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Picture Fuzzy Multigroups

This paper investigates picture fuzzy multisets and introduces the concept of picture fuzzy multigroups, establishing their key algebraic properties.

Original authors: Taiwo O. Sangodapo

Published 2026-07-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Taiwo O. Sangodapo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to describe a group of people, but you don't just want to say "they are here" or "they are not here." You want to capture the messy, complex reality of human opinion and membership.

This paper by Taiwo O. Sangodapo is about building a new, super-detailed mathematical tool to handle that complexity. It's called Picture Fuzzy Multigroups.

Here is the breakdown of what the paper does, using simple analogies.

1. The Evolution of "Fuzzy" Thinking

To understand this paper, you have to see how the math got more complicated over time:

Crisp Sets (The Old Way): Imagine a guest list for a party. You are either on the list or you aren't. It's black and white.
Fuzzy Sets (The First Upgrade): Imagine a guest list where you can be "sort of" invited. Maybe you have a 70% chance of showing up. This is the "Fuzzy Set."
Intuitionistic Fuzzy Sets (The Second Upgrade): Now, imagine you can say "I'm 70% likely to come, but 20% likely to stay home." You have a "Yes" and a "No," but they don't have to add up to 100%. The rest is "maybe."
Picture Fuzzy Sets (The Third Upgrade): This is where the paper starts. Imagine a vote with four options: Yes, Neutral (I don't care), No, and Refusal (I refuse to vote). The "Picture" set tracks all three active opinions (Yes, Neutral, No) at once.
Multisets (The "Counting" Upgrade): Usually, math treats a list of items as a set (unique items). But in real life, things repeat. If you have a bag of marbles, you might have three red ones. A "Multiset" allows you to count how many times an item appears.

The Paper's Big Idea: The author combines the "Picture" idea (Yes/Neutral/No) with the "Multiset" idea (counting repetitions). He calls this a Picture Fuzzy Multiset.

2. The "Group" Concept

In math, a Group is a collection of things (like numbers or shapes) that follow specific rules when you mix them together (like adding or multiplying). For example, if you mix two numbers in a group, you get another number in the same group.

The paper asks: What happens if our "Group" is made of these complex Picture Fuzzy Multisets?

3. The New Rules (The "Multigroup")

The author introduces the Picture Fuzzy Multigroup (PFMG). Think of this as a club where membership isn't just a simple "in" or "out." Instead, every member has a complex profile:

Positive Count: How strongly they belong (Yes).
Neutral Count: How unsure they are (Neutral).
Negative Count: How strongly they don't belong (No).

The paper establishes the "laws of physics" for this new type of club. For example:

The Mixing Rule: If you take two members and "mix" them (perform the group operation), the resulting member's "Yes" score must be at least as high as the weaker of the two original "Yes" scores.
The Mirror Rule: If you flip a member (take their inverse), their "Yes" score shouldn't drop below their original score.

4. The "Cut" Analogy (Simplifying the Complex)

One of the most useful tools in the paper is the Cut Set.

Imagine you have a very complex, multi-layered cake (the Picture Fuzzy Multigroup). It's too complicated to eat all at once. The "Cut Set" is like slicing the cake at a specific height.

If you slice it high up, you only get the "very strong" members.
If you slice it lower, you get a bigger group that includes "okay" members.

The paper proves a crucial fact: No matter how you slice this complex cake, the piece you get is always a standard, simple "crisp" group. This means that even though the math is fancy and complex, if you look at it through a specific filter, it behaves like a normal, predictable group.

5. The "Core" of the Group

The paper also defines a special subset called $D^*$ . Think of this as the "VIP Lounge" or the "Core Team."

These are the members who have the exact same "Yes," "Neutral," and "No" scores as the group's leader (the identity element).
The paper proves that this VIP Lounge is also a valid, standard group on its own.

Summary

In plain English, this paper is a mathematical blueprint for handling groups where membership is complicated, repeated, and multi-faceted.

It takes the idea of a "group" (a structured collection) and upgrades it to handle:

Ambiguity: (Yes, Neutral, No).
Repetition: (Counting how many times something appears).
Structure: (Ensuring that when you mix these complex items, the rules still hold up).

The author doesn't just invent this; he proves that these new "Picture Fuzzy Multigroups" behave logically. He shows that if you look at them closely enough (using "cuts"), they turn into normal, reliable groups, giving mathematicians a way to use this complex tool with confidence.

Technical Summary: Picture Fuzzy Multigroups

Problem Statement and Context
The paper addresses the need to extend algebraic structures within the framework of fuzzy set theory to handle more complex uncertainty. The authors trace the evolution of set theory from Zadeh's Fuzzy Sets (FS) to Atanassov's Intuitionistic Fuzzy Sets (IFS), and subsequently to Cuong and Kreinovich's Picture Fuzzy Sets (PFS), which incorporate positive, neutral, and negative membership degrees. Parallel to this, the concept of multisets (where elements can occur multiple times) was introduced by Yager and extended to Intuitionistic Fuzzy Multisets. While Cao et al. had previously established Picture Fuzzy Multisets (PFMS) as a generalization of both Fuzzy Multisets and Intuitionistic Fuzzy Multisets, and Shinoj and Sunil had defined Intuitionistic Fuzzy Multigroups, a formal definition and algebraic analysis of Picture Fuzzy Multigroups (PFMG) were lacking. This paper aims to fill that gap by introducing the concept of PFMGs and establishing their fundamental algebraic properties.

Methodology
The study employs a deductive mathematical approach grounded in abstract algebra and fuzzy logic. The methodology proceeds as follows:

Foundational Definitions: The paper begins by reviewing necessary preliminaries, including definitions of Fuzzy Sets, Intuitionistic Fuzzy Sets, and Picture Fuzzy Sets. It specifically defines Picture Fuzzy Multisets (PFMS) over a crisp group $C$ , characterized by three count functions: Positive Count Membership ( $PC_M$ ), Neutral Count Membership ( $NeC_M$ ), and Negative Count Membership ( $NC_M$ ).
Structural Operations: The authors define operations on PFMS, including the inverse ( $D^{-1}$ ) and the composite ( $D \circ E$ ) of two picture fuzzy multisets. These operations are defined using lattice operations (supremum $\vee$ and infimum $\wedge$ ) applied to the membership counts of elements and their inverses within the group.
Definition of PFMG: A Picture Fuzzy Multiset $D$ is defined as a Picture Fuzzy Multigroup if its membership counts satisfy specific inequalities regarding the group operation and inverses. Specifically, the positive and neutral counts must be non-decreasing under the group operation and inverses, while the negative count must be non-increasing.
Theoretical Proofs: The paper utilizes these definitions to prove a series of theorems regarding the properties of PFMGs, including closure under inversion, behavior under composition, and the relationship between a PFMG and its level cuts.

Key Contributions and Results
The primary contribution of the paper is the formal introduction of Picture Fuzzy Multigroups and the subsequent derivation of their algebraic properties. The specific results established include:

Properties of Inverse and Composition: The paper proves that the inverse of a PFMG is also a PFMG and establishes the relationship $(D \circ E)^{-1} = E^{-1} \circ D^{-1}$ . It further provides alternative formulations for the composite operation using unique group elements.
Fundamental Inequalities: Theorem 3.3 demonstrates that for any PFMG $D$ , the membership counts at the identity element $e$ are maximal for positive and neutral counts and minimal for the negative count (i.e., $PC_M(e) \geq PC_M(g)$ , $NeC_M(e) \geq NeC_M(g)$ , and $NC_M(e) \leq NC_M(g)$ ). It also proves that $D^{-1} \supseteq D$ .
Equivalent Conditions: Theorem 3.4 provides an equivalent condition for a PFMS to be a PFMG, stating that $D$ is a PFMG if and only if the membership conditions hold for the expression $g_1 g_2^{-1}$ .
Cut Sets and Subgroups: The paper introduces the $(d, r, s, t)$ -cut set for a picture fuzzy multiset. Theorem 3.5 proves that if $D$ is a PFMG, its $(d, r, s, t)$ -cut set is a crisp subgroup of the underlying group $C$ .
Level Subgroups: Theorem 3.6 defines the set $D^*$ consisting of elements where the membership counts equal those of the identity element. The paper proves that $D^*$ forms a crisp subgroup of $C$ .

Significance and Claims
The paper modestly claims that its significance lies in extending the existing work on Intuitionistic Fuzzy Multigroups (Shinoj and Sunil) to the broader context of Picture Fuzzy Sets. By introducing PFMGs, the authors provide a new algebraic structure that can model uncertainty with greater granularity, accounting for positive, neutral, and negative opinions simultaneously within a multiset environment. The establishment of properties such as the subgroup nature of cut sets and the behavior of inverses and compositions lays the necessary theoretical groundwork for future applications of these structures in algebraic systems involving complex, multi-faceted uncertainty. The paper does not propose specific real-world applications or experimental validations but focuses strictly on the mathematical consistency and properties of the introduced concept.