1.2.45 Tensor Arity Definition

Tensor arity defines the number of indices required to index a tensor, fundamental in understanding its structure and operations in algebra.

Tensor Arity Definition is the characterization of a tensor by the number of arguments it accepts when regarded as a multilinear function, using the term arity in the same sense it carries in logic and computer science, where the arity of a function or relation denotes the number of inputs it takes. For a tensor viewed as a multilinear map from a product of copies of a vector space and its dual to the base field, the arity is precisely the total number of vector and covector arguments the map requires before it returns a scalar.

Formal Definition

Let $V$ be a vector space over a field $F$ , with dual $V^{*}$ . A tensor of type $(r, s)$ , regarded as a multilinear map

T : \underset{r}{⏨} V^{*} \times \dots \times V^{*} \times \underset{s}{⏨} V \times \dots \times V \to F

has arity

arity (T) = r + s

equal to the total number of arguments the map consumes before producing its scalar output. Under this reading, a tensor is treated first and foremost as a function, and its arity is the same kind of invariant used to classify ordinary functions of several variables: a function of no arguments is a constant, a function of one argument is unary, a function of two arguments is binary, and so on, with the pattern continuing to tensors of arbitrary arity.

Arity in Familiar Cases

Arity Zero

A tensor of arity zero corresponds to the case $r = s = 0$ , a map that takes no arguments at all and simply returns a fixed scalar — that is, a scalar itself, the nullary case of the general pattern.

Arity One

A tensor of arity one takes a single argument, either a covector, if $r = 1$ and $s = 0$ , corresponding to a vector acting on covectors through the natural pairing, or a vector, if $r = 0$ and $s = 1$ , corresponding to a linear functional. Both are unary maps in the arity sense, despite differing in the kind of argument they accept.

Arity Two and Beyond

A tensor of arity two is a bilinear map, taking two arguments jointly, and includes bilinear forms, linear operators presented in mixed valence, and inner products among its instances. Tensors of arity three or higher are genuinely multilinear maps of three or more arguments, such as the multiplication tensor encoding an algebra's product as a bilinear-to-linear map, or curvature-type tensors requiring several vector inputs simultaneously.

Arity Does Not Track Argument Kind

A key feature of arity, as distinct from type or valence, is that it counts arguments without regard to whether each is a vector or a covector. A tensor of type $(2, 0)$ and a tensor of type $(1, 1)$ both have arity two, since both accept exactly two arguments in total, even though the arguments differ in kind between the two cases and the tensors are not interchangeable without additional structure such as a metric. Arity is therefore the coarsest of the standard tensor invariants: it discards more information than order or type, both of which coincide numerically with arity in total count but, in the case of type, retain the split between contravariant and covariant slots that arity ignores entirely.

Arity and Currying

Because a multilinear map of arity $n$ is linear in each of its arguments separately, fixing all but one argument of a tensor produces a linear map of arity one in the remaining slot. This is the tensor-algebra analogue of currying a function of several variables into a chain of single-argument functions, and it underlies the identification of a tensor of type $(r, s)$ with a linear map from one tensor product space into another obtained by "peeling off" one argument at a time, reducing the arity of the remaining map by one at each step.

Role Within Tensor Algebra

Arity provides the functional, argument-counting perspective on a tensor's complexity, complementing the algebraic perspective given by order, type, and degree. While type and valence retain the distinction between contravariant and covariant slots, and order and degree track only the total factor count within a graded tensor algebra, arity is the term most directly borrowed from the general theory of functions and relations, useful whenever a tensor is treated primarily as a multilinear map to be evaluated on a specified number of arguments rather than as an element of an abstractly constructed tensor product space.