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

In this paper, the authors extend the concept of a t-intuitionistic fuzzy set to PMS-subalgebras of PMS-algebras. The authors define the t-intuitionistic fuzzy PMS-subalgebra of a PMS-algebra and show that any intuitionistic fuzzy PMS-subalgebra of a PMS-algebra is a t-intuitionistic fuzzy PMS-subalgebra. The authors provide the condition for an intuitionistic fuzzy set in a PMS-algebra to be a t-intuitionistic fuzzy PMS-subalgebra. The authors use their (α,β) level cuts to characterize the t-intuitionistic fuzzy PMS-subalgebras of PMS-algebra. The authors investigate whether the homomorphic images and inverse images of t-intuitionistic fuzzy PMS-subalgebras are also t-intuitionistic fuzzy PMS-subalgebras. Furthermore, the authors show that the homomorphic images and inverse images of the nonempty (α,β) level cuts of the t-intuitionistic fuzzy PMS-subalgebras of a PMS-algebra are again PMS-subalgebras of a PMS-algebra. Finally, the authors show that the Cartesian product of the t-intuitionistic fuzzy PMS-subalgebras of a PMS-algebra is itself a t-intuitionistic fuzzy PMS-subalgebra and characterize it in terms of its (α,β) level cuts


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 (1986Atanassov ( , 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 decisionmaking 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.Iseki and Tanaka (1978) introduced a class of abstract algebra called BCK-algebra.Iseki (1980) introduced another class of abstract algebra called BCI-algebra as a generalization of BCK-algebra.Selvam and Nagalakshmi (2016) introduced a new concept, called PMS-algebra, which is related to several classes of algebra such as BCI-algebra, BCK-algebras, TM-algebra and so on.The concept of a fuzzy PMS-subalgebra of a PMS-algebra was introduced by Selvam and Nagalakshmi in 2016.The study of intuitionistic fuzzification of PMS-subalgebra and PMS-ideal of PMS-algebras was done by Derseh et al. in (2021Derseh et al. in ( , 2022)).The authors (2022) also studied the intuitionistic Q-fuzzy PMSsubalgebra of a PMS-algebra and the intuitionistic Q-fuzzy PMS-ideals of a PMS-algebra.The notion of t-intuitionistic fuzzy subalgebra has been studied in several algebraic structures (Barbhuiya 2015;Gulzar et al. 2020;Sharma 2012;Shuaib et al. 2019Shuaib et al. , 2020)).However, as far as we know, no study has been conducted on the t-intuitionistic fuzzification of the PMS-subalgebra of a PMS-algebra.This motivated us to develop the t-intuitionistic fuzzy PMS-subalgebras in PMS-algebras.
In this article, we apply the concept of a t-intuitionistic fuzzy set to PMS-subalgebras in PMS-algebras.We define the t-intuitionistic fuzzy PMS-subalgebra of a PMS-algebra and show that an intuitionistic fuzzy PMS-subalgebra of a PMS-algebra is a t-intuitionistic fuzzy PMS-subalgebra.We give the condition for an intuitionistic fuzzy set in a PMS-algebra to be a t-intuitionistic fuzzy PMS-subalgebra.We characterize the t-intuitionistic fuzzy PMS-subalgebras of PMS-algebra by using their ± ² , ( ) level cuts.We investigate whether the homomorphic images and inverse images of t-intuitionistic fuzzy PMS-subalgebras are also t-intuitionistic fuzzy PMS-subalgebras.Moreover, we prove that the homomorphic images and inverse images of the nonempty ± ² , ( ) level cuts of the t-intuitionistic fuzzy PMS-subalgebras of a PMS-algebra are again PMS-subalgebras of a PMS-algebra.Finally, we demonstrate that the Cartesian product of the t-intuitionistic fuzzy PMS-subalgebras of a PMS-algebra is again t-intuitionistic fuzzy PMS-subalgebra and subsequently characterize it in terms of its (α, β) cuts.

PReLIMeNARIeS
In this section, we consider some basic definitions, results and important concepts of PMS-algebras that are needed for our work.Definition 2.1 (Selvam and Nagalakshmi, 2016) A PMS-algebra is a nonempty set X with a constant 0 and a binary operation * of type (2, 0) satisfying the following axioms: ) the following properties hold for all x y z X , , ∈ , i.
( ) defines the degree of membership of the element x in X .Definition 2.5 (Atanassov, 1986(Atanassov, , 1989) ) An intuitionistic fuzzy set A in a nonempty set X is an object having the form A      0 1 define the degree of membership and the degree of nonmembership respectively, with the condition 0 1 (Atanassov, 1986(Atanassov, , 1989) ) Let A and B be two intuitionistic fuzzy subsets of the set X , where A

Definition 2.7 (Derseh et al., 2021) An intutionistic fuzzy subset
, and ν ν ν , .Then the t-intuitionistic fuzzy set (t-IFS) A t in a nonempty set X is an object , where the function µ      0 1 denote the degree of membership and degree of nonmembership, respectively such t h a t µ µ , for all x X ∈ .
Note: For the sake of Simplicity we shall use the symbol ) and ν ν ν

t -INTUITIONISTIC FUZZy PMS-SUBALGeBRS OF A PMS-ALGeBRA
In this section, we study the notion of t -intuitionistic fuzzy PMS-subalgebra defined on intuitionistic fuzzy subset of X and show that every intuitionistic fuzzy PMS-subalgebra is also t-intuitionistic fuzzy PMS-subalgebra, but the converse need not be true.We also establish many fundamental results.
In this and the next sections X and Y denote PMS-algebra unless otherwise specified.
Proof.Let x y X , .∈ such that x y ≤ .Then x y * = 0 .By Definition 2.1 ( i ) and Proposition

. 3 ( iv ) , w e h a v e µ µ µ µ
Proof.Let A be an IF PMS-SA of X and x y X , ∈ .Then by the definition of t-IFS, x y ⇒ ( ) ) is a PMS-algebra with the following table Define the intuitionistic fuzzy subset µ ν

Thus, µ µ µ
, .By Remark 2 1 and ν ( ) sets respectively defined by , to the structures of PMS-algebras based on t-intuitionistic fuzzy sets and serve as a foundation for future studies in the fuzzification of PMS-algebra.To gain additional novel results, we will extend this notion to t-Q intuitionistic fuzzy PMS-subalgebras, t-intuitionistic multi-fuzzy, and anti-multifuzzy PMS-subalgebras in our future works.Furthermore, we will develop the Neuro-fuzzy algebraic structures for PMS-subalgebras of PMS-algebras.
Let X and Y be any two PMS-algebras, for every x y u (Gulzar et al., 2020) Let A t be t -IFS of X w.r.t IFS A. Then the α β , ( ) -cut of and B are IF PMS-SAs of Z .Thus by Theorem 3.4 A t and B t are t-IF PMS-SAs of Z for t ∈ If we take x = 2 and y = 3 , then µ µ ν The converse of above Theorem need not be neccassarily true.This fact is shown by the following Example: The intersection of any two t-IF PMS-SAs of X is also a t-IF PMS-SA of X. Proof.Let A t and B t be any two t-IF PMS-SAs of a PMS-algebra X and x y X Therefore A t is a t -IF PMS-SA of X .Theorem 3.11