This paper studies the ``usefulness'' of initial conditions for
distributed algorithms on anonymous networks.
In the literature, several initial
conditions such as making one vertex a leader, giving the number of
vertex to each vertices, and so on, have been considered.
In this paper, we study a relation between the initial condition by
considering transformation algorithm from one initial
condition to another.
For such transformation algorithms, we consider in this paper, both
deterministic and randomized distributed algorithms.
For each deterministic and randomized transformation type, we show
that the relation induces an infinite lattice structure among
equivalence classes of initial conditions.