R is called commuting regular ring (resp. semigroup) if for each x,y $in$ R there exists a $in$ R such that xy = yxayx. In this paper, we introduce the concept of commuting $pi$-regular rings (resp. semigroups) and study various properties of them.

‎a new method for the defuzzification of fuzzy numbers is developed in this paper. it is well-known, defuzzification methods allow us to find aggregative crisp numbers or crisp set for fuzzy numbers. but different fuzzy numbers are often converted into one crisp number. in this case the loss of essential information is possible. it may result in inadequate final conclusions, for example, expert...

r is called commuting regular ring (resp. semigroup) if for each x,y $in$ r thereexists a $in$ r such that xy = yxayx. in this paper, we introduce the concept of commuting$pi$-regular rings (resp. semigroups) and study various properties of them.

Consider a d-regular directed graph, G, on n vertices determined by d permutations 1; : : : ; d 2 Sn. Each word, w, over the alphabet = f 1; 2; : : : ; dg determines a walk at each vertex, and these walks give rise to a permutation of the vertices (which is simply the product of the permutations). Such a word maps a distinct r-tuple of vertices, (v1; : : : ; vr) to another distinct r-tuple of v...