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
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
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
Then [equivalent to][sub.F] is a congruence relation
Then relation [~.sub.K] is an congruence relation