6.6.1 Tensor Arity Input Slot Count

Tensor Arity Input Slot Count is the count k of input positions in a tensor's domain V₁ × ... × V_k, singled out explicitly as counting only the arguments the multilinear map receives and never the value it produces, since a tensor as ordinarily defined, T : V₁ × ... × V_k → F, always has exactly one output, a scalar in F, regardless of how large k becomes. This input/output distinction, taken for granted in most discussions of tensor arity, becomes worth stating explicitly once tensors are compared to more general multi-input, multi-output functions for which the two counts can differ independently.

Separating Input Count From Output Count

A Tensor Always Has Exactly One Output

By definition, a tensor of the ordinary kind treated throughout tensor algebra is a map into the field F, a single scalar output; no matter how many input arguments k a tensor accepts, its output count is fixed at 1, never 0 and never more than 1.

Why This Distinction Is Usually Silent

Because the output count is invariably 1 for a tensor, ordinary discussions of tensor arity conflate "arity" with "input slot count" without any loss, since there is no second number — output count — to distinguish it from; the distinction only becomes visible once tensors are compared to more general functions where output count is not fixed at 1.

Diagram Contrasting Input-Only Counting With General Functions

Counting Input Slots Precisely

Input Slot Count as the Number of Domain Factors

The input slot count is exactly the number of factors in the domain product V₁ × ... × V_k; each factor, whether it is V itself (a covariant argument slot) or the dual V* (a contravariant argument slot), contributes one to the input slot count regardless of which of the two it is, since the input/output distinction cuts across, and is independent of, the covariant/contravariant distinction studied under type classification.

Every Slot, Regardless of Variance, Is an Input

Both upper and lower index positions are input slots in this counting; a type (p, q) tensor has p + q input slots total, distributed between contravariant and covariant roles, but every one of them is, without exception, a position the map receives a value into, not a position from which a value is produced.

Input Slot Count in Composite Constructions

Input Slot Count Under the Tensor Product

follows the same addition rule as arity and order, since forming the tensor product concatenates the two input lists while the output remains a single scalar throughout.

Input Slot Count After Contraction

Contraction removes exactly two input slots (one upper, one lower) while continuing to produce a single scalar output, so the output count of 1 is preserved by contraction even as the input slot count decreases by 2; contraction never touches the output side of the map because there is no output side with more than one slot to touch.

Contrast With General Multi-Output Maps

Where the Distinction Becomes Necessary

Multilinear maps with more than one output, mapping into a vector space of dimension greater than 1 rather than into the scalar field F directly, do exist in broader multilinear algebra and are naturally described by an input slot count and a separate output slot count; a tensor of the kind treated here is the special case of such a map with output slot count fixed at 1.

Recovering a Tensor as a Special Case

Any multilinear map with input count k and output dimension m > 1 can be viewed, after choosing a basis for the output space, as m separate scalar-valued multilinear maps, each an ordinary tensor with input slot count k and output count 1; the input slot count concept generalizes cleanly to this setting, while remaining exactly what "arity" already meant for the ordinary, single-output tensors this concept is built to describe.

Why the Input Slot Count Matters

Precision When Comparing to General Multilinear Maps

Stating explicitly that tensor arity refers only to the input side becomes essential precisely when tensors are discussed alongside more general multi-output multilinear maps, preventing an unstated assumption about output count from causing confusion when the two kinds of object are compared side by side.

A Foundation for Consistent Arity Arithmetic

Confirming that the output count is always 1 and never participates in the arithmetic of tensor combination clarifies why every arity-based formula encountered elsewhere — addition under the tensor product, reduction under contraction — is entirely a statement about input slots, with the constant output count of 1 correctly omitted from the arithmetic as a fixed, non-varying quantity.