6.6 Tensor Arity Classification

Tensor Arity Classification is the classification of tensors by arity, borrowing the function-theoretic vocabulary of nullary, unary, binary, ternary, and n-ary directly from the general theory of functions and applying it to a tensor viewed as a multilinear map T : V₁ × ... × V_k → F, with arity equal to the number k of arguments the map accepts. It is numerically identical to order classification's slot count in every case, but frames that same number using the terminology and conceptual toolkit of function arity — currying, composition, and multicategorical structure — rather than the terminology of index counting.

Arity as Function-Theoretic Vocabulary

The Standard Arity Ladder

Arity classification borrows its names directly from the general vocabulary used for functions of any kind, not specifically tensors:

A scalar is a nullary tensor, taking no arguments; a vector or covector is unary; a matrix-like tensor is binary; and so on, matching order classification's scalar/vector/matrix hierarchy term for term but using words drawn from general function theory instead of tensor-specific naming.

Diagram Aligning Arity Terms With Order Names

Arity and Currying

Reducing Arity by Partial Application

Currying, the standard operation of fixing one argument of a multi-argument function and leaving the rest open, applies directly to a tensor viewed as a multilinear map: fixing one of its k arguments produces a new tensor of arity k - 1.

Full Currying Down to Nullary

Currying all k arguments of a tensor one at a time, supplying a specific vector at each step, eventually produces a nullary tensor — a scalar — the value of the fully evaluated multilinear map; arity classification frames this stepwise reduction explicitly as a sequence of arity-reducing operations, k → k-1 → ... → 1 → 0.

Arity Under Composition and the Tensor Product

Arity Addition Under the Tensor Product

matches order classification's addition rule exactly, since combining a k₁-ary map with a k₂-ary map into one map over the concatenated argument list produces a (k₁+k₂)-ary map.

Arity Reduction Under Contraction

Contraction, in arity language, is the operation of composing a k-ary tensor with a canonical binary pairing map applied to two of its arguments and summing, an operation that reduces arity by exactly 2, matching the order-classification rule for contraction stated in arity terms.

Arity in a Categorical and Multicategorical Setting

Tensors as Morphisms in a Multicategory

In a multicategory, morphisms are permitted to take several inputs at once rather than exactly one, and a k-ary tensor corresponds directly to a multimorphism with k inputs and one output; arity classification is, in this setting, simply the classification of multimorphisms by their number of input slots, with tensors as a specific, concrete instance of the general pattern.

Composition of Arities

Composing multimorphisms in a multicategory combines their arities according to how their inputs and outputs are connected, generalizing the tensor product's arity-addition rule and the contraction's arity-reduction rule into the single, more abstract notion of multicategorical composition.

Distinguishing Arity Classification From Order Classification

Same Number, Different Conceptual Lineage

Arity and order report the identical integer k for any given tensor; the distinction is entirely in which body of established vocabulary and technique is being invoked — arity classification draws on the general theory of functions, currying, and multicategories, while order classification draws on the tensor-specific tradition of index counting and naming (scalar, vector, matrix, higher-order).

When the Arity Framing Is Preferred

The arity framing becomes preferable whenever a discussion is explicitly about functional or categorical properties of tensors — currying, composition, or multicategorical structure — since these operations and their properties are already well developed in the general theory of arity and transfer to tensors with no modification beyond specialization to multilinear maps.

Why Arity Classification Matters

Connecting Tensor Algebra to General Function Theory

Framing a tensor's order as an arity connects tensor algebra directly to the broader mathematical and computational vocabulary already developed for functions of any kind, allowing techniques such as currying and compositional reasoning, well understood in that broader context, to be imported into tensor algebra without needing to be reinvented from scratch.

Supporting a Uniform Treatment Alongside Non-Tensor Multilinear Objects

Because arity classification is stated in terms that apply to any multilinear map, not specifically to tensors built from a fixed vector space V, it supports a uniform treatment of tensors alongside other multilinear objects — such as multilinear maps between different, unrelated vector spaces — that order classification's tensor-specific vocabulary does not equally naturally accommodate.