1
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ The phrase "lattice on a set" seems to be quite rare. Would "distributive lattice on a set X" naturally be read as implying that it must contain X and the empty set? (Of course one could specify that it does to be sure.) $\endgroup$ – David M Sep 6 '19 at 8:03
  • 1
    $\begingroup$ @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. $\endgroup$ – Mark Kamsma Sep 6 '19 at 8:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.