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In graph theory, the hypergraph [22] extends the traditional graph structure by allowing edges to connect multiple vertices, and this concept is further broadened by the superhypergraph [174,176]. Additionally, several types of uncertain graphs have been explored, including fuzzy graphs [136, 153], neutrosophic graphs [35, 36], and plithogenic graphs [66, 75, 185]. This study explores the SuperHyperGraph, Single-Valued Neutrosophic Quasi SuperHyperGraph, and Plithogenic Quasi SuperHyperGraph, analyzing their relationships with other graph classes. Future work will define the Semi Superhypergraph, Multi Superhypergraph, Pseudo Superhypergraph, Mixed Superhypergraph, and Bidirected Superhypergraph and examine their connections to existing classes in hypergraphs and graphs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of ♭, ℓ-neutrosophic subsemigroup (NSS), neutrosophic left ideal(NLI), neutrosophic right ideal(NRI), neutrosophic ideal (NI), neutrosophic bi-ideal(NBI), (ϵ, ϵ ∨ q)-neutrosophic ideal, neutrosophic bi-ideal of an ordered Γ-semigroups and discuss some of their properties. The concept of ♭, ℓ-neutrosophic ideal is a new extension of neutrosophic ideal over ordered Γ-semigroups Z. A non-empty subset ξ♭ is a (♭, ℓ)-NSS (NLI, NRI, NBI, (1,2)-ideal) of Z. Then the lower level set Δ♭ is an subsemigroup (LI, RI, BI, (1, 2) - ideal) of Z, where Δ♭ = {ρ ∈ Z|Δ(ρ) > ♭}, Ψ♭ = {ρ ∈ Z|Δ(ρ) > ♭} and 0♭ = {ρ ∈ Z|Δ(ρ) < ♭}. A subset ξ = [Δ,Ψ,0] is a (♭, ℓ) - NSS[NLI, NRI, NBI, (1, 2) - ideal] of Z if and only if each non-empty level subset ξt is a subsemigroup [LI, RI, BI, (1, 2) - ideal] of Z for all t ∈ (♭, ℓ]. Every (ϵ, ϵ ∨ q)NBI of Z is a (♭, ℓ)NBI of Z, but converse need not be true and examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]