaviors of such systems in terms of rational formal power series. This generalizes a classical result of Schützenberger. Using the previous results, ...

We extend the concept of CDF-series to the context of several variables, and show that the series solution of first order differential equations y′ = x(t, y) and functional equation y = x(t, y), with x CDF in two variables, are CDF-series. We also give many effective closure properties for CDF-series in several variables. 1. CDF-series in one variable We present in this paper, new properties an...

We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product and a restricted star which is applied only to series for which the elements in the support all hav...

A notion of semilinearity is introduced for formal power series as a natural generalization of semilinear languages over a commutative monoid. Some closure properties of the semilinear languages are established also for formal power series. In this way, a classical result due to Eilenberg and Sch utzenberger and some results due to Ginsburg are extended to series. We also prove that Parikh's t...

Let K = C((t)) be the field of formal Laurent series, and let O = C[[t]] be the ring of formal power series. In this paper, we present a version of the Matsuki correspondence for the affine Grassmannian Gr = G(K)/G(O) of a connected reductive complex algebraic group G. Our main statement is an anti-isomorphism between the orbit posets of two subgroups of G(K) acting on Gr. The first subgroup is...