I have been looking for information on (finite) set families $\mathcal F$ such that if $X,Y \in \mathcal F$ then $X \,\triangle \,Y \in \mathcal F$.
Are these kind of families (possibly with extra conditions) studied? If, so, in what areas? My interests are primarily combinatorial, but other areas are of interest as well.
I have found the binary matroid that has some connection to symmetric differences, but is not quite what I am looking for.
The following two papers might be useful, but note that their focus is not on finite sets. However, you might be able to find something more appropriate by searching for papers that cite one of these two papers.
Edward Marczewski [Szpilrajn], Concerning the symmetric difference in the theory of sets and in Boolean algebras, Colloquium Mathematicum 1 #3 (1948), 200-202.
Henry Helson, On the symmetric difference of sets as a group operation, Colloquium Mathematicae 1 #3 (1948), 203-205.