  Earlier papers \cite{VB2022,VB2023a,VB2023b} introduced the notions of a core
and an index of a relation (an index being a special case of a core). A limited
form of the axiom of choice was postulated -- specifically that all partial
equivalence relations (pers) have an index -- and the consequences of adding
the axiom to axiom systems for point-free reasoning were explored. In this
paper, we define a partial ordering on relations, which we call the
\textsf{thins} ordering. We show that our axiom of choice is equivalent to the
property that core relations are the minimal elements of the \textsf{thins}
ordering. We also characterise the relations that are maximal with respect to
the \textsf{thins} ordering. Apart from our axiom of choice, the axiom system
we employ is paired to a bare minimum and admits many models other than
concrete relations -- we do not assume, for example, the existence of
complements; in the case of concrete relations, the theorem is that the maximal
elements of the \textsf{thins} ordering are the empty relation and the
equivalence relations. This and other properties of \textsf{thins} provide
further evidence that our axiom of choice is a desirable means of strengthening
point-free reasoning on relations.
