Abstract. We present a methodology for characterization and an approach for computing the solutions of fuzzy linear systems with LR fuzzy variables. As solutions, notions of exact and approximate solutions are onsidered. We transform the fuzzy linear system into a orresponding linear crisp system and a constrained least squares problem. If the corresponding crisp system is incompatible, then the fuzzy LR system lacks exact solutions. We how that the fuzzy LR system has an exact solution if and only if the corresponding crisp system is compatible (has a solution) and the solution of the corresponding least squares problem is equal to zero. In this case, the exact solution is determined by the solutions of the two esponding problems. On the other hand, if the corresponding crisp system is compatible and the optimal value of the corresponding constrained least squares problem is nonzero, then we characterize approximate solutions of the fuzzy system by solution of the least squares problem. Also, we characterize solutions by defining an appropriate membership function so that an exact solution is a fuzzy LR vector having the membership function value equal to one and, when an exact solution does not exist, an approximate solution is a fuzzy LR vector with a maximal membership function value. We propose a class of algorithms based on ABS algorithm for solving the LR fuzzy systems. The proposed algorithms can also be used to solve the extended dual fuzzy linear systems. Finally, we show that, when the system has more than one solution, the proposed algorithms are flexible enough to compute special solutions of interest. Several examples are worked out to demonstrate the various possible scenarios for the solutions of fuzzy LR linear systems.

Abstract. The fuzzy linear regression model with fuzzy input-output data and crisp coefficients is studied in this paper. A linear programming model based on goal programming is proposed to calculate the regression coefficients. In contrast with most of the previous works, the proposed model takes into account the centers of fuzzy data as an important feature as well as their spreads in the procedure of constructing the regression model. Furthermore, the model can deal with both symmetric and non-symmetric triangular fuzzy data as well as trapezoidal fuzzy data which have rarely been considered in the previous works. To show the efficiency of the proposed model, some numerical examples are solved and a simulation study is performed. The computational results are compared with some earlier methods.

Abstract. In this paper, we define the notions of fuzzy congruence relations and fuzzy convex subalgebras on a commutative residuated 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 subalgebras on a commutative residuated lattice. Then we study fuzzy convex subalgebras of an integral commutative residuated lattice and will prove that fuzzy filters and fuzzy convex subalgebras of an integral commutative residuated lattice coincide.

Abstract. Fuzzy bi-ideals play an important role in the study of ordered semigroup structures. The purpose of this paper is to initiate and study the intiuitionistic fuzzy bi-ideals in ordered semigroups and investigate the basic theorem of intuitionistic fuzzy bi-ideals. To provide the characterizations of regular ordered semigroups in terms of intuitionistic fuzzy bi-ideals and to discuss the relationships of left(resp. right and completely regular) ordered semigroups in terms intuitionistic fuzzy bi-ideals.

Abstract. This paper presents characterizations of M-fuzzifying matroids by means of two kinds of fuzzy operators, called the M-fuzzifying derived operators and M-fuzzifying difference derived operators.

Abstract. In this paper, the concept of local base with stratified structure in I-topological vector spaces is introduced. We prove that every I-topological vector space has a balanced local base with stratified structure. Furthermore, a new characterization of I-topological vector spaces by means of the local base with stratified structure is given.

Abstract. The aim of this paper is the study of the sequence of join spaces and fuzzy subsets associated with a hypergroupoid. In this paper we give some properties of the membership function $ \ widetilde \ mu_{ \ otimes } $ corresponding to the direct product of two hypergroupoids and we determine the fuzzy grade of the hypergroupoid < H × H , $ \ otimes $ > in a particular case.

Abstract. In this paper, we introduce the concepts of 2-isometry, collinearity, 2-Lipschitz mapping in 2-fuzzy 2-normed linear spaces. Also, we give a new eneralization of the Mazur-Ulam theorem when X is a 2-fuzzy 2-normed linear space or $ \ Im (X) $ is a fuzzy 2-normed linear space, that is, the Mazur-Ulam theorem holds, when the 2-isometry mapped to a 2-fuzzy 2-normed linear space is affine.

Abstract. Let $ S $ be an ordered semigroup. A fuzzy subset of S is an arbitrary mapping from S into [0, 1], where [0, 1] is the usual interval of real numbers. In this paper, the concept of fuzzy generalized bi-ideals of an ordered semigroup S is introduced. Regular ordered semigroups are characterized by means of fuzzy left ideals, fuzzy right ideals and fuzzy (generalized) bi-ideals. Finally, two main theorems which characterize regular ordered semigroups and intraregular ordered semigroups in terms of fuzzy left ideals, fuzzy right ideals, fuzzy bi-ideals or fuzzy quasi-ideals are given. The paper shows that one can pass from results in terms of fuzzy subsets in semigroups to ordered semigroups. The corresponding results of unordered semigroups are also obtained.

Abstract. In this paper, we introduce the notion of an action $ Y_X $ as a generalization of the notion of a module, and the notion of a norm $ \vt: Y_X \ to F $, where $ F $ is a field and $ \ vartriangle (xy) \ vartriangle(y') = $ $ \ vartriangle(y) \ vartriangle (xy') $ as well as the notion of fuzzy norm, where $ \ vt: Y_X \ to [0, 1] \ subseteq { \ bf R} $ , with $ \ bf R $ the set of all real numbers. A great many standard mappings on algebraic systems can be modeled on norms as shown in the examples and it is seen that $ \ mathrm {Ker} \ vt = \ {y| \ vt(y)=0 \ } $ has many useful properties. Some are explored, especially in the discussion of fuzzy norms as they relate to the complements of subactions $ N_X $ of $ Y_X $.

Abstract. In this paper we enumerate fuzzy subgroups, up to a natural equivalence, of some finite abelian p-groups of rank two where p is any prime number. After obtaining the number of maximal chains of subgroups, we count fuzzy subgroups using inductive arguments. The number of such fuzzy subgroups forms a polynomial in p with pleasing combinatorial coefficients. By exploiting the order, we label the subgroups of maximal chains in a special way which enables us to count the number of fuzzy subgroups.