Direct Product of Picture Fuzzy Subgroups

This paper investigates the concept of picture fuzzy subgroups by introducing their direct product and establishing several characterizations of this product using $(r, s, t)$-cut sets.

The Gist

Imagine you are trying to organize a massive, chaotic voting system. In the real world, when people vote, they don't just say "Yes" or "No." Sometimes they say "Maybe," sometimes they say "I refuse to vote," and sometimes they just stay silent.

This paper by Taiwo O. Sangodapo is about creating a mathematical framework to handle that kind of messy, human decision-making, and then figuring out what happens when you combine two different groups of voters together.

Here is the breakdown of the paper using simple analogies:

1. The Problem: The "Three-Color" Ballot

Traditional math (Classical Set Theory) is like a light switch: it's either ON or OFF.
Later, Fuzzy Logic was invented, which is like a dimmer switch: it can be 50% ON or 80% ON.

But the author introduces Picture Fuzzy Sets, which is like a three-color voting card. For every person (or object) in a group, you have to assign three scores:

• Green (Positive): How much do they agree? (e.g., "Vote Yes")
• Yellow (Neutral): How much are they undecided? (e.g., "Abstain")
• Red (Negative): How much do they disagree? (e.g., "Vote No")

The Rule: The sum of these three colors can't exceed 100%. If someone is 60% Green and 30% Yellow, they can only be 10% Red.

2. The Goal: The "Fuzzy Subgroup"

In math, a Group is a collection of things that can be combined (like adding numbers or mixing ingredients) following specific rules. A Subgroup is a smaller club inside that group that follows the same rules.

A Picture Fuzzy Subgroup is a "fuzzy club." Not everyone is 100% a member. Some are "mostly" members, some are "barely" members, and some are "sort of" members. The paper asks: If I have two of these fuzzy clubs, what happens if I smash them together?

3. The Main Event: The "Direct Product" (The Marriage of Clubs)

The core of the paper is the Direct Product. Imagine you have:

• Club A: A group of voters in Ibadan.
• Club B: A group of voters in Lagos.

The Direct Product is creating a new, giant club where every member is a pair: (One person from Ibadan + One person from Lagos).

The paper asks: If Club A is a valid fuzzy club and Club B is a valid fuzzy club, is the new "Super Club" (A × B) also a valid fuzzy club?

The Answer: Yes! The paper proves that if you take two valid fuzzy clubs and combine them, the result is automatically a valid fuzzy club.

4. The Secret Weapon: The "Cut Sets" (The Flashlight)

How did the author prove this? He used a clever trick called $(r, s, t)$-cut sets.

Think of a Picture Fuzzy Set as a foggy landscape. It's hard to see the edges clearly because everyone has different levels of membership.

• The Cut Set is like a flashlight or a filter.
• You set the brightness of the flashlight to a specific level ($r, s, t$).
• Anything that is "bright enough" (meets the threshold) stays visible. Anything too dim disappears.

The Magic: The author shows that if you shine this flashlight on the "Super Club" (the Direct Product), the people you see are exactly the same as shining the flashlight on Club A and Club B separately and then combining those visible people.

Because the "visible" parts (the crisp sets) are easy to understand and follow the rules, the author proves that the "foggy" whole (the fuzzy set) must also follow the rules.

5. The Special Cases: Normal Subgroups and Conjugates

The paper also looks at two special scenarios:

• Normal Subgroups: These are clubs where the order of mixing members doesn't matter (A + B = B + A). The paper proves that if you combine two "order-independent" fuzzy clubs, the result is also "order-independent."
• Conjugate Subgroups: These are clubs that are essentially the same, just viewed from a different angle (like looking at a sculpture from the left vs. the right). The paper proves that if you combine two pairs of "mirror-image" clubs, the resulting super-clubs are also mirror images of each other.

Summary: Why Does This Matter?

This paper is like building a mathematical safety net for complex decision-making.

By proving that you can combine these "fuzzy" groups without breaking the rules, the author gives scientists and engineers a reliable tool. They can now model complex systems—like medical diagnoses (where a patient might be "mostly sick," "slightly healthy," and "uncertain") or voting systems (with yes/no/abstain/refusal)—and combine data from different sources without losing the mathematical structure.

In a nutshell: The paper says, "If you have two groups of people making fuzzy, three-colored decisions, and you combine them into one big team, the math still works perfectly, provided you look at them through the right 'flashlight' (cut sets)."

Technical Summary

Here is a detailed technical summary of the paper "Direct Product of Picture Fuzzy Subgroups" by Taiwo O. Sangodapo.

1. Problem Statement

The paper addresses a gap in the algebraic theory of fuzzy sets. While the concepts of Fuzzy Sets (Zadeh, 1965), Intuitionistic Fuzzy Sets (Atanassov, 1984), and Picture Fuzzy Sets (Cuong & Kreinovich, 2013) are well-established, the specific algebraic structure of Picture Fuzzy Subgroups (PFSG) and their behavior under direct products requires further characterization.

Previous works (e.g., Dogra & Pal, Sangodapo & Onasanya) established the definition of PFSGs and their properties via $(r, s, t)$-cut sets. However, the behavior of the direct product of two picture fuzzy subgroups—specifically how the membership degrees (positive, neutral, negative) interact in the product space and under what conditions the product retains subgroup properties—had not been fully explored. The paper aims to define the direct product of PFSGs and establish necessary and sufficient conditions for these products to form valid subgroups and normal subgroups.

2. Methodology

The author employs a rigorous algebraic approach grounded in Picture Fuzzy Set (PFS) theory and Group Theory. The core methodology involves:

• Definitions: Formalizing the Cartesian product of two Picture Fuzzy Sets (PFS) $P$ and $Q$ defined on groups $G$ and $G^*$. The membership functions for the product $P \times Q$ are defined using:
  • Positive membership ($\sigma$): Maximum (supremum) of individual memberships.
  • Neutral membership ($\tau$): Maximum (supremum) of individual memberships.
  • Negative membership ($\eta$): Minimum (infimum) of the maximums of individual memberships (specifically $\min(\eta_P(p) \vee \eta_Q(q))$).
• Cut Set Analysis: The primary tool for characterization is the $(r, s, t)$-cut set. The paper utilizes the property that a PFS is a subgroup if and only if its $(r, s, t)$-cut sets are crisp subgroups for all valid thresholds.
• Logical Deduction: Theorems are proven by establishing equivalences between the properties of the fuzzy sets and their corresponding crisp cut sets, leveraging the known properties of direct products in classical group theory.

3. Key Contributions and Results

A. Definition of Direct Product

The paper defines the direct product $P \times Q$ of two PFSs $P$ (on $G$) and $Q$ (on $G^*$) as a PFS on the Cartesian product group $G \times G^*$.

• Membership Functions:
  • $\sigma_{P \times Q}(p, q) = \max(\sigma_P(p), \sigma_Q(q))$
  • $\tau_{P \times Q}(p, q) = \max(\tau_P(p), \tau_Q(q))$
  • $\eta_{P \times Q}(p, q) = \min(\eta_P(p) \vee \eta_Q(q))$ (Note: The paper uses $\wedge$ for min and $\vee$ for max in the definition logic).

B. Characterization via Cut Sets (Theorem 3.1)

A fundamental result establishes that the $(r, s, t)$-cut set of a direct product is the direct product of the individual cut sets:
$$C_{r,s,t}(P \times Q) = C_{r,s,t}(P) \times C_{r,s,t}(Q)$$
This bridges the gap between fuzzy algebra and classical algebra, allowing crisp group properties to be lifted to the fuzzy domain.

C. Preservation of Subgroup Properties (Theorem 3.2 & 3.3)

• Subgroups: If $P$ and $Q$ are Picture Fuzzy Subgroups (PFSG) of $G$ and $G^*$ respectively, then $P \times Q$ is a PFSG of $G \times G^*$.
• Normal Subgroups: If $P$ and $Q$ are Picture Fuzzy Normal Subgroups (PFNSG), then $P \times Q$ is a PFNSG of $G \times G^*$.
• Proof Strategy: These results are derived directly from Theorem 3.1 and the known fact that the direct product of classical subgroups (or normal subgroups) is a subgroup (or normal subgroup).

D. Necessary Conditions for Identity Elements (Theorem 3.4)

The paper identifies critical constraints regarding the identity elements ($e_1, e_2$) of the groups. If $P \times Q$ is a PFSG, then at least one of the following must hold:

1. The membership values of the identity in $Q$ dominate the values of any element in $P$ (i.e., $\sigma_Q(e_2) \geq \sigma_P(g)$, etc.).
2. The membership values of the identity in $P$ dominate the values of any element in $Q$.
   This implies that for the product to be a valid subgroup, one of the component fuzzy sets must be "dominant" at the identity level relative to the other.

E. Inverse Implications (Theorem 3.5 & Corollary 3.2)

The paper investigates the converse: If $P \times Q$ is a PFSG, does it imply $P$ and $Q$ are PFSGs?

• Theorem 3.5: If $P \times Q$ is a PFSG and specific dominance conditions regarding the identity of $G^*$ are met (specifically $\sigma_P(g) \leq \sigma_Q(e_2)$), then $P$ is guaranteed to be a PFSG of $G$.
• Corollary 3.2: Generally, if $P \times Q$ is a PFSG, then either $P$ is a PFSG of $G$ or $Q$ is a PFSG of $G^*$ (or both).

F. Conjugacy in Direct Products (Theorem 3.6)

The paper extends the concept of Picture Fuzzy Conjugate Subgroups (PFCSG). It proves that if $P_1, P_2$ are conjugate in $G$ and $Q_1, Q_2$ are conjugate in $G^*$, then their direct products $P_1 \times Q_1$ and $P_2 \times Q_2$ are conjugate in $G \times G^*$.

4. Significance and Impact

• Theoretical Advancement: This work extends the algebraic structure of fuzzy groups to the more complex Picture Fuzzy environment, which accounts for positive, neutral, and negative opinions (crucial for voting systems and medical diagnosis).
• Methodological Utility: By proving that the $(r, s, t)$-cut set of a product is the product of the cut sets, the paper provides a powerful tool for researchers. It allows them to solve complex fuzzy problems by reducing them to classical crisp problems.
• Application Potential: The results are directly applicable to decision-making systems involving multiple agents or criteria (modeled as groups) where neutrality and refusal are significant factors. The conditions for normality and conjugacy are essential for constructing quotient structures in picture fuzzy algebra, which are necessary for advanced applications in cryptography and coding theory within fuzzy environments.

In summary, Sangodapo successfully establishes the algebraic framework for the direct product of picture fuzzy subgroups, proving that these structures preserve subgroup and normal subgroup properties under specific conditions and providing a clear characterization via cut sets.