‎Let $tilde{F}=(Q,S,tilde{R},Z,omega,tilde{delta},‎ F_1,F_2)$‎ ‎be a general fuzzy automaton and the set of its states be a group.‎ ‎The aim of this paper is the study of applications of a group in a‎ ‎general fuzzy automaton‎. ‎For this purpose‎, ‎we define the concepts‎ ‎of fuzzy normal kernel of a general fuzzy automaton‎, ‎fuzzy kernel‎ ‎of a general fuzzy automaton‎, ‎adjustable‎, ‎multip...

We associate a graph NG with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of NG and its induced subgraph on G\nil(G), where nil(G) = {x ∈ G | 〈x, y〉 is nilpotent for all y ∈ G}. For any finite group G, we prove that NG has eithe...

Let P be an arbitrary set of primes. The P -nilpotent completion of a group G is defined by the group homomorphism η : G → G P̂ where G P̂ = invlim(G/ΓiG)P . Here Γ2G is the commutator subgroup [G,G] and ΓiG the subgroup [G,Γi−1G] when i > 2. In this paper, we prove that P -nilpotent completion of an infinitely generated free group F does not induce an isomorphism on the first homology group with...

Let G $G$ be a finite group admitting coprime automorphism α $\alpha$ . J ( ) $J_G(\alpha )$ denote the set of all commutators [ x , ] $[x,\alpha ]$ where $x$ belongs to an -invariant Sylow subgroup We show that $[G,\alpha is soluble or nilpotent if and only any generated by pair elements orders from nilpotent, respectively.

The application of conjunctive aggregation operators in fuzzy control systems with 2 inputs is discussed and the effect of the choice of a continuous t-norm operator in inference phase for TSK systems is computed. It is shown that the choice of a continuous t-norm modelling AND connective in antecedent part of fuzzy rules can be reduced just to strict or nilpotent t-norm. Moreover, it is shown ...

We provide a new and simple proof to show that finite group in which every non-nilpotent maximal subgroup is normal solvable.

In this paper, we define the concept of a bipolar fuzzy cosets of a bipolar fuzzy HX subgroup, bipolar anti fuzzy HX subgroup, bipolar fuzzy middle coset of a bipolar fuzzy HX subgroup and bipolar anti fuzzy HX subgroup, conjugate bipolar fuzzy HX subgroup, conjugate bipolar anti fuzzy HX subgroup and discussed some of its properties with the examples. Further we define the level subsets and lo...

In this paper, the notion of generalized regular fuzzy irresolute, generalized regular fuzzy irresolute open  and generalized regular fuzzy irresolute closed maps in fuzzy  topological spaces are introduced and studied. Moreover, some separation axioms and $r$-GRF-separated sets are established. Also, the relations between generalized regular fuzzy continuous maps and generalized regular fuzzy ...

We know that the existence of Kähler structure on a compact complex manifold imposes certain homological or even homotopical restrictions on its underlining topological manifold. Hodge theory is of central importance in this line. There have been recently certain extensions and progresses in this area of research. Among them is the field of Kähler groups, in which the main subject to study is t...