This paper first improves Chen and Hsieh’ s definition of generalized Fuzzy Numbers, which makes it the generalization of definition of Fuzzy Numbers. Secondly, in terms of the generalized Fuzzy Numbers set, we introduce two different kinds of orders and arithmetic operations and metrics based on the -cutting sets or generalized -cutting sets, so that the generalized Fuzzy Numbers are integrated into a partial ordering set, a semi-ring and a complete non-separable metric space. Again, through two isomorphism theorem, the relationship between generalized Fuzzy number space and Fuzzy number space is established. Finally, as an application of generalized Fuzzy Numbers, the concept of continuous generalized Fuzzy number-valued functions is introduced. And it is pointed out that many results of trapezoidal generalized Fuzzy Numbers can also be extended to generalized Fuzzy Numbers.

In this paper, the concept of Fuzzy Convex subgroup (resp. Fuzzy Convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. Also, the Fuzzy Convex subgroup (resp. Fuzzy Convex lattice-ordered subgroup) generated by a Fuzzy subgroup (resp. Fuzzy subsemigroup) is characterized. Furthermore, the Fundamental Homomorphism Theorem is established. Finally, it is proved that the class of all Fuzzy Convex lattice-ordered subgroups of a lattice-ordered group G forms a complete Heyting sublattice of the lattice of Fuzzy subgroups of G.

In this paper, we define the notions of Fuzzy congruence relations and Fuzzy Convex sub algebras on a commutative resituated lattice and we obtain some related results. In particular, we will show that there exists a one to one correspondence between the set of all Fuzzy congruence relations and the set of all Fuzzy Convex subalgcbras on a commutative residuatcd lattice. Then we study Fuzzy Convex sub algebras of an integral commutative residuated lattice and will prove that Fuzzy filters and Fuzzy Convex suhalgebras of an integral commutative residuated lattice coincide.

Relative comparison of Fuzzy Numbers plays an important role in the domain of the Fuzzy multi-criteria and multi-attribute decision making analysis. For making a comparison between two Fuzzy Numbers, beyond the determination of their order, the concept of inequality is no longer crisp. That is, an inequality becomes Fuzzy in the sense of representing partial belonging or degree of membership when one makes a cardinal comparison between two or more hypothetical Fuzzy Numbers. It means that we need to calculate m (mÎ[0,1]) in the Fuzzy inequalities £m and ³m among two normal Fuzzy Numbers. In this paper we propose a method for capturing the membership degree of Fuzzy inequalities through discretizing the membership axis (m) into equidistant intervals. In this method, the two discretized Fuzzy Numbers are compared point by point and at each point the degree of preferences is identified. To show its validity, this method is examined against the essential properties of Fuzzy number ordering methods in [Wang, X. and E.E. Kerre, Reasonable properties for the ordering of Fuzzy quantities (I). Fuzzy Sets and Systems, 2001. 118(3): p. 375-385]. Furthermore it is compared numerically with some of the celebrated Fuzzy number ranking methods. The results, which provides promising outcomes, may come in useful in the domain of Fuzzy multi-criteria or multi-attribute decision making analysis, and more importantly, for Fuzzy mathematical programming with Fuzzy inequality constraints.

Stock portfolio problems are one of the most relevant real-world problems. In this study, we discuss the portfolio's risk amount, rate of risk-return, and expected return rate under a Fermatean Fuzzy environment. A linear programming problem is used to formulate a Fermatean Fuzzy portfolio. The Fermatean Fuzzy portfolio is converted to a deterministic form using the score function. Lingo software is used to solve these deterministic portfolio problems. The main feature of this model is that investors can select a risk coefficient to enhance predicted returns and customize their strategies according to their circumstances. An example is offered that illustrates the effectiveness and dependability of the proposed approach.