In this paper we introduce fuzzy congruence and fuzzy strong congruence of semi-hypergroups as well as polygroups. We obtain some results in this respect. We also prove that FC(H), the set of all fuzzy strong congruences on a polygroup H , consists a complete lattice and this lattice is isomorphic to FN (H), the lattice of fuzzy normal subpolygroups.

We study in detail the blow-up procedure described in [BTW01]. We obtain a structure theorem for coreless polygroups as a double quotient space G/H, and a polygroup chunk theorem. Seeking to remove the arbitrary parameter needed for the blow-up, we find canonical ∅-invariant groupoids G > H analogous to G and H above, and show that H contains precisely all the arbitrary choices related to the b...

The class of  $n$-ary polygroups is a certain subclass of $n$-ary hypergroups, a generalization of D{"o}rnte $n$-arygroups and  a generalization of polygroups. The$beta^*$-relation and the $gamma^*$-relation are the smallestequivalence relations on an $n$-ary polygroup $P$ such that$P/beta^*$ and $P/gamma^*$ are an $n$-ary group and acommutative $n$-ary group, respectively. We use the $beta^*$-...

Fuzzy n-ary hypergroups were introduced as suitable generalizations of fuzzy polygroups and a special case of fuzzy n-ary hyper groups. The aim of this paper is to introduce the notion of fuzzy n-ary factor polygroups of a polygroup. Based on the fuzzy n-ary factor group, we also study the product structures of the generating fuzzy factor n-ary groups. At the end of the paper, we prove the fund...

The combination of two elements in a group structure is an element, while, hypergroup, the non-empty set. use hypergroups appears mainly certain subclasses. For instance, polygroups, which are special subcategory hypergroups, used many branches mathematics and basic sciences. On other hand, multi-fuzzy set, element universal set may occur more than once with possibly same or different membershi...

In this paper, by using the notion of fuzzy subsets, the concept of F-permutation is introduced. Then by applying this notion the concepts of presentation of an F-polygroup, graph of an F-permutation and F-permutation matrices are investigated.

Applications of hypergroups have mainly appeared in special subclasses. One of the important subclasses is the class of polygroups. In this paper, we study the notions of nilpotent and solvable polygroups by using the notion of heart of polygroups. In particular, we give a necessary and sufficient condition between nilpotent (solvable) polygroups and fundamental groups.