6.1.4 Tensor Slot Counting Scope

Tensor Slot Counting Scope is the delimitation of the subject matter that belongs to the raw combinatorial act of counting a tensor's argument slots, viewing a tensor as a multilinear map and asking only how many arguments it takes and how that count behaves under the basic algebraic operations, prior to and independent of any naming or classification built on top of that count. This scope statement fixes slot counting as pure arithmetic on the number of arguments, and separates it from order classification, type classification, and valence classification, each of which attaches names, splits, or arrangements to a slot count that is itself computed according to the rules given here.

What Falls Inside This Scope

A Tensor as a Multilinear Map With Countable Arguments

Within scope is the view of a tensor T of the appropriate kind as a multilinear map

taking some number k of vector or covector arguments and returning a scalar in the field F; slot counting is simply the determination of this k for a given tensor, with no attention paid yet to which arguments are vectors and which are covectors.

Counting Rules for the Elementary Building Blocks

Within scope are the base cases fixing the slot count of the elementary objects a tensor algebra is built from: a scalar has 0 slots, a vector or covector has exactly 1 slot, and every compound tensor's slot count is derived from these base cases by the combination rules below rather than assigned independently.

Slot Count Arithmetic Under Tensor Product

is within scope as the addition rule governing the tensor product: forming A ⊗ B simply appends the argument list of B after that of A, so slot counts add.

Slot Count Arithmetic Under Contraction

is within scope as the subtraction rule governing contraction: pairing one argument slot against another and summing removes exactly two slots from the count, regardless of which two slots were chosen.

A Worked Counting Diagram

Slot Count as an Arity, Not a Structure

Within scope is the emphasis that a slot count is a bare integer arity with no attached information about which slots are contravariant, which are covariant, or in what order they appear; two tensors with the same slot count may differ completely once type or valence is examined, but slot counting itself does not distinguish them.

What Falls Outside This Scope

Attaching the Name "Order" to a Slot Count

Once a slot count is computed, giving it the name "order" and attaching conventional labels such as scalar, vector, or matrix belongs to tensor order classification, a naming layer built on top of, but outside, the pure counting arithmetic addressed here.

Splitting the Count Into Upper and Lower Totals

Determining how many of the counted slots are contravariant and how many are covariant belongs to tensor type classification and requires information — variance — that slot counting alone does not track.

Arranging the Count Into an Ordered Sequence

Recording the specific left-to-right arrangement of upper and lower slots belongs to tensor valence classification, which likewise requires variance information absent from a bare slot count.

Numerical Rank of a Represented Matrix

The linear-algebraic rank of a matrix obtained by representing a type (1,1) tensor in a basis is a wholly different numerical invariant from the slot count 2, and is excluded from this scope.

Boundary Cases Addressed Within This Scope

The Slot Count of a Scalar

A scalar is within scope as the base case of zero slots, the identity element for slot-count addition under the tensor product, since slots(A ⊗ scalar) = slots(A) + 0 = slots(A).

Non-Negativity and Integrality of Slot Counts

Within scope is the fact that a slot count is always a non-negative integer, since it arises only from the base cases of 0 and 1 combined by addition and by subtraction of 2 at a time under contraction, a subtraction that is only ever performed when at least 2 slots remain.

Slot Counting Across Iterated Products

is within scope as the direct extension of the addition rule to an n-fold tensor power of a single vector space, giving a slot count that grows linearly in n even though the dimension of the underlying space grows exponentially.

Purpose of Fixing This Scope

Supplying the Arithmetic Foundation for All Classification Schemes

By isolating slot counting as pure, variance-free arithmetic, this scope supplies the common numerical foundation that order, type, and valence classification each build upon, ensuring that the more elaborate classification schemes do not need to restate or redefine how counts combine under product and contraction.

Preventing Confusion Between Counting and Naming

Keeping slot counting strictly separate from the naming conventions of order classification prevents the informal habit of calling a slot count itself an "order" from leaking into contexts where the bare arithmetic, not the classification name, is what is actually being used.