On t-Intuitionistic Fuzzy PMS-Subalgebras of a PMS Algebra

1. Introduction

In 1965, Zadeh introduced the idea of a fuzzy set as the generalization of the crisp set for describing uncertainty in our universe. Rosenfeld (1971) introduced the concept of fuzzy subgroups and established some related results. Atanassov (1986, 1989) developed the theory of an intuitionistic fuzzy set as an extension of a fuzzy set for describing uncertainties more efficiently. Since then, several authors have applied the idea of an intuitionistic fuzzy set to different algebraic structures. Biswas (1989) studied intuitionistic fuzzy subgroups of a group using the concept of intuitionistic fuzzy sets. Peng (2012) introduced the notion of intuitionistic fuzzy B-algebras in B-algebra and investigated various aspects of their homomorphic image and inverse image. Jana et al. (2015) investigated several properties of G-subalgebras of G-algebras using the concept of intuitionistic fuzzy sets.

Many researchers have also applied the idea of an intuitionistic fuzzy set for describing uncertainties in real-life situations. Yu and Li. (2022) proposed a novel intuitionistic fuzzy goal programming method for heterogeneous multi-attribute decision making under multi-source information. Yu et al. (2021) developed a new and unified intuitionistic fuzzy multi-objective linear programming model for portfolio selection problems to solve multi-objective decision problems with hesitation degrees and reduce the complexity of the nondeterministic polynomial-hard problems. Li and Wan (2017) developed an effective method for solving intuitionistic fuzzy multi-attribute decision-making problems with incomplete weight information. Bhaumik et al. (2017) studied a matrix game with triangular intuitionistic fuzzy numbers as payoffs and used robust ranking approaches to rank fuzzy numbers in order to solve the matrix game. Moreover, to deal with complex decision-making problems in which the membership and non-membership degrees of fuzzy concepts cannot be expressed with exact numerical values due to a lack of information in many real-life situations, Atanassov and Gargov (1989) developed an interval-valued intuitionistic fuzzy set characterized by interval-valued membership and non-membership functions rather than real numbers. Using the concept of an interval-valued intuitionistic fuzzy set, Wei et al. (2021) developed and applied an information-based score function of the interval-valued intuitionistic fuzzy set to multiattribute decision-making to overcome the limitations of existing ranking methods and rank the interval-valued intuitionistic fuzzy set well. Their results demonstrated that the information-based score function is more reasonable than existing ranking methods. Li (2011) developed the representation theorem and extension principles for interval-valued intuitionistic fuzzy sets based on the concept of level sets of interval-valued intuitionistic fuzzy sets.

Sharma (2012) developed the concept of the t-intuitionistic fuzzy set as an extension of the intuitionistic fuzzy set to deal with uncertainty and vagueness and then introduced the idea of t-intuitionistic fuzzy subgroups and t-intuitionistic fuzzy quotient groups, as well as t-intuitionistic fuzzy subring of a ring. Shuaib et al. (2020) introduced the notion of η-intuitionistic fuzzy subgroup over η-intuitionistic fuzzy subset and studied some algebraic aspects of η-intuitionistic fuzzy subgroups. Barbhuiya (2015) introduced the concepts of t-intuitionistic fuzzy subalgebra and t-intuitionistic fuzzy normal subalgebra of a BG-algebra, and then investigated the homomorphic image and inverse image of both t-intuitionistic fuzzy subalgebra and t-intuitionistic fuzzy normal subalgebra of a BG-algebra.