AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring

In this work, we develop a theory of asymptotic growth for quasi-filtrations in the framework of commutative semirings. A quasi-filtration g = (Gn)n ∈ ℕ ∪ {+∞} is a family of submonoids of a semiring B that satisfies the conditions of a quasi-graduation and is also decreasing for indices n ≥ 1. This structure generalizes the notion of a filtration by using submonoids instead of the more restrictive semi-ideals. In this context, we extend the Samuel number wf(J) of a pair of semi-ideals (I, J) to a quasi-filtration f and a submonoid J, denoted wf(J), and also to a pair of quasi-filtrations (f, g), which we will denote wf(g). The central question is the existence of generalized Samuel number, denoted w̅f(g), which is defined as the limit of the ratio wf(Gn) ⁄ n as n tends to infinity. Our main result provides a positive answer to this question under certain conditions. We demonstrate that if the quasi-filtration f is regular, then wf(J) exists for any submonoid J, and the generalized Samuel number w̅f(g) is well-defined for any quasi-filtration g that is Approximable by Powers of submonoids (AP). Finally, we study the algebraic properties of this number. We prove that this number is non-negative and positively homogeneous under certain conditions. These results constitute an essential step towards the development of analytical tools for non-symmetrizable algebraic structures.