6.22.1 Tensor Order from Type Pair

Tensor Order from Type Pair is the deterministic function that takes a tensor's type — the ordered pair (p, q) recording its contravariant and covariant slot counts — and returns its order, always by the single rule n = p + q, with the emphasis here on this being a genuine function: every type pair maps to exactly one order, though many different type pairs can map to the same order. This viewpoint treats "order" as derived data computed from the more fundamental type, rather than as an independently specified attribute of a tensor.

The Mapping from Type to Order

A Function on Pairs of Non-Negative Integers

Formally, the assignment can be written as a function f on pairs of non-negative integers,

taking the type (p, q) ∈ ℕ × ℕ as input and returning the order n ∈ ℕ as output. Because addition of non-negative integers always produces a unique non-negative integer, f is total and single-valued: every type pair has exactly one associated order, and this order can be computed immediately without reference to any specific tensor or vector space.

Worked Examples of the Mapping

The type (2, 1) maps to order 3; the type (0, 4) maps to order 4; the type (1, 1) maps to order 2; the type (0, 0) maps to order 0. Each of these computations is nothing more than addition, which is the entire content of computing order from a type pair — no additional structural information about the tensor is required beyond the two counts themselves.

The Mapping Is Not Injective

Multiple Types Sharing One Order

The function from type to order is not one-to-one: the type pairs (2, 1) and (1, 2), for instance, both map to order 3, yet they denote tensors in different spaces — V ⊗ V ⊗ V* versus V ⊗ V* ⊗ V* — with different transformation behavior. Knowing that the order-from-type-pair mapping sends both to 3 does not identify which space is meant; recovering the type from the order alone is impossible in general.

Counting the Preimage

For a fixed order n, the preimage of n under this mapping — the set of type pairs (p, q) with p + q = n — has exactly n + 1 elements, namely (0, n), (1, n−1), ..., (n, 0). This count is a direct combinatorial consequence of the mapping being simple addition over non-negative integers with a fixed sum.

Diagram of the Mapping

Why the Mapping Direction Matters

Order from Type Is Immediate; Type from Order Is Underdetermined

Because the mapping (p, q) ↦ p + q collapses distinct type pairs onto the same order value, order-from-type is a well-posed computation while type-from-order is not; the latter requires additional information beyond the order itself, namely the actual split between contravariant and covariant slots. This asymmetry is the reason type is treated as the primary classification of a tensor and order as a secondary quantity computed from it.

Role in Systematizing Tensor Terminology

The order-from-type mapping is what justifies referring to a tensor unambiguously as "type (p, q)" while treating "order n" as a convenient but coarser descriptor: any statement made using only the order, such as "this is an order-two tensor," is implicitly a statement about a whole family of n + 1 possible types, and precision requires stating the type pair whenever the contravariant/covariant distinction is relevant to the discussion at hand.