# Boolean algebras and measures without complements

A (concrete) Boolean algebra $$\mathcal{B}$$ on a set $$X$$ is any nonempty set of subsets of $$X$$ that is closed under union, intersection, and complements relative to $$X$$. With some redundancy, this implies that it is characterized by the following axioms:

1. $$\varnothing, X \in \mathcal{B}$$.
2. (Union) If $$E, F \in \mathcal{B}$$, then $$E \cup F \in \mathcal{B}$$.
3. (Intersection) If $$E, F \in \mathcal{B}$$, then $$E \cap F \in \mathcal{B}$$.
4. (Complement) If $$E \in \mathcal{B}$$, then $$X \setminus E \in \mathcal{B}$$.

Is there a standard name for the more general structure in which axiom 4 is dropped? For example, let $$X = \mathbb{N}$$, and let $$\mathcal{B}$$ contain precisely $$X$$ and its finite subsets. This satisfies 1-3 but not 4.

In addition, a vector measure $$\mu$$ on a Boolean algebra $$\mathcal{B}$$ is a map $$\mu \colon \mathcal{B} \to V$$, where $$V$$ is a vector space, such (i) $$\mu(\varnothing)=0$$; and (ii) whenever $$E$$ and $$F$$ are disjoint, $$\mu(E \cup F) = \mu(E)+\mu(F)$$.

This definition makes sense if $$\mathcal{B}$$ is only required to satisfy axioms 1-3, but it seems to be misleading to call such a mapping on the more general structure a `measure'. So is there a standard name for "measures" that are not required to be defined on Boolean algebras?

If there aren't standard names, are there any names that would sound natural and reasonable to mathematicians?

EDIT: The first question has been answered, but the second (about measures) remains open.

• @David Alternative phrasing could be "a distributive lattice in $\mathcal{P}(X)$". For me lattices are bounded, so yes it should contain $X$ and $\emptyset$. But as you said, it may be good to just define this. I was just pointing out that the structure mentioned in the question does have a standard name. – Mark Kamsma Sep 6 '19 at 8:10