[Paper Review] On Intuitionistic Fuzzy Magnified Translation in Semigroups

Sujit Kumar Sardar, Manasi Mandal+1 more|arXiv (Cornell University)|Jan 18, 2011

Fuzzy and Soft Set Theory7 references3 citations

TL;DR

This paper introduces intuitionistic fuzzy magnified translation in semigroups, generalizing fuzzy translation and multiplication via Atanassov’s intuitionistic fuzzy sets. It characterizes regular, intra-regular, and left/right regular semigroups using this new concept, showing that intuitionistic fuzzy translation and multiplication are special cases of magnified translation.

ABSTRACT

The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. S.K Sardar and S.K. Majumder unified the idea of fuzzy translation and fuzzy multiplication of Vasantha Kandasamy to introduce the concept of fuzzy magnified translation in groups and semigroups. The purpose of this paper is to intuitionistically fuzzify(by using Atanassov's idea) the concept of fuzzy magnified translation in semigroups. Here among other results we obtain some characterization theorems of regular, intra-regular, left(right) regular semigroups in terms of intuitionistic fuzzy magnified translation.

Motivation & Objective

• To extend the concept of fuzzy magnified translation to intuitionistic fuzzy sets in semigroups.
• To investigate the properties of intuitionistic fuzzy magnified translation in the context of semigroup theory.
• To characterize regular, intra-regular, and left/right regular semigroups using intuitionistic fuzzy magnified translation.
• To demonstrate that intuitionistic fuzzy translation and multiplication are special cases of the proposed magnified translation.
• To establish connections between algebraic semigroup properties and intuitionistic fuzzy ideal structures.

Proposed method

• Define intuitionistic fuzzy magnified translation as $ A_{etaeta}^{C}(x) = eta \cdot \mu_A(x) + \alpha $ for membership and $ \nu_{A_{eta\alpha}^{C}}(x) = \beta \cdot \nu_A(x) - \alpha $ for non-membership, with constraints on $ \alpha $ and $ \beta $.
• Introduce intuitionistic fuzzy subsemigroups, bi-ideals, (1,2)-ideals, and left/right ideals in semigroups using membership and non-membership functions.
• Establish closure properties: intuitionistic fuzzy magnified translation preserves intuitionistic fuzzy right/left ideals, bi-ideals, and (1,2)-ideals.
• Use the operation $ A \circ B $ defined via sup-inf composition to model semigroup multiplication in the intuitionistic fuzzy context.
• Apply characteristic theorems linking algebraic semigroup properties (e.g., regularity) to the behavior of intuitionistic fuzzy magnified translations.
• Utilize characteristic functions of ideals to reduce abstract results to concrete set-theoretic conditions, proving equivalence via sup-inf equations.

Experimental results

Research questions

• RQ1 How can the concept of fuzzy magnified translation be extended to intuitionistic fuzzy sets in semigroups?
• RQ2 What are the closure properties of intuitionistic fuzzy magnified translation with respect to intuitionistic fuzzy subsemigroups and ideals?
• RQ3 How can intuitionistic fuzzy magnified translation be used to characterize regular, intra-regular, and left/right regular semigroups?
• RQ4 In what way do intuitionistic fuzzy translation and multiplication emerge as special cases of the proposed magnified translation?
• RQ5 What is the relationship between the sup-inf composition of intuitionistic fuzzy magnified translations and the algebraic structure of semigroups?

Key findings

• Intuitionistic fuzzy magnified translation preserves the structure of intuitionistic fuzzy right and left ideals, bi-ideals, and (1,2)-ideals in semigroups.
• A semigroup is regular if and only if for every intuitionistic fuzzy right ideal $ A $ and left ideal $ B $, the composition $ A_{\beta\alpha}^{C} \circ B_{\beta\alpha}^{C} = A_{\beta\alpha}^{C} \cap B_{\beta\alpha}^{C} $.
• The condition $ \alpha = 0 $ is necessary for the characteristic function of a left or right ideal to satisfy the magnified translation identity.
• The sup-inf equations derived from the composition operation yield $ \sup_{x=yz} \min\{\chi_R(y), \chi_L(z)\} = 1 $ and $ \inf_{x=yz} \max\{\overline{\chi}_R(y), \overline{\chi}_L(z)\} = 0 $, implying $ x \in RL $ and thus $ R \cap L = RL $.
• Intuitionistic fuzzy translation and multiplication are special cases of magnified translation when $ \beta = 1 $ and $ \alpha = 0 $, respectively.
• Theorems and corollaries on fuzzy magnified translation in the literature are generalized and extended to the intuitionistic fuzzy setting, with analogues established for all relevant results.