You seem to believe that the right to property as you envision it is inherent in the matter of the world and has nothing to do with the State. In order for us to be able to choose to hold property laws in the way we do - a social choice that could take many different forms, and in fact does across countries and cultures and time periods within the U.S. (can I assume you're from the U.S.?) - we have a State to provide enforcement of the "right" to property (and money as a medium of exchange for property, etc.). Sure, you are welcome in a (purely imaginary, of course) state of anarchy to try to beat up anyone who tries to take what you declare to be your property, or to find someone who holds the same beliefs as you do about the meaning of contracts and then to cooperate via a contract to achieve whatever ends you want to agree on. In other words, there are social mechanisms for enforcing a group's view of property outside of the bounds of a modern State. But if you apply your algebra-as-social theory model to the modern State in which you presumably live, you have, congratulations, enslaved everyone who helps to defend your property for you.
But wait! We pay police officers, the National Guard, or maybe your private security team, or whomever to take care of our property for us. Some people are willing to provide those services, and expressing that there is a "right" to receive those services does not imply that anyone will be forced to provide them. Just so, a "right to healthcare" or a "right to Internet access" is a formulation not necessarily premised on the use of conscription. In fact, I suspect the vast majority of people(s) who talk about these kinds of rights do not intend for them to be meant in that way - and it is not an algebraic fact that rights (with the magical exception of certain kinds of property, of course) entail slavery.
Of course, I have expressed an opinion, based though it is on logic and facts, with which it is possible to disagree and still use logic. Your algebra demonstration was perfect logic. Your flaw was to assume that human interactions are premised on the exact definitions and perfect logical properties of abstract mathematics such that a demonstration of logic applied to an ethical/social/political question is unproblematically appropriate.