The general procedure they present associates to every strongly local vertex operator algebra V a conformal net AV acting on the Hilbert space completion of V, and prove that the

isomorphism class of AV does not depend on the choice of the scalar product on V.

We shall always identify a tree shape, an ordered tree shape, a cladogram, or an ordered cladogram, with its

isomorphism class, and in particular we shall make henceforth the abuse of language of saying that two of these objects, T, T', are the same, in symbols T = T', when they are (only) isomorphic.

where Irrep(G) denotes the

isomorphism classes of irreducible representations of G, V is a representative of its

isomorphism class of irreducible representation and [H.sup.V.sub.[alpha]] is the isotypical subspace associated to V.

As E [??] E ', ([GAMMA], x, E) and ([GAMMA], x, E') cannot be elements of the same

isomorphism class in S.

As E [??] E', ([GAMMA], x, E) and ([GAMMA], x, E') cannot be elements of the same

isomorphism class in S.

* the identity morphism of a pomonoid S is the

isomorphism class [S] of the Pos-prodense biposet [sub.S][S.sub.S].

Let T!prime^ be a theory obtained from T by removing all but a single model from each

isomorphism class of the models of T.

Suppose A is an artinian ring, with indecomposable A-modules {[P.sub.[alpha]] |[alpha] [member of] I} (representatives from each

isomorphism class for some index set I).

In Theorem 5.10 we show that for each

isomorphism class D of skew diagrams, the number [r.sub.[lambda]] (D), of [lambda]-removable diagrams in D, can be expressed as a polynomial with rational coefficients in variables [r.sub.[lambda]] (C), where C runs over the set of all

isomorphism classes of connected skew diagrams of size [absolute value of C] [less than or equal to] [absolute value of D].

Lemma 3.2 If exactly one representative of every

isomorphism class of cubic connected graphs up to n - 2 vertices is given, then applying bundled triangle insertion to one member of each equivalence class of extensible sets that leads to a cubic connected graph on n vertices generates exactly one representative for every

isomorphism class of cubic connected graphs on n vertices that contain reducible triangles.

By a classification we mean a cross-section for the

isomorphism classes, i.e., a set of objects in which precisely one representative for each

isomorphism class occurs.

But, in general, the Hopf algebra R(G) fails to determine the

isomorphism class of G [11, p.