  The first steps towards linearisation of partial orders and equivalence
relations are described. The definitions of partial orders and equivalence
relations (on sets) are formulated in a way that is standard in category theory
and that makes the linearisation (almost) automatic. The linearisation is then
achieved by replacing sets by coalgebras and the Cartesian product by the
tensor product of vector spaces. As a result, definitions of orders and
equivalence relations on coalgebras are proposed. These are illustrated by
explicit examples that include relations on colagebras spanned by grouplike
elements (or linearised sets), the diagonal relation, and an order on a
three-dimensional non-cocommutative coalgebra. Although relations on coalgebras
are defined for vector spaces, all the definitions are formulated in a way that
is immediately applicable to other braided monoidal categories.
