In this paper, we considered two types of semigroup say bA and Mn[0,1] for the given directed graph D=(V,A) and mainly try to show that the above two semigroups are isomorphic to each other. Further we study the class of reduced idempotent in bA coincide with the class of partial order relation on a vertex set of disubgraph of D.

 In single objective optimization problems it is easy to find a metric that allows different solutions to be compared and ranked even if the optimum is not known. In a multiobjective optimization (MOO), however, a Pareto front must be considered rather than a single optimal point. A large number of methods for solving MOO problems have been developed. To compare these methods rigorously, or to measure the performance of a particular MOO algorithm quantitatively, a variety of performance metrics have been proposed. This paper presents a new performance metric based on Ideal and nadir points that should enable a designer to either monitor the quality of an observed Pareto solution set as obtained by a multiobjective optimization method, or compare the quality of observed Pareto solution sets as reported by different multiobjective optimization methods, also measuring solution quality are useful during execution of a heuristic procedure, namely as stopping rules. Numerical analysis is used to demonstrate the calculation of this metric for an observed Pareto solution set.. The results clearly show that our performance metric gives a quick and good means of assessing progress towards true Pareto optimal solution.

A description of weighted composition operators generated  by a sequence and a mapping on Lorentz-sequence spaces  is presented.