Hypergroups that have at least one identity element and where each element has at least one inverse are called regular hypergroup. In this regards, for a regular hypergroup $H$, it is shown that there exists a correspondence between the set of all strongly regular relations on $H$ and the set of all normal subhypergroups of $H$ containing $S_{\beta}$. More precisely, it has been proven that for every strongly regular relation $\rho$ on $H$, there exists a unique normal subhypergroup of $H$ containing $S_{\beta}$, such that its quotient is a group, isomorphic to $H/\rho$. Furthermore, this correspondence is extended to a lattice isomorphism between them.

A graph $G$ with a vertex set $V$ and an edge set $E$ is called regular if the degree of every vertex is the same. A quasi-regular graph is a graph whose vertices have one of two degrees $r$ and $r-1$, for some positive integer $r$. A graph $G$ is said to be self-complementary if $G$ is isomorphic to it's complement $\overline{G}$. In this paper we give a new method for construction of regular and quasi-regular self-complementary graph.

Let S be an ordered semi group. A fuzzy subset of S is an arbitrary mapping from S into [0,1], where [0,1] is the usual interval of real numbers. In this paper, the concept of fuzzy generalized bi-ideals of an ordered semi group S is introduced. Regular ordered semi groups are characterized by means of fuzzy left ideals, fuzzy right ideals and fuzzy (generalized) bi-ideals. Finally, two main theorems which characterize regular ordered semi groups and intraregular ordered semi groups in terms of fuzzy left ideals, fuzzy right ideals, fuzzy bi-ideals or fuzzy quasi-ideals are given. The paper shows that one can pass from results in terms of fuzzy subsets in semi groups to ordered semi groups. The corresponding results of unordered semi groups are also obtained.

In this paper we introduce the concepts of special regular clean elements and regular clean decomposition in a ring R. These concepts lead us to the notion of special regular clean ring. We prove that for a special regular clean element a = e + r ∈ R and unit u ∈ R then au is a special regular clean if u is an inner inverse of e. We establish that an abelian ring R is a special regular clean ring if and only if the twisted power series ring R[[x, σ]] is a special regular clean ring. We also study various characterizations of special clean and special regular clean rings.

In this paper, it is proven that a semigroup is regular and RGC_n-commutative if, and only if, it is a spined product of a commutative Clifford semigroup and a right regular band