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Published online by Cambridge University Press: 24 October 2008
The following considerations form the basis of the work on generalised integrals with which I have been engaged for some years. Their intimate connection with Alexander's new notation for combinatory topology encourages me to publish them separately.
* Alexander, , Annals of Mathematics, vol. 31 (1930).CrossRefGoogle Scholar
† Alexandroff, , “Simpliziale Approximationen in der allgemeinen Topologie,” Math. Ann. vol. 96 (1925), pp. 488–511.Google Scholar
* Lefschetz, , Topology (1930), p. 15.Google Scholar
† A simplex is thus defined in accordance with the definition first suggested by Newman as the (ordered) set of its vertices. Cf. Newman, , Amsterdam Proc. vol. 29 (1926), pp. 611–626.Google Scholar
* We may summarise these differentiation rules by saying that the effect p x on any complex is equivalent to that of first rearranging all the products so that x occurs first in all the products in which it is present, and then of differentiating in the ordinary way.
† I.e. lowers by 1 the degree of a polynomial in the x's.
‡ We have here a different meaning of the word “component.” It is hoped that this will not cause any confusion in the present note.
* This identity expresses the fact that σ is a linear homogeneous form in the Δ, since p Δ is a differentiation operator.
We have made use of the term “vector” for a function of x having n + 1 components g 0 (x), g 1 (x), …, g n(x); it would have been more correct to use the term “simplex,” for an odd permutation of these components changes all our signs, and nothing more. The analogy with our differentiation operators then becomes more close. A function-simplex is the formal product of its components, each of these being counted of degree – 1; thus the product of a function-simplex by a complex of same dimension is a pure number; a linear combination of function-simplexes is a function-complex. We should then be able to say that every f (σ) is a product of a function-complex by an ordinary complex.
This might, however, cause a confusion of notation, for in our function-complexes the coefficients would not be integers.
† In its n-dimensional form.