  We introduce the notion of an orbit series in a quandle. Using this notion we
define four families of quandles based on finiteness conditions on their orbit
series. Intuitively, the classes tOS and tOSn correspond to finitary
compositions of trivial quandles while the classes OS and OSn correspond to
finitary compositions of connected quandles. We study properties of these four
families of quandles and explore their relationships with several previously
studied families of quandles: reductive, n-reductive, locally reductive,
n-locally reductive, and solvable quandles.
