Suppose that there exists an extremal ray R' [subset] [bar.NE](X) such that [K.sub.X] R' < 0 and the associated contraction

morphism [phi] := [Cont.sub.R']: X [right arrow] Y is a fibration to a lower dimensional variety Y.

214) that the coproduct of the objects A and B of a category b is a triple (A [??] B, [alpha], [beta]), where A [??] B is an object in b and [alpha] : A [right arrow] A [??] B, [beta] :B [beta] A [??] B are

morphisms of the category b such that for every object X in b and every pair of

morphisms f : A [right arrow] X, g : B [right arrow] X of b there exists a unique

morphism [theta] : A [??] B [right arrow] X of b such that [theta] [theta] [alpha] = f and [theta] [??] [beta] = g.

(The d-stands for "distributive".) The self-dual conspansion

morphism corresponds to the conspansive semimodel of M, which models derivation and is associated with the nonterminal semilanguage Ls; it evolves discretely and in just the way that reality appears to evolve in the quantum-theoretic picture.

Let (A, *) be a Maltsev algebra and [alpha] : A [right arrow] A an algebra

morphism. Then [A.sub.[alpha]] := (A, [[,,].sub.[alpha]], [alpha]) is a multiplicative Hom-Lts, where [[x, y, z].sub.[alpha]] = [alpha](2xy * z-yz * x-zx * y), for all x, y, and z in A.

A left (A, [alpha])-Hom-module consists of (M, [mu]) in [??]([M.sub.k]) together with a

morphism [psi] : A [cross product] M [right arrow] M, [psi](a [cross product] m) = a x m such that

We begin by introducing an informal category of fractions in which the objects are compact ENRs and a

morphism a/p from X to Y is a pair (a, p) consisting of a fibrewise manifold p : [~.X] [right arrow] X, with fibres closed manifolds of some dimension, m say, and a map a : [~.X] [right arrow] Y.

A bijective function h : [V.sub.1] [right arrow] [V.sub.2] is called an m-polar

morphism or m-polar h-morphism if there exists two numbers [l.sub.1] > 0 and [l.sub.2] > 0 such that [p.sub.i] [omicron] [W.sub.2](h(u)) = [l.sub.1][p.sub.i] [omicron] [W.sub.1](u), [for all]u [member of] [V.sub.1], [p.sub.i] [omicron] [F.sub.2](h(u)h(v)) = [l.sub.2][p.sub.i] [omicron] [F.sub.1](uv), [for all]uv [member of] [E.sub.1], i = 1, 2, ..., m.

A

morphism [phi] : ([X.sub.1], [f.sub.1]) [right arrow] ([X.sub.2], [f.sub.2]) is a map [phi] : [X.sub.1] [right arrow] [X.sub.2] such that [f.sub.2][phi] = [phi][f.sub.1].

A family of A-modules [{[P.sub.i]}.sub.i[member of]I], indexed by a directed set I, is called the direct system if, for any pair i < j, there is a

morphism [r.sup.i.sub.j]: [P.sub.i] [right arrow] [P.sub.j] such that (i) [r.sup.i.sub.i] = Id [P.sub.i]; (ii) [r.sup.i.sub.j] [omicron] [r.sup.j.sub.k] = [r.sup.i.sub.k], i < j < k.

(ii) each

morphism is a mapping f: (X, [A.sub.X]) [right arrow] (Y, [A.sub.Y]) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A mapping [mu] : A* [right arrow] B* is a

morphism if [mu](wv) = [mu](w)[mu](v) for all w,v [member of] A*.

Define a linear functional [mathematical expression not reproducible] by a linear

morphism,