4.12.3 Tensor Reduced Arity Result

Tensor Reduced Arity Result is the outcome object produced by a tensor multilinear partial evaluation operation, classified specifically by its arity, the number of arguments it still requires, once some arguments have already been absorbed by fixing a subset of slots. Where the tensor remaining slot map describes this outcome as a function and the tensor fixed slot evaluation describes the input data that produced it, the reduced arity result focuses on the outcome purely as an object of a definite, smaller arity, situating it within the classification of multilinear maps by how many arguments they take.

Arity and Its Reduction

Arity of the Original Tensor

A type (p, q) tensor T on a vector space V has arity p + q, meaning it requires exactly p + q arguments, p covectors and q vectors, before it produces a scalar. Arity is a purely numerical invariant of the tensor's type, counting total argument slots without regard to which are contravariant and which are covariant.

Arity After Fixing m Slots

If m of the p + q slots are fixed through partial evaluation, the reduced arity result has arity exactly p + q - m:

where |S| denotes the number of slots fixed, that is, m. This reduction is exact and depends only on how many slots were fixed, not on which specific values were used to fix them.

Classifying the Reduced Arity Result by Type

Splitting the Reduction Between Contravariant and Covariant Slots

The reduced arity result is not merely characterized by its total arity p + q - m but by its full type (p - k, q - (m - k)), where k is the number of fixed slots that were contravariant; two reduced arity results with the same total arity but different splits between remaining contravariant and covariant slots are considered different kinds of objects, even though they require the same number of arguments.

Arity Zero: The Scalar Case

When m = p + q, every slot has been fixed, and the reduced arity result has arity zero; an arity-zero result is not a function at all in any meaningful sense but simply an element of the base field, coinciding with the tensor multilinear evaluation result obtained from full evaluation.

Arity One: Linear Functionals and Vectors

When exactly one slot remains open, the reduced arity result has arity one and is a linear map: if the open slot is covariant, the result is a linear functional on V; if the open slot is contravariant, the result is an element of V itself once identified through its action on covectors, illustrating how the arity-one case recovers the ordinary notion of a linear map from a higher-arity tensor.

Arity Reduction as an Ordered Process

Monotonic Decrease Under Successive Fixing

Each additional slot fixed through partial evaluation decreases the arity of the result by exactly one, regardless of which slot is chosen or what value is used to fix it, so the arity of the reduced result decreases monotonically and predictably as more arguments are supplied, reaching zero exactly when every slot has been filled.

Arity as an Invariant Independent of Fixing Order

Because arity depends only on the count of slots fixed and not on the order in which they were fixed, any sequence of partial evaluations that fixes the same total number of slots produces a reduced arity result of the same arity, even if the intermediate reduced arity results obtained along the way differ from one sequence to another in which specific slots remain open at each step.

The Reduced Arity Result as a Tensor Product Element

Membership in a Smaller Tensor Product Space

A reduced arity result of type (p - k, q - (m - k)) corresponds to a unique element of the tensor product space built from p - k copies of V and q - (m - k) copies of V*, which is strictly smaller in dimension than the tensor product space containing the original tensor T, since it involves fewer total factors.

Dimension Count of the Result Space

If V has dimension n, the space containing the reduced arity result has dimension n^{(p+q-m)}, in contrast to the dimension n^{(p+q)} of the space containing the original tensor T, reflecting the exponential decrease in the size of the ambient tensor product space as arity is reduced.

Practical Significance of Tracking Arity

Determining What Further Arguments Are Needed

Knowing the arity of a reduced arity result immediately determines how many further arguments must be supplied before a scalar evaluation result can be obtained, which is essential bookkeeping whenever a tensor is applied incrementally across multiple stages, as happens when slots are filled one at a time rather than all at once.

Matching Reduced Results to Expected Operators

Recognizing that a reduced arity result has arity one or two allows it to be identified directly with familiar objects, a linear map or a bilinear form respectively, connecting the general framework of tensor partial evaluation to the more elementary operators typically introduced before tensors are considered in full generality.

Diagrammatic Summary

The diagram shows the arity of the reduced result stepping down by one with each additional slot fixed, from the full arity p + q of the original tensor down to arity zero, at which point a scalar remains.