AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅<sub>φ</sub>(θ) andValuative Reduction

Kouadjo Pierre Brou; Edjabrou Ulrich Blanchard Kablam; Grah Gelin Bogui

doi:doi:10.11648/j.pamj.20251406.13

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AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅_φ(θ) andValuative Reduction

Kouadjo Pierre Brou^*

, Edjabrou Ulrich Blanchard Kablam

, Grah Gelin Bogui

Published in Pure and Applied Mathematics Journal (Volume 14, Issue 6)

Received: 19 September 2025 Accepted: 7 October 2025 Published: 10 November 2025

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Abstract

In this work, we develop an asymptotic theory for axe-filtrations within the broader framework of semi-modules, aiming to generalize the classical theory of filtrations from rings to these algebraic structures where addition is not necessarily invertible. This extension is crucial as semi-modules and semirings appear naturally in fields like geometry and computer science. Our first contribution is the introduction of the concept of an axe-filtration on a semi-module, which adapts the notion of a sequence of powers of an ideal in a ring. We then define a generalized Samuel number, denoted v̅_φ(θ), designed to measure the relative asymptotic growth between two distinct axe-filtrations, φ and θ. The main result of this paper establishes a fundamental theorem: the existence of this Samuel number is guaranteed under the condition that one axe-filtration, φ, is a valuative reduction of the other, θ. This key finding extends the classical results of D. Rees by connecting the asymptotic behavior of these generalized filtrations to the well-established theory of discrete valuations. By doing so, we lay the groundwork for a robust asymptotic theory applicable to semi-modules. This work provides new analytical tools for studying non-symmetrizable algebraic structures and opens avenues for further research into the geometric and algebraic properties of systems described by semirings.

Published in	Pure and Applied Mathematics Journal (Volume 14, Issue 6)
DOI	10.11648/j.pamj.20251406.13
Page(s)	176-181
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Semiring, Semi-module, Filtration, Axe-filtration, Generalized Samuel Number, Valuative Reduction, Commutative Algebra

References

[1]	K. P. Brou and E. D. Beche, Deep Classification of a Generalization of Ring Filtration in Commutative Algebra, International Journal of Algebra, Vol. 19 (2025), no. 2, pp. 79-88. https://doi.org/10.12988/ija.2025.91962
[2]	E. D. Akeke, P. K. Ayegnon, and H. Dichi, Extension of Samuel Number to Bifiltrations, Southeast Asian Bulletin of Mathematics, Vol. 45 (2021), pp. 143-153.
[3]	E. D. Akeke, S. Ouattara, and P. Ayegnon, Another generalized Samuel number b̅_f(g) on a semi-ring, JP J. Algebra Number Theory Appl., Vol. 36 (2015), no. 2, pp. 123-139.
[4]	E. D. AkekeandP. Ayegnon, Some aspects of generalized Samuel numbers and quasi graduations on a semi-ring, Pioneer J. Algebra Number Theory Appl., Vol. 5 (2013), no. 1, pp. 17-28.
[5]	Y. M. Diagana and S. Ouattara, Extension of the Samuel numbers of two distinctive types to quasi-graduations on a semi-ring, Afr. Math. Ann. (AFMA), Vol. 2 (2011), pp. 119-128.
[6]	S. Ouattara, E. D. Akeke, and P. Ayegnon, Generalized Samuel numbers v̅_φ (θ)and w̅_φ (θ)onasemi-module, Afr. Math. Ann. (AFMA), Vol. 2 (2011), pp. 175-189.
[7]	S. Ouattara, E. D. Akeke, and P. Ayegnon, Another generalized Samuel number â_f(g) on a semi-ring, JP J. Algebra Number Theory Appl., Vol. 19 (2010), no. 2, pp. 185-201.
[8]	M. Lejeune-Jalabert and B. Teissier, Integral closure of ideals and equisingularity, Ann. Fac. Sci. Toulouse Math., Vol. 17 (2008), pp. 781-859. https://doi.org/10.5802/afst.1202
[9]	P. Ayegnon, Filtrations on a set and Samuel numbers, Afr. Mat., Ser. III, Vol. 16 (2005), pp. 139-144.
[10]	P. Ayegnon, Extensions to filtrations of Samuel numbers associated with ideals, Communication in Algebra, Vol. 22 (1994), no. 9, pp. 3249-3263.
[11]	P. Ayegnon and D. Sangare, Generalized Samuel numbers and AP-filtrations, Journal of Pure and Applied Algebra, Vol. 65 (1990), pp. 1-13.
[12]	D. Sangare, On various generalizations of the formula v̅_{f_n} = 1/n v̅_f to pseudo-valuations associated with a f iltration, Department of Mathematics, University of Abidjan, Ivory Coast, (1984).
[13]	J. W. Petro, Some results in the theory of pseudo valuations, Ph.D. dissertation, State University of Iowa, Iowa City, (1961).
[14]	D. Rees, Variations associated with a local ring (1), Proc. London Math. Soc., Series 3, Vol. 5 (1955), pp. 107-128.

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Brou, K. P., Kablam, E. U. B., Bogui, G. G. (2025). AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction. Pure and Applied Mathematics Journal, 14(6), 176-181. https://doi.org/10.11648/j.pamj.20251406.13

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ACS Style

Brou, K. P.; Kablam, E. U. B.; Bogui, G. G. AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction. Pure Appl. Math. J. 2025, 14(6), 176-181. doi: 10.11648/j.pamj.20251406.13

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AMA Style

Brou KP, Kablam EUB, Bogui GG. AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction. Pure Appl Math J. 2025;14(6):176-181. doi: 10.11648/j.pamj.20251406.13

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@article{10.11648/j.pamj.20251406.13,
  author = {Kouadjo Pierre Brou and Edjabrou Ulrich Blanchard Kablam and Grah Gelin Bogui},
  title = {AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {6},
  pages = {176-181},
  doi = {10.11648/j.pamj.20251406.13},
  url = {https://doi.org/10.11648/j.pamj.20251406.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251406.13},
  abstract = {In this work, we develop an asymptotic theory for axe-filtrations within the broader framework of semi-modules, aiming to generalize the classical theory of filtrations from rings to these algebraic structures where addition is not necessarily invertible. This extension is crucial as semi-modules and semirings appear naturally in fields like geometry and computer science. Our first contribution is the introduction of the concept of an axe-filtration on a semi-module, which adapts the notion of a sequence of powers of an ideal in a ring. We then define a generalized Samuel number, denoted v̅φ(θ), designed to measure the relative asymptotic growth between two distinct axe-filtrations, φ and θ. The main result of this paper establishes a fundamental theorem: the existence of this Samuel number is guaranteed under the condition that one axe-filtration, φ, is a valuative reduction of the other, θ. This key finding extends the classical results of D. Rees by connecting the asymptotic behavior of these generalized filtrations to the well-established theory of discrete valuations. By doing so, we lay the groundwork for a robust asymptotic theory applicable to semi-modules. This work provides new analytical tools for studying non-symmetrizable algebraic structures and opens avenues for further research into the geometric and algebraic properties of systems described by semirings.},
 year = {2025}
}

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TY  - JOUR
T1  - AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction

AU  - Kouadjo Pierre Brou
AU  - Edjabrou Ulrich Blanchard Kablam
AU  - Grah Gelin Bogui
Y1  - 2025/11/10
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251406.13
DO  - 10.11648/j.pamj.20251406.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 176
EP  - 181
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251406.13
AB  - In this work, we develop an asymptotic theory for axe-filtrations within the broader framework of semi-modules, aiming to generalize the classical theory of filtrations from rings to these algebraic structures where addition is not necessarily invertible. This extension is crucial as semi-modules and semirings appear naturally in fields like geometry and computer science. Our first contribution is the introduction of the concept of an axe-filtration on a semi-module, which adapts the notion of a sequence of powers of an ideal in a ring. We then define a generalized Samuel number, denoted v̅φ(θ), designed to measure the relative asymptotic growth between two distinct axe-filtrations, φ and θ. The main result of this paper establishes a fundamental theorem: the existence of this Samuel number is guaranteed under the condition that one axe-filtration, φ, is a valuative reduction of the other, θ. This key finding extends the classical results of D. Rees by connecting the asymptotic behavior of these generalized filtrations to the well-established theory of discrete valuations. By doing so, we lay the groundwork for a robust asymptotic theory applicable to semi-modules. This work provides new analytical tools for studying non-symmetrizable algebraic structures and opens avenues for further research into the geometric and algebraic properties of systems described by semirings.
VL  - 14
IS  - 6
ER  -

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Author Information

Kouadjo Pierre Brou

Applied Fundamental Sciences Department, Nangui Abrogoua University, Abidjan, Ivory Coast

Contact Email

http://orcid.org/0009-0000-5895-2264
Edjabrou Ulrich Blanchard Kablam

Applied Fundamental Sciences Department, Nangui Abrogoua University, Abidjan, Ivory Coast

Contact Email

http://orcid.org/0009-0009-1464-5049
Grah Gelin Bogui

Applied Fundamental Sciences Department, Nangui Abrogoua University, Abidjan, Ivory Coast

Contact Email

http://orcid.org/0009-0008-7475-2156

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APA Style

Brou, K. P., Kablam, E. U. B., Bogui, G. G. (2025). AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction. Pure and Applied Mathematics Journal, 14(6), 176-181. https://doi.org/10.11648/j.pamj.20251406.13

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ACS Style

Brou, K. P.; Kablam, E. U. B.; Bogui, G. G. AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction. Pure Appl. Math. J. 2025, 14(6), 176-181. doi: 10.11648/j.pamj.20251406.13

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AMA Style

Brou KP, Kablam EUB, Bogui GG. AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction. Pure Appl Math J. 2025;14(6):176-181. doi: 10.11648/j.pamj.20251406.13

Copy | Download

@article{10.11648/j.pamj.20251406.13,
  author = {Kouadjo Pierre Brou and Edjabrou Ulrich Blanchard Kablam and Grah Gelin Bogui},
  title = {AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {6},
  pages = {176-181},
  doi = {10.11648/j.pamj.20251406.13},
  url = {https://doi.org/10.11648/j.pamj.20251406.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251406.13},
  abstract = {In this work, we develop an asymptotic theory for axe-filtrations within the broader framework of semi-modules, aiming to generalize the classical theory of filtrations from rings to these algebraic structures where addition is not necessarily invertible. This extension is crucial as semi-modules and semirings appear naturally in fields like geometry and computer science. Our first contribution is the introduction of the concept of an axe-filtration on a semi-module, which adapts the notion of a sequence of powers of an ideal in a ring. We then define a generalized Samuel number, denoted v̅φ(θ), designed to measure the relative asymptotic growth between two distinct axe-filtrations, φ and θ. The main result of this paper establishes a fundamental theorem: the existence of this Samuel number is guaranteed under the condition that one axe-filtration, φ, is a valuative reduction of the other, θ. This key finding extends the classical results of D. Rees by connecting the asymptotic behavior of these generalized filtrations to the well-established theory of discrete valuations. By doing so, we lay the groundwork for a robust asymptotic theory applicable to semi-modules. This work provides new analytical tools for studying non-symmetrizable algebraic structures and opens avenues for further research into the geometric and algebraic properties of systems described by semirings.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - AXE-Filtrations of Semi-modules, Generalized Samuel Number v̅φ(θ) andValuative Reduction

AU  - Kouadjo Pierre Brou
AU  - Edjabrou Ulrich Blanchard Kablam
AU  - Grah Gelin Bogui
Y1  - 2025/11/10
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251406.13
DO  - 10.11648/j.pamj.20251406.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 176
EP  - 181
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251406.13
AB  - In this work, we develop an asymptotic theory for axe-filtrations within the broader framework of semi-modules, aiming to generalize the classical theory of filtrations from rings to these algebraic structures where addition is not necessarily invertible. This extension is crucial as semi-modules and semirings appear naturally in fields like geometry and computer science. Our first contribution is the introduction of the concept of an axe-filtration on a semi-module, which adapts the notion of a sequence of powers of an ideal in a ring. We then define a generalized Samuel number, denoted v̅φ(θ), designed to measure the relative asymptotic growth between two distinct axe-filtrations, φ and θ. The main result of this paper establishes a fundamental theorem: the existence of this Samuel number is guaranteed under the condition that one axe-filtration, φ, is a valuative reduction of the other, θ. This key finding extends the classical results of D. Rees by connecting the asymptotic behavior of these generalized filtrations to the well-established theory of discrete valuations. By doing so, we lay the groundwork for a robust asymptotic theory applicable to semi-modules. This work provides new analytical tools for studying non-symmetrizable algebraic structures and opens avenues for further research into the geometric and algebraic properties of systems described by semirings.
VL  - 14
IS  - 6
ER  -

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