T-Norm Based Picture Fuzzy Subgroup Homomorphisms

Intelligent and Fuzzy Systems, İstanbul, Türkiye, 7 - 09 Temmuz 2026, ss.1-10, (Tam Metin Bildiri)

Fuzzy sets assign membership values α(x) ∈ [0, 1] to elements.

Intuitionistic fuzzy sets add non-membership β(x) with α(x) +

β(x) ≤ 1. Picture fuzzy sets incorporate neutral membership γ(x) satisfying

α(x) + γ(x) + β(x) ≤ 1, capturing approval, neutrality, and rejection.

Fuzzy subgroups adapt group theory to uncertainty, but existing

definitions use fixed min/max operators.

Triangular norms (t-norms) provide flexible aggregation. This work introduces

T -picture fuzzy groups by incorporating t-norms into picture

fuzzy subgroup conditions, unifying classical (T = min), intuitionistic,

and picture fuzzy groups. With T = min and S = max, we recover Dogra

and Pal’s definition.

We provide an information fusion interpretation where (α, γ, β) represent

belief, uncertainty, and disbelief, and t-norms act as fusion operators. Homomorphism

theory is developed including images, preimages, kernels,

and quotients. The kernel of a T -picture fuzzy homomorphism is a normal

subgroup. A First Isomorphism Theorem gives G/KerT

B (ϕ) ∼=

ϕ(A)

for surjective ϕ and normal A. Deeper results include level cut representations

and fusion closure theorems. A multi-source decision-making

example illustrates the framework.