In this paper a first step in classifying the FUZZY NORMAL subgroups of a finite group is made. Explicit formulas for the number of distinct FUZZY NORMAL subgroups are obtained in the particular cases of symmetric groups and dihedral groups.

In this paper, NORMAL equations for singular FUZZY linear systems is proposed. The principal application of the NORMAL equations is to systems of equations that are inconsistent. The results are explained by solving some numerical examples.

In most manufacturing processes, each product may contain a variety of quality characteristics which are of the interest to be optimized simultaneously through determination of the optimum setting of controllable factors. Although, classic experimental design presents some solutions for this regard, in a FUZZY environment, and in cases where the response data follow non-NORMAL distributions, the available methods do not apply any more. In this paper, a general framework is introduced in which NORTA inverse transformation technique and FUZZY goal programming are used to deal with non-NORMALity distribution of the response data and the fuzziness of response targets respectively. Moreover, the presented framework uses a simulation approach to investigate the effectiveness of the determined setting of controllable factors obtained from statistical analysis, for optimization of sink mark index, deflection rate and volumetric shrinkage in plastic molding manufacturing processes. The accuracy of the proposed method is verified through a real case study.

In this paper, we introduce some properties of an intuitionistic NORMAL FUZZY m-subgroup of m- group with m-homomorphism and isomorphism. We study the image, the pre-image and the inverse mapping of the intuitionistic NORMAL FUZZY m-subgroups.

In this paper, we analyze the connection between R-hypermodules and graphs by associating a graph with an R- hypermodule through a NORMAL FUZZY subhypermodule. We investigate the graph's properties, including connectedness, completeness, Eulerian and Hamiltonian characteristics. By defining a regular relation based on the FUZZY  subhypermodule, we study how algebraic properties of R-hypermodules induence the associated graph. This work contributes to the understanding of FUZZY algebraic structures and their graphical representations, with potential applications in computer science and network theory.

In this paper, two related quotient structures are investigated utilizing the concept of coset. At first, a new hypervector space F/V = (F/V,\circ,\circledcirc,K) is created, which is composed of all cosets of a bipolar FUZZY soft set (F;A) over a hypervector space V . Then it will be shown that dim F/V = dim V/W, where the quotient hypervector space V/W includes all cosets of an especial subhyperspace W of V. Also, three bipolar FUZZY soft sets over the quotient hypervector space V/W are presented and in this way some new bipolar FUZZY soft hypervector spaces are defined.

In this paper we investigate FUZZY tree automata. FUZZY tree automata extend FUZZY automata. In FUZZY automata we deal with strings and in FUZZY tree automata we utilize trees instead of strings. We define a recognizable step FUZZY tree language and by a theorem we show that recognizable FUZZY tree language is recognizable step FUZZY tree language. Then, we define a deterministic FUZZY tree automaton and support of a FUZZY tree language and show that support of a FUZZY tree language can be recognized by a FUZZY tree automaton. We explain these concepts by examples. We also show that for a recognizable step mapping, there exists a deterministic FUZZY tree automaton that can recognize it, and vice versa. In fact we prove that the class of recognizable FUZZY tree languages is equal to the class of NORMAL recognizable step mappings. The obtained theorems generalize the results of weighted automata to tree automata.

In this research, new representations of basic functions are proposed based on the new types of FUZZY partition and a subNORMAL generating function. The generalized uniform FUZZY partitions in subNORMAL case, i.e. in case a generating function K is not NORMAL (generalized NORMAL case), and simpler form of FUZZY transform (FzT) components based on these new representations of the generalized uniform FUZZY partitions are indicated. The main properties of a new uniform FUZZY partition are suggested. New theorems and lemmas are proved.

The aim of this paper is the study of applications of a group in a general FUZZY automaton. For this purpose, we de ne the concepts of FUZZY NORMAL kernel of a general FUZZY automaton, FUZZY kernel of a general FUZZY automaton, adjustable, multiplicative. Then we obtain the relationships between them...