mohammad ali dehghanizadeh

Crossed polysquares are defined by Dehghanizadeh, Davvaz and Alp. Their properties and the generalization of results from intersecting squares to intersecting polysquares have been expressed and proved by them with the help of fundamental relations.In the following, the concept of intersected polymodules and $\Gamma$-equivalent, intersected polymodules of polygroups are introduced and some properties are obtained from it. In addition, the concept of hypermultiplying fiber and crossed polysquares in the homotopy form of kernels has been studied. These results have extended the results related to intersecting squares to intersecting polysquares. In this article, homotopy crossed polysquares are studied as homotopies of cokernel, then the image of a crossed polymodule is considered and some results are proved that show the correspondence between crossed polymodules and crossed polysquares. In the continuation of the studies, we can check the results about the crossed 2-squares and then the crossed 2-polysquares. In addition, concepts about intersecting n-squares and intersecting n-polysquares can be expanded and studied.

In this article, we introduce the soft subhypergroupoid andsoft action hypergroupoid and study their properties. We consider thecategory of soft hypergroupoids whose objects are soft hypergroupoidsand morphisms are soft hypergroupoid homomorphisms. Also we con-sider the concept of soft crossed hypermodules and the category of softcrossed hypermodules. We show that a soft hg-hypergroupoid, can beobtained from each soft crossed hypermodule and a soft crossed hyper-module, can be obtained from each soft hg-hypergroupoid. In addition,we show that soft hg-hypergroupoids and soft crossed hypermodules,are equivalent categories.

One of the most important topics in the theory of fuzzy groups is the classification of fuzzy subgroups of a finite group. Enumeration of fuzzy subgroups has been done in the theory of fuzzy groups using the equivalence relation defined by Murali and Makamba. Using this idea, Neeraj and Sharma enumerated Intuitionistic fuzzy subgroups of a finite Abelian group by concept of double pinned flags.We are in thisWe do this in another way for some finite Abelian and non-Abelian groups. In fact, with the help of a suitable equivalence relation on Intuitionistic fuzzy subgroups and the lattice of subgroups, we obtain the enumeration of Intuitionistic fuzzy subgroups of finite groups of order 12. Also, we obtain the enumeration of Intuitionistic fuzzy subgroups of finite groups $Z_{p^k}\times Z_q$ and $Z_{p^k}$ with a different method from the article [9]. Finally, with the help of the results obtained in this paper, we present the number of Intuitionistic fuzzy subgroups on a group with order less than 16 in a table.