6.6.3 Tensor Arity Tensor Order Relation

Tensor Arity Tensor Order Relation is the statement that arity classification and order classification report the exact same integer k for every individual tensor, without exception and without any restriction analogous to the homogeneity condition required to relate degree and order; the two schemes are total, unconditional relabelings of one another, differing only in which body of vocabulary — function theory or index-counting tradition — is used to name the shared number k. This unconditional agreement distinguishes the arity–order relation sharply from the degree–order relation, which holds only on homogeneous elements of the graded tensor algebra.

Why the Relation Holds Without Exception

Both Schemes Are Defined on the Same Object

Order classification and arity classification are both defined directly on an individual tensor T, considered as a multilinear map T : V₁ × ... × V_k → F; order counts the index positions in its component expression, and arity counts the argument positions in its functional definition, but both counts are performed on the very same object T, with no intermediate structure (such as an ambient graded algebra) that could introduce a case where the two counts diverge.

Contrast With the Degree–Order Relation's Homogeneity Requirement

The degree–order relation only holds for elements of T(V) that are homogeneous, because a general, inhomogeneous element of T(V) has no single order at all, only several order-classified parts; no analogous complication arises for arity, since every individual tensor, by the very fact of being an individual tensor rather than a general sum living in T(V), already has a single, fixed number of arguments to count.

Diagram Contrasting an Unconditional Relation With a Conditional One

The Nature of the Difference Between the Two Schemes

A Difference of Vocabulary and Emphasis, Not of Value

Because the two schemes always agree numerically, the only real content in distinguishing them is which conceptual toolkit is being invoked: order classification draws on the tensor-specific naming tradition (scalar, vector, matrix-like, higher-order) built directly around index notation, while arity classification draws on the general theory of functions (nullary, unary, binary, k-ary) built around currying, composition, and multicategorical structure.

Interchangeability in Ordinary Use

Given the unconditional agreement, any statement made using order-classification vocabulary about a specific tensor can be restated using arity-classification vocabulary with no loss or gain of content, and vice versa; the choice between the two is a matter of which surrounding discussion — index manipulation or functional/categorical reasoning — the statement is embedded in.

Where a Careless Analogy With Degree Could Mislead

The Temptation to Impose a Homogeneity-Style Caveat

Because the degree–order relation, discussed elsewhere, requires a homogeneity condition, there is a temptation to assume some similar caveat must apply to the arity–order relation as well; this temptation should be resisted, since arity, unlike degree, is not a property relative to an ambient graded algebra and has no analogous notion of "inhomogeneous arity" that could break the correspondence.

Confirming There Is No Hidden Restriction

Checking the definitions directly confirms the absence of any hidden restriction: arity is defined purely from the multilinear-map structure of a single tensor, exactly as order is defined purely from its index structure, so there is no step in either definition that could fail or become ambiguous for some tensors and not others.

Consequences of the Unconditional Relation

Arity-Based Arithmetic Automatically Matches Order-Based Arithmetic

Since the two numbers always coincide, every arithmetic rule stated for order — addition under the tensor product, reduction by 2 under contraction — automatically holds in identical form for arity, with no need for a separate proof or a separate check of edge cases; the rules are the same rules, restated in different words.

A Reliable Bridge for Translating Between Literatures

Sources that describe tensors using the language of multilinear-map arity and sources that describe tensors using the language of index-counted order can be compared directly and without qualification, term for term, using this relation, since no case exists in which the two literatures' numbering would disagree for the same tensor.

Why This Relation Matters

Establishing a Fully Reliable Correspondence

Confirming that the arity–order relation is total and unconditional, in contrast to the necessarily conditional degree–order relation, gives a fully reliable basis for treating "order" and "arity" as synonyms whenever the context is an individual tensor, with no residual doubt about edge cases that a more careful analogy with degree classification might otherwise raise.

Clarifying the Genuine Source of the Distinction Between Schemes

Locating the difference between order and arity classification entirely in vocabulary and conceptual emphasis, rather than in any numerical disagreement, clarifies that choosing one scheme over the other is a stylistic and contextual decision, never a decision with mathematical consequences for the tensor being described.