  It is well known that "bad" quotient spaces (typically: non-Hausdorff) can be
studied by associating to them the groupoid C*-algebra of an equivalence
relation, that in the "nice" cases is Morita equivalent to the C*-algebra of
continuous functions vanishing at infinity on the quotient space. It was
recently proposed by A. Connes and W.D. van Suijlekom that a similar procedure
for relations that are reflexive and symmetric but fail to be transitive (i.e.
tolerance relations) leads to an operator system. In this paper we observe that
such an operator system carries a natural product that, although in general
non-associative, arises in a number of relevant examples. We relate this
product to truncations of (C*-algebras of) topological spaces, discuss some
geometric aspects and a connection with positive operator valued measures.
