Convergence of powers and Canonical form of s-transitive intuitionistic fuzzy matrix

The theory of fuzzy sets is used in various Mathematical fields. Zadeh [1] 1965 developed the concept of fuzzy sets which is the basis of fuzzy Mathematics. Since then various researchers worked on the development of fuzzy set theory. Atanassov [2,3,4,5,6,7] has given idea about intuitionistic fuzzy sets. Im and Lee [8] studied about the determinant of square intuitionistic fuzzy matrices (IFMs). Pal et.al [9] discussed (IFMs). Pal and Shyamal [10] defined distance between (IFMs). Bhowmik and Pal [11,12] discussed few properties of (IFMs) , intuitionistic circulant fuzzy matrices and generalized (IFMs). Meenakhsi and Gandhimathi [13] developed intuitionistic fuzzy relational equations. Sriram and Murugadas [14,15] developed the concept of semiring and sub-inverse of (IFMs). Murugadas and Lalitha [16,17, 18] applied implication operators and defined sub-inverse, g-inverse and decomposition of (IFMs). The authors [19] have studied reduction of rectangular (IFM). The theory of IFM is very important for the study of intuitionistic fuzzy relations. Thomason [20] studied about the convergence powers of fuzzy matrix. He provided the sufficient condition for convergence of fuzzy matrix. Buckley [21] Ran and Liu [22] and Gregory et al. [23] after using max-min operation of fuzzy matrix obtained only two results, either the fuzzy matrix convergences to idempotent matrices or oscillates to finite period. Hashimoto [24] explored the convergence of the power of a fuzzy transitive matrix. Lur et al. [25] studied about convergence of powers for a fuzzy matrix by using maxmin and max-arithmetic mean operations. Kolodziejczyk [26] discussed convergence of powers of s-transitive fuzzy matrix. Xin [27] studied the convergence of powers of controllable fuzzy matrix. He also showed that controllable fuzzy matrix oscillate with period equal 2. Nola [28] worked on the convergence of powers of reciprocal fuzzy matrices and deduced some properties. Kolodziejczyk [29] examined canonical form of s-transitive fuzzy matrix by using max-min transitive fuzzy matrix. Chenggong [30] discussed canonical form of the s-transitive matrices over lattices. Hashimoto [31] studied canonical form of the transitive fuzzy matrix. He reduced a transitive fuzzy matrix into the sum of a nilpotent fuzzy matrix and a symmetric fuzzy matrix. Lee and Jeong [32] studied some properties of canonical form of transitive IFM. An interesting problem in the theory of IFM is the convergence of the powers and canonical form of s-transitive IFM. Many authors worked on this problem. The purpose of this paper is to discuss the convergence of the powers and canonical form of the IFM.