Clearly, transposition is an

equivalence relation in [\[Z.sub.n]\.sup.m].

The right adjoint G is left exact and consequently preserves the monomorphims and the

equivalence relations. Now let m : X' [right arrow] X be a monomorphism in C such that the monomorphism U(m) is normal to the

equivalence relation R in D, namely such that we have a discrete fibration in D:

An

equivalence relation [equivalent to] defined on [A.sup.*] is compatible with the Schutzenberger involution if for all u, v [member of] [A.sup.*], u [equivalent to] v implies [u.sup.#] = [v.sup.#].

The results of within-class relational preference tests documented the degree to which participants selected the comparison from the 1-node

equivalence relation relative to comparisons from the 1- to 5-node transitive relations.

Definition 3.7 : Let U be an universe, R be an

equivalence relation on U and F be a fuzzy set in U and if the collection [[tau].sub.F](F) = {[0.sub.N], [1.sub.N], [F.bar](F), [bar.F](F), [B.sub.F](F)} forms a topology then it is said to be a fuzzy nano topology.

Intuitively, when S is an

equivalence relation, this fundamental theorem gives a way to compute [R.sub.s], by computing [mathematical expression not reproducible].

If (u, v) [member of] R, it indicates that the objects u and v belong to the same equivalence class with the

equivalence relation R; they are indiscernible.

The

equivalence relation IND(B) constitutes the partition of U, denoted by U/IND(B) and often abbreviated to U/B.

The addition operation defined by Definition 10 is a group operation over the set of fuzzy number equivalence classes F/[phi] up to the

equivalence relation in Definition 7.

One of the main results of [6] gives a sufficient condition: the Mal'tsev property, that is, 2-permutability RS = SR of internal

equivalence relations R, S on the same object.

Let U be a finite domain, let Z be an interval set on U, and let R be an

equivalence relation on U.