Why would an abstract mathematical system with only one object in its collection be not very useful?

Try to draw a table similar to the table showing Boolean negation but using only one literal value.

You will see that any table that defines an operator in such a system will have only one entry in the leftmost column showing the only literal value available. The literal value of any operator will be the literal value of the only object available in the system.

Thus everything about the abstract mathematical system with only one object is contained in one very simple table. We tend to want more richness in our mathematical systems, so we declare such a system to be not very useful.

In fact the only use I can think of for such a system is that its existence allows me to ask an interesting question like this--author.