6.6.5 Tensor Arity Notation Context

Tensor Arity Notation Context is the set of notational conventions used to write a tensor in functional, arity-based form — blank-slot notation, curried chain notation, and juxtaposition notation — together with the contextual information each convention must carry to remain unambiguous about which of a tensor's k slots is being referred to, filled, or left open. Where order-based notation is chiefly concerned with managing the sheer number of index symbols as order grows, arity-based notation is chiefly concerned with representing the functional, blank-and-fill structure of a multilinear map and with making clear, at every point in a calculation, which slots are currently open.

Blank-Slot Notation

Writing an Unevaluated Tensor With Explicit Blanks

A k-ary tensor with no arguments yet supplied is commonly written with an explicit placeholder, often a bullet or underscore, in each of its k positions:

with the number of blanks visually communicating the arity k at a glance, without requiring any index symbols to be introduced at all.

Mixing Filled and Open Slots in the Same Expression

shows a 3-ary tensor with the first and third slots filled by specific vectors v and w, and the second slot left open, directly encoding an evaluation context within the notation itself: the positions of the blanks tell the reader exactly which slot still awaits an argument.

Diagram of Blank-Slot Notation Encoding Context

Curried Chain Notation

Writing Sequential Application as Chained Parentheses

An alternative to filling several slots inside one set of parentheses is to curry the tensor one argument at a time, writing T(v)(w)(u) to mean "first apply T to v, producing a lower-arity tensor, then apply the result to w, and so on."

Why the Chained Form Matches the Currying Operation Exactly

Curried chain notation is notation whose structure literally mirrors the currying operation itself: each successive pair of parentheses corresponds to exactly one arity-reducing application, so the number of chained parenthesis groups in the notation equals the number of arguments supplied so far, and the arity remaining equals k minus that count.

Juxtaposition Notation

Suppressing Parentheses Entirely for Common Low-Arity Cases

For very low arity, notation sometimes suppresses parentheses and function-call syntax altogether: a bilinear form is sometimes written ⟨v, w⟩ using angle brackets rather than T(v, w), and a linear operator applied to a vector is sometimes written by simple juxtaposition, Av, rather than A(v).

The Trade-Off of Juxtaposition Notation

Juxtaposition notation is compact and familiar for arity 1 and 2, matching long-established conventions from linear algebra, but it does not generalize gracefully to higher arity, since juxtaposing three or more objects without any separating punctuation quickly becomes ambiguous about which objects are arguments and which is the tensor itself.

Context Required to Disambiguate Each Notational Style

Blank-Slot Notation Requires No Additional Context Beyond the Blanks

Because blank-slot notation places its blanks directly in position, the notation is self-contained: reading which positions are filled and which are blank requires no external context beyond the expression itself.

Curried Notation Requires Knowing the Slot-Filling Order

Curried chain notation, by contrast, requires the reader to know, or to have been told elsewhere, which slot each successive application fills; if a tensor's slots are not filled strictly left to right, the chain notation alone does not indicate which specific slot each parenthesis group is targeting, and additional context must supply that information.

Juxtaposition Notation Requires Knowing the Established Convention

Juxtaposition notation requires the reader to already know, from established convention in the surrounding context, that Av means "the operator A applied to the vector v" rather than some other operation entirely, since the notation itself carries no explicit marker of function application at all.

Why These Notational Contexts Matter

Choosing Notation to Match the Evaluation Context Being Described

Blank-slot notation is well suited to describing a tensor with a specific, fixed pattern of filled and open slots at a single point in a calculation, while curried notation is well suited to describing a sequence of successive evaluations performed one after another; selecting the notation that matches the actual evaluation context being described keeps a calculation both accurate and easy to follow.

Preventing Ambiguity When Combining Notational Styles

Because each notational style carries a different implicit contract about what context is required to interpret it, mixing styles within a single expression without care — for example, combining a curried chain with juxtaposition — risks introducing genuine ambiguity about which slots have been filled, making consistent use of a single notational style within one continuous calculation the safer practice.