One theme is that equality helps to bred respect. The members of the island have worked out a system in which all individuals and all families are on equally footing. Everyone works the same amount of time and everyone makes the same amount of money. Because their is no disparity, there is no need for resentement, fear or jealousy. Therefore, all the islanders respect one another, which is demonstrated by the careful rationing of food and failure to argue and fight.
The importance of duty is another theme of the story. As shown above, all the islanders do their duty accordingly. They work their allotted time in their restaurant and serve their community members without complaint. They work together for the common good, and not their own, because they understand that it is their duty to do so under their constitution. Likewise, the professor - after returning home - resists the appeals of the mayor and the President of the United States to tell the tale of his trip. Having made an oath to the Explorer's Club, the professor knows it is his duty to tell them first, and hd eagerly abides by that duty.