A , nite group G, in which two randomly chosen subgroups H and K commute, has been classi, ed by Iwasawa in 1941. It is possible to de, ne a probabilistic notion, which \measures the distance" of G from the groups of Iwasawa. Here we introduce the generalized subgroup commutativity degree gsd(G) for two arbitrary sublattices S(G) and T(G) of the lattice of subgroups L(G) of G. Upper and lower bounds for gsd(G) are shown and we study the behaviour of gsd(G) with respect to subgroups and quotients, showing new numerical restrictions.

Picture fuzzy set (PFS) is a novel concept for dealing with uncertainty and a generalization of the traditional fuzzy set (FS) and intuitionistic fuzzy set (IFS) and can easily manage the uncertain nature of human thoughts by incorporating the positive, neutral, negative and refusal membership degrees of an object. Many conceptual ideas on PFSs have been developed so far and applied in diversified fields. In this paper, the concept of picture fuzzy sublattices and picture fuzzy ideals are developed and some of their associated properties are established in detail. Moreover, the sum and product of two picture fuzzy ideals are introduced with their properties. Finally, some properties of picture fuzzy ideals under lattice homomorphism are explored.

Uninorms and nullnorms are special 2-uninorms. In this work, we construct 2-uninorms on bounded lattices. Let L be a bounded lattice with a nontrivial element d. Given two uninorms U1 and U2, de ned on sublattices [0; d] and [d; 1], respectively, this paper presents two methods for constructing binary operators on L which extend both U1 and U2. We show that our  rst construction is a 2-uninorm on L if and only if U2 is conjunctive and our second construction is a 2-uninorm on L if and only if U1 is disjunctive. Moreover, we prove that the two 2-uninorms are, respectively, the weakest and the strongest 2-uninorm among all 2-uninorms, the restrictions of which on [0; d]2 and [d; 1]2 are respectively U1 and U2.

The discovery of graphene and its remarkable electronic and magnetic properties has initiated great research interest in this material. Furthermore, there are many derivatives in these graphene related materials among which graphene nanoribbons and graphene nanofragments are candidates for future carbon-based nanoelectronics and spintronics. Theoretical studies have shown that magnetism can arise in various situations such as point defects, disorder and reduced dimensionality. Using a mean field Hubbard model, we studied the appearance of magnetic textures in zero-dimensional graphene nanofragments and one-dimensional graphene zigzag nanoribbons. Among nanofragments, triangular shape, bowtie and coronene were studied. We explain how the shape of these materials, the imbalance in the number of atoms belonging to the graphene sublattices, the existence of zero-energy states and the total and local magnetic moments were related. At the end, we focused on the effects of a model disorder potential (Anderson-type), and illustrate how density of states of zigzag nanoribbons was affected.

In this research the magnetoelastic properties of Nd6Fe13-xCoxCu (x=0 and 1) compound are investigated. Analysis of X-ray diffraction patterns indicates that a single phase is formed approximately for x=0 and the x=1 sample is multi-phase. The lattice parameters were decreased, the Néel temperature of main phase and the Curie temperature of impurity phase was increased with Co content. Due to the presence of Nd2Fe17-yCoy ferromagnetic phase in the sample with x=1 the change of the anisotropy and increase of exchange effects was observed in the magnetic and magnetoelastic measurements. Thermal expansion, longitudinal (ll) and transverse (lt) magnetostriction were measured using the strain gauge method in the selected temperatures range of 80- 500 K under applied magnetic fields up to 1.5 T. Anomaly and Invar effect are observed in the linear thermal expansion and a (T) curves at the Néel temperature. The linear spontaneous magnetostriction decreases sharply by approaching the Néel temperature and also shows the short-range magnetic ordering effects when antiferromagnetic to paramagnetic transition occurs. In the low field region, the absolute values of anisotropic magnetostriction of Nd6Fe13Cu compound are small and then start to increase with applied magnetic field. Each isofield curve of the anisotropic magnetostriction passes through a minimum and then approaches to zero with increasing temperature. This magnetostriction compensation arises from the difference in the magnetoelastic coupling constants of the sublattices in this compound.