AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number <i>w̅<sub>f</sub>(g)</i>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 = (G<sub>n</sub>)<sub>n ∈ ℕ ∪ {+∞}</sub> 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 w<sub>f</sub>(J) of a pair of semi-ideals (I, J) to a quasi-filtration f and a submonoid J, denoted w<sub>f</sub>(J), and also to a pair of quasi-filtrations (f, g), which we will denote w<sub>f</sub>(g). The central question is the existence of generalized Samuel number, denoted <i> w̅<sub>f</sub>(g)</i>, which is defined as the limit of the ratio w<sub>f</sub>(G<sub>n</sub>) ⁄ 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 w<sub>f</sub>(J) exists for any submonoid J, and the generalized Samuel number <i> w̅<sub>f</sub>(g)</i> 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.