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...

Sequences of numbers abound in combinatorics whose generating functions are algebraic over the rational functions. Examples include Catalan and related numbers, numbers of words expressing an element in a free group, and diagonal coe cients of 2-variable rational generating functions (Furstenberg's theorem). Algebraicity is of of practical as well as theoretical interest, since it guarantees an...

We study perfect multiple coverings in translation invariant graphs with vertex set $\mathbb{Z}^2$ using an algebraic approach. In this approach we consider any such covering as a two-dimensional binary configuration which then express two-variate formal power series. Using known results, conclude that has non-trivial periodizer, is, there exists non-zero polynomial whose product the series pre...

Let k be the algebraic closure of Fp and K be the local field of formal power series with coefficients in k. The aim of this paper is the description of the set Yn of conjugacy classes of series of order p for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic p which are invertible and of finite order p for the composition la...

This paper presents an approach to defining distances between nonlinear and hybrid dynamical systems based on formal power series theory. The main idea is that the input-output behavior of a wide range of dynamical systems can be encoded by rational formal power series. Hence, a natural distance between dynamical systems is the distance between the formal power series encoding their input-outpu...