AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number <i>w&#773;<sub>f</sub>(g)</i>to Quasi-filtrations on a Semiring

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

doi:doi:10.11648/j.pamj.20251406.11

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

Kouadjo Pierre Brou^*

, Edjabrou Ulrich Blanchard Kablam

, Grah Gelin Bogui

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

Received: 24 August 2025 Accepted: 15 September 2025 Published: 10 November 2025

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Abstract

In this work, we develop a theory of asymptotic growth for quasi-filtrations in the framework of commutative semirings. A quasi-filtration g = (G_n)_{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 w_f(J) of a pair of semi-ideals (I, J) to a quasi-filtration f and a submonoid J, denoted w_f(J), and also to a pair of quasi-filtrations (f, g), which we will denote w_f(g). The central question is the existence of generalized Samuel number, denoted w̅_f(g), which is defined as the limit of the ratio w_f(G_n) ⁄ 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_f(J) exists for any submonoid J, and the generalized Samuel number w̅_f(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.

Published in	Pure and Applied Mathematics Journal (Volume 14, Issue 6)
DOI	10.11648/j.pamj.20251406.11
Page(s)	161-165
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

Commutative Ring, Semi-ring, Submonoid, Filtration, Quasi-filtration, Generalized Samuel Number, Regular Quasi-filtration, AP Quasi-filtration

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. Diehl, 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 b̂_f(g) on a semi-ring, JP J. Algebra Number Theory Appl., Vol. 36 (2015), no. 2, pp. 123-139.
[4]	E. D. Akeke and P. 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-gradings on a semi-ring, Afr. Mathematiques. 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) on a semi-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, Cloîture intégrale des tétaux et eïguïsingularité, Ann. Fac. Sci. Toulouse Math., Vol. 17 (2008), pp. 781-859.
[9]	P. Ayegnon, Filtrations on a set and Samuel numbers, Afr. Mat., Sér. III, Vol. 16 (2005), pp. 139-144.
[10]	P. Ayegnon and H. Diehl, Extensions to filtrations of the Samuel numbers associated to ideals, Communication in Algebra, Vol. 22 (1994), no. 9, pp. 3249-3263.
[11]	P. Ayegnon and D. Sangare, Generalized Samuel numbers and AF-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 filtration, Department of Mathematics, University of Abidjan, Ivory Coast, (1984).
[13]	J. W. Petro, Some results in the theory in 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). AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring. Pure and Applied Mathematics Journal, 14(6), 161-165. https://doi.org/10.11648/j.pamj.20251406.11

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Brou, K. P.; Kablam, E. U. B.; Bogui, G. G. AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring. Pure Appl. Math. J. 2025, 14(6), 161-165. doi: 10.11648/j.pamj.20251406.11

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Brou KP, Kablam EUB, Bogui GG. AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring. Pure Appl Math J. 2025;14(6):161-165. doi: 10.11648/j.pamj.20251406.11

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@article{10.11648/j.pamj.20251406.11,
  author = {Kouadjo Pierre Brou and Edjabrou Ulrich Blanchard Kablam and Grah Gelin Bogui},
  title = {AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {6},
  pages = {161-165},
  doi = {10.11648/j.pamj.20251406.11},
  url = {https://doi.org/10.11648/j.pamj.20251406.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251406.11},
  abstract = {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.
},
 year = {2025}
}

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TY  - JOUR
T1  - AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring

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.20251406.11
DO  - 10.11648/j.pamj.20251406.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 161
EP  - 165
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251406.11
AB  - 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.

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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Brou, K. P., Kablam, E. U. B., Bogui, G. G. (2025). AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring. Pure and Applied Mathematics Journal, 14(6), 161-165. https://doi.org/10.11648/j.pamj.20251406.11

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ACS Style

Brou, K. P.; Kablam, E. U. B.; Bogui, G. G. AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring. Pure Appl. Math. J. 2025, 14(6), 161-165. doi: 10.11648/j.pamj.20251406.11

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AMA Style

Brou KP, Kablam EUB, Bogui GG. AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring. Pure Appl Math J. 2025;14(6):161-165. doi: 10.11648/j.pamj.20251406.11

Copy | Download

@article{10.11648/j.pamj.20251406.11,
  author = {Kouadjo Pierre Brou and Edjabrou Ulrich Blanchard Kablam and Grah Gelin Bogui},
  title = {AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {6},
  pages = {161-165},
  doi = {10.11648/j.pamj.20251406.11},
  url = {https://doi.org/10.11648/j.pamj.20251406.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251406.11},
  abstract = {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.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - AStudy of Quasi-filtrations on a Semiring and Extension of the Generalized Samuel Number w̅f(g)to Quasi-filtrations on a Semiring

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.20251406.11
DO  - 10.11648/j.pamj.20251406.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 161
EP  - 165
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251406.11
AB  - 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.

VL  - 14
IS  - 6
ER  -

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