Whenever a public employer or the employing head of a board, commission, or department in the state service or in the services of any subordinate jurisdiction or whenever any union, association, or group of public employees of a public unit of government decides by vote of such group that a grievance or matter in dispute which has been considered between the employer and the employee cannot be settled amicably and without disruption of the public service, it is the duty of the employer or the employees or both to request in writing of the chief executive officer of the unit of government, namely the governor in the state service, the chairman of the county board or commission in county services, and the mayor or city manager of the city in city services, that it is the wish of one or both parties concerned in such dispute or grievance that the provisions of this chapter be made use of by the designation by such officer of two members of a mediation board, one to be recognized as a representative of the employer in the public service and one to be a bona fide employee who is a member of the union, association, or group of employees and who is interested in the point of view of such group of employees. The two persons so designated shall upon their appointment forthwith meet and agree upon a third person who must be chairman of the mediation board.