For [tau] to be a

congruence relation on X(I), we must show that for all (x,yI) /= (0, 0) in X(I), (a,bI)[tau] (c, dI) implies that (a,bI) * (x,yI) [tau] (c,dI) * [x,yl) and [x,yI) * (a, bI) [tau] (x,yI) * (c, dI).

Then [[equivalent].sub.I] is a

congruence relation on X.

We also briefly touch on zero-divisor free, zero-sum free, additively cancellative, and multiplicatively cancellative (m, n) semirings and the

congruence relation on (m, n)-semirings.

"X [equivalent to] Y([[theta].sub.D]) iff (X [disjunction] Y) [conjunction] [D.sub.1] = X [conjunction] Y, for some [D.sub.1] [less than or equal to] D" is

congruence relation.

(2) follows from the following

congruence relation of polynomials modulo the ideal ([x.sub.s] + [epsilon][x.sub.t] - z):

Let X be a SU-algebra, I be an ideal of X and ~ be a

congruence relation on X.

It follows by transitivity of the

congruence relation.

A

congruence relation R on G is a collection of equivalence relations [R.sub.a,b] on hom(a, b), a, b [member of] ob(G),chosen so that if (s, s') [member of] [R.sub.a,b] and (t, t') [member of] [R.sub.b,c], then (ts, t's') [member of] [R.sub.a,c] for all a, b, c [member of] ob(G).

The induction steps arise from the definition of structural congruence as a

congruence relation; they include rules allowing structural congruence to be applied to each subterm of each process constructor, plus symmetry and transitivity.

Let X be a SU-algebra, I be an ideal of X and = be a

congruence relation on X.

Then [equivalent to][sub.F] is a

congruence relation on A.

Then relation [~.sub.K] is an

congruence relation on A.