On fuzzy subgroups and fuzzy ideals under triangular norms

In this communication, we explicitly point out that the principal results of Liu 1982 basically deduced from the definition of binary operation ° on the set F(X) of all fuzzy subsets of X, also hold if one uses the weaker definition of product under triangular norm °_t. Fuzzy ideals with respect to the triangular norms are also defined.     

On the product of two Chernikov subgroups

It is shown that if a group G = AB is the product of two subgroups A and B, each of which has an abelian subgroup of index at most 2 satisfying the minimum condition and such that one of the subgroups A or B is of dihedral type, then G is abelian-by-finite with minimum condition.     

On some properties of pronormal subgroups

New results on tight connections among pronormal, abnormal and contranormal subgroups of a group have been established. In particular, new characteristics of pronormal and abnormal subgroups have been obtained.     

A study of normal fuzzy subgroups and characteristic fuzzy subgroups of a fuzzy group

In this paper, we investigate the properties of normal fuzzy subgroups of a fuzzy group. We also introduce the notion of a characteristic fuzzy subgroup of a fuzzy group, of which we provide level subset and strong level subset characterizations. Then we prove that a characteristic fuzzy subgroup of a fuzzy group is a normal fuzzy subgroup. Besides we prove that the commutator subgroup generated by the commutator of two normal fuzzy subgroups of a fuzzy group is contained in their intersection. Finally, we construct many new lattices and sublattices of fuzzy subgroups and normal fuzzy subgroups of a given fuzzy group. We also construct lattices of characteristic fuzzy subgroups possessing sup-property.     

On some rigid subgroups of semisimple Lie groups

We show that certain discrete subgroups of semisimple Lie groups satisfy rigidity properties and that a subclass of these discrete groups are actually of finite covolume.     

基于模糊乘积有限自动机的覆盖关系

对Mealy-型模糊有限自动机乘积结构作了进一步的研究,并且对覆盖关系作了细致的刻画,推广了原有的覆盖概念.针对Mealy-型这类模糊有限自动机,通过性质考察了此覆盖概念的合理有效性,新的覆盖概念在乘积自动机间建立了更多的联系.特别证明了直积、级联积、圈积三种乘积之间的覆盖关系.得到了一些乘积自动机覆盖关系的传递性质.     

On groups which are the product of two solvable subgroups

All the groups in this article are finite. A group G is said to be factorizable if G = AB with A,B proper subgroups of G . The general problem is to obtain information on structure of a factorizable group G = AB, under suitable restrictions on A and B . Recently E.Pisman [1] described the composition factors of factorizable group G = AB, where A and B are solvable groups of relatively prime orders.The aim of this paper is to prove the following more general result.     

On some chains of fuzzy sets

A partial order relation σ is defined in the set $\text{F}$(X) of the fuzzy sets in X. If this ordering is induced in the subset F(X) of the measurable fuzzy sets in the set X with totally finite positive measure, then fσg implies that the entropy of the fuzyy set f is not less than the entropyof g. By means of this ordering a lattice L on $\text{F}$(X) is defined and a lattice structure is induced in the set of infinite chains in L. Furthermore the set F′(X) of the fuzzy sets of F(X) which assume value in a finite subset of the real interval [0,1] is considered and the following properties are stated: any chain of elements of F′(X) is an infinite sequence of functions convergent in the mean to an integrable function, and the entropy is a valuation of bounded variation on the sublattice of L whose elements are in F′(X). The chains on L can offer a model of a cognitive process in a fuzzy environment when their elements are determined by a sequence of decisions. The limit property traduces the determinism of a such procedure.     

A characterization of fuzzy subgroups

An ordinary subgroup of a group G is (1) a subset of G, (2) closed under the group operation. In a fuzzy subgroup it is precisely these two notions that lose their deterministic character. A fuzzy subgroup μ of a group (G,·) associates with each group element a number, the larger the number the more certainly that element belongs to the fuzzy subgroup. The closure property is captured by the inequality μ(x · y)?T(μ(x), μ(y)). In A. Rosenfeld's original definition, T was the function ‘minimum’. However, any t-norm T provides a meaningful generalization of the closure property. Two classes of fuzzy subgroups are investigated. The fuzzy subgroups in one class are subgroup generated, those in the other are function generated. Each fuzzy subgroup in these classes satisfies the above inequality with T given by T(a, b) = max(a + b ?1, 0). While the two classes look different, each fuzzy subgroup in either is isomorphic to one in the other. It is shown that a fuzzy subgroup satisfies the above inequality with $\text{T =}\text{‘minimum’}$ if and only if it is subgroup generated of a very special type. Finally, these notions are applied to some abstract pattern recognition problems.     

On finite groups with some subgroups of prime indices

We establish the solvability of each finite group whose every proper nonmaximal subgroup lies in some subgroup of prime index.     

On the direct product of fuzzy topological groups

The connections between some countability properties of a topological space (X, $\text{T}$) and its generated fuzzy topological space (X, ω($\text{T}$)) are investigated.     

Semidirect products OF fuzzy subgroups

The semidirect product of two groups is an important construction in group theory. In this paper we define the semidirect product of fuzzy subgroups and give conditions for it to be a fuzzy subgroup. This extends the work of H. Sherwood on direct products of fuzzy subgroups. We then give an example where a fuzzy subgroup of a semidirect product is a semidirect product of fuzzy subgroups.     

On the supersoluble residual of a product of subnormal supersoluble subgroups

We give a sufficient condition for supersolubility of a finite group that is a product of two subnormal supersoluble subgroups. We prove that the supersoluble residual of such a group is equal to its nilpotent residual. Also we apply these results to finite groups that are a product of two subnormal p-supersoluble subgroups.     

On some kinds of fuzzy connected spaces

In this paper we introduce new results in fuzzy connected spaces. Among the results obtained we can mention the good extension of local connectedness. Also we prove that in a T ₁-fuzzy compact space the notions c-zero dimensional, strong c-zero dimensional and totally c_i-disconnected are equivalent.     

On some families of cooperative fuzzy games

In a fuzzy cooperative game the players may choose to partially participate in a coalition. A fuzzy coalition consists of a group of participating players along with their participation level. The characteristic function of a fuzzy game specifies the worth of each such coalition. This paper introduces well-known properties of classical cooperative games to the theory of fuzzy games, and studies their interrelations. It deals with convex games, exact games, games with a large core, extendable games and games with a stable core.     

On some characterizations of preinvex fuzzy mappings

Jeyakumar (Methods Oper. Res. 55:109–125, 1985) and Weir and Mond (J. Math. Anal. Appl. 136:29–38, 1988) introduced the concept of preinvex function. The preinvex functions have some interesting properties. For example, every local minimum of a preinvex function is a global minimum and nonnegative linear combinations of preinvex functions are preinvex. Invex functions were introduced by Hanson (J. Math. Anal. Appl. 80:545–550, 1981) as a generalization of differentiable convex functions. These functions are more general than the convex and pseudo convex ones. The type of invex function is equivalent to the type of function whose stationary points are global minima. Under some conditions, an invex function is also a preinvex function. Syau (Fuzzy Sets Syst. 115:455–461, 2000) introduced the concepts of pseudoconvexity, invexity, and pseudoinvexity for fuzzy mappings of one variable by using the notion of differentiability and the results proposed by Goestschel and Voxman (Fuzzy Sets Syst. 18:31–43, 1986). Wu and Xu (Fuzzy Sets Syst 159:2090–2103, 2008) introduced the concepts of fuzzy pseudoconvex, fuzzy invex, fuzzy pseudoinvex, and fuzzy preinvex mapping from \(\mathbb{R}^{n}\) to the set of fuzzy numbers based on the concept of differentiability of fuzzy mapping defined by Wang and Wu (Fuzzy Sets Syst. 138:559–591, 2003). In this paper, we present some characterizations of preinvex fuzzy mappings. The necessary and sufficient conditions for differentiable and twice differentiable preinvex fuzzy mapping are provided. These characterizations correct and improve previous results given by other authors. This fact is shown with examples. Moreover, we introduce additional conditions under which these results are valid.     

On the some normal subgroups of the extended modular group

In this paper, we give the group structures and the signatures of some normal subgroups of the extended modular group Π containing the principal congruence subgroup Γ(12).