IJO - International Journal of Mathematics (ISSN: 2992-4421 )

ON STABILITY IN SEPARATIVE SEMIGROUP

Otobong Johnson Tom — 2025-09-30

Stability, as introduced by Koch and Wallace (1956), has long stood as a central
notion in semigroup theory, ensuring that inclusions of principal ideals collapse into
equalities and that left and right structures align harmoniously. Separativity, on
the other hand, generalises cancellativity while retaining algebraic regularity, and
Burmistrovich's decomposition theorem revealed that every separative semigroup
can be expressed as a semilattice of cancellative semigroups. Yet, whether separative semigroups inherit stability in the sense of Koch and Wallace has remained
unresolved. In this work, we close this gap: we prove that semilattices of cancellative semigroups are stable, and hence every separative semigroup is inherently
stable. This result elevates stability from a supplementary condition to a built-in
feature of separative semigroups, oering a unied perspective that strengthens the
foundations of semigroup theory and deepens its structural coherence.

Stability in Semigroups of Bounded Linear Operators: Bridging Algebraic and Analytic Notions

Otobong Johnson Tom — 2025-09-30

Stability is a cornerstone in the theory of semigroups, shaping the study of evo-
lution equations, operator theory, and algebraic structures. Yet, algebraic and ana-
lytic perspectives on stability have traditionally developed in isolation. This paper
builds a novel bridge between the two. Tom, Udoaka and Udoa (2025) established
that every strongly continuous (C0) semigroup of bounded linear operators is stable
in the sense of Koch and Wallace (KW), a universal algebraic property that forces
Green's relations to collapse (D = J = L = R). This recognition is new in operator
semigroup theory, where stability has typically been studied only in analytic terms.
We further provide precise spectral conditions under which KW-stability aligns with
analytic stability notionsstrong, asymptotic, exponential, and uniformthereby
unifying algebraic semigroup stability with spectral/operator-theoretic stability. Il-
lustrative examples, including the translation, right shift, heat, and damped wave
semigroups, demonstrate the stability gap and the exact conditions under which
the two approaches coincide. The study is signicant because it supplies a uni-
versal structural property of operator semigroups, a spectral criterion for analytic
decay, and practical insights for evolution equations, control design, and numerical
discretization.

A Mathematical Model on Cholera Outbreak with Vaccination

Bolanle Adeola Olokuntoye — 2025-10-02

This study develops and analyzes a cholera transmission model of SIRB type (Sus-
ceptible–Infected–Recovered–Bacteria) that incorporates vaccination. The main
objective is to investigate the threshold conditions under which cholera can either
be eradicated or persist in the community. The model formulation captures both
direct person-to-person transmission and indirect infection through contaminated
water. Using standard dynamical systems techniques, the disease-free equilibrium
(DFE) and endemic equilibrium (EE) were derived. The next-generation matrix
approach was then applied to obtain the basic reproduction number, R0, which
serves as the central threshold parameter governing disease dynamics. The analysis
showed that the DFE is locally asymptotically stable whenever R0 < 1, implying
cholera elimination under effective interventions, while the EE exists and is locally
stable when R0 > 1, confirming sustained disease persistence. These results em-
phasize the importance of vaccination and improvements in sanitation as essential
strategies to reduce R0 below unity and achieve long-term cholera control

CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS

Thomas, Henry Sylvester — 2025-10-08

Diphtheria is a bacterial infectious disease that can lead to severe complications and even deaths. This work presents the Caputo-Fabrizio Fractional derivatives of the aged-structured deterministic model of diphtheria infection. The existence and the uniqueness of the solution of the model are investigated and established using the contraction principle. The stability of the model is investigated with the help of the well-known Ulem-Hyers and the generalized Ulem-Hyers theorems. Analyzing the model using the Laplace Adomian Decomposition Methods,the system’s analytical solution, in the form of an infinite series that converges quickly to it exact value is obtained.