The board of trustees of the Central state university shall elect, fix the compensation of, and have the right to remove the president, who shall be the chief executive officer of the university, and elect, fix the compensation of, and remove such number of professors, teachers, and other employees as is necessary. No trustee, or his relative by blood or marriage, is eligible to a professorship or position in the university, the compensation for which is payable out of the state treasury or a university fund. No officer, teacher, or other employee of the university is eligible to membership on the board of trustees of such university while occupying any of said positions.