In the usual way to construct  from  we define an equivalence relation on the set  and define the following operations on

While the addition is natural, the multiplication is not so natural.

In the present paper, we construct  from  using the natural operations on matrices. More generally, we begin to look for sufficient conditions on a pair of a ring  and a semiring S yielding that  can be constructed from S.

rings, foundation, semirings, cancellative semirings.