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:
- $\varnothing, X \in \mathcal{B}$.
- (Union) If $E, F \in \mathcal{B}$, then $E \cup F \in \mathcal{B}$.
- (Intersection) If $E, F \in \mathcal{B}$, then $E \cap F \in \mathcal{B}$.
- (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.
Best Answer
A Boolean algebra is a distributive lattice with negation. Dropping the negation part, you are just left with the distributive lattice part. So for your first question: a (concrete) distributive lattice would be the right name.