The Fuzzy Lacunary I− Convergent of Γ Space Defined by Modulus (communicated by Naim Braha)

The concept of fuzzy sets and fuzzy set operations were first introduced by Zadeh [30], fuzzy logic has become an important area of research in various branches of Mathematics such as metric and topological spaces, theory of functions, approximation theory etc. Subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets. The concept of fuzzyiness has been applied in various fields such as Statistics, Cybernetics, Artificial intelligence, Operation research, Decision making, Agriculture, Weather forecasting, Quantum physics. Similarity relations of fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming etc. The concept of ideal convergence as a generalization of statistical convergence, and any concept involving statistical convergence plays a vital role in pure mathematics and also in other branches of science involving mathematics, especially in information theory, computer science, biological science, dynamical systems, geographic information systems, population modelling, and motion planning in robotics. The notion of I− convergence initially introduced by Kostyrko et al. Later on, it was further investigated from the sequence space point of view and linked with the summability theory by S̆alát et al., Tripathy and Hazarika, Kumar, Hazarika and Savas, Khan and Ebdullah, Khan et al, Khan and Tabassum, Das et al., and many other authors. Let (xmn) be a double sequence of real or complex numbers. Then the series ∑∞ m,n=1 xmnis called a double series. The double series ∑∞ m,n=1 xmnis said to be convergent if and only if the double sequence (Smn)is convergent, where