What kind of a structure do the utilities that egoists (on an individual level) and utilitarians (on a wider scale) want to maximize have? A standard approximation is that utilities are like real numbers. They have an order structure, so that we can compare utilities, an additive structure, so we can add utilities, and a multiplicative structure, so we can rescale them with probabilities. But that is insufficiently general. We want to allow for cases such as that any amount of value V2 swamps any amount of value V1. Thus, Socrates thought that any amount of virtue is better to have than any amount of pleasure. The structure of the real numbers won't allow that to happen.
A natural generalization is to note that the multiplicative structure of the space of utilities was overkill. We don't need to be able to multiply utilities by utilities. That operation need not make sense. We simply need to be able to multiply utilities by probabilities. Since probabilities are real numbers, a structure that will allow us to do that is that of a partially ordered vector space. However, we should not impose more structure on the utilities than there really is. It makes sense to multiply a utility by a probability in order to represent the value of such-and-such a chance at the utility. And since we have an additive structure on the utilities, we can make sense of multiplying a utility by a number greater than 1. E.g., 2.5U=U+U+(0.5)U. But it is not clear that it always makes conceptual sense to negate utilities. While it makes sense to think of a certain degree of pain as the negative of a certain degree of pleasure, it is not clear that such a negation operation is available in general.
Getting rid of the spurious structure of multiplying utilities by a negative number, and removing the unnecessary multiplication by numbers greater than 1, we get naturally get a structure as follows. Utilities are a partially ordered set with an operation + on them and there is an action of the commutative multiplicative monoid [0,1] on the utilities, with the order, addition and action all compatible.
A further generalization is that [0,1] may not be the best way to represent probabilities in general. So generalize that to a commutative monoid (with multiplicative notation). We now have this. A utility space is a pair (P,U) where P is a commutative monoid with multiplicatively written operation and an action on U, U is a commutative semigroup with an additively written operation + and a partial order ≤, where the operations, action and orders satisfy:
- (xy)a=x(ya) for x,y∈P and a∈U
- x(a+b)=xa+xb for x∈P and a,b∈U
- If a≤b, then xa≤xb for a,b∈U and x∈P
- If a≤b and c∈U, then a+c≤b+c.
I keep on going back and forth on whether U really should have an addition operation, though. I do not know if utilities can be sensibly added.