12.1.1 Tensor Algebraic Operation Scope

Tensor Algebraic Operation Scope defines the boundaries and applicability of tensor operations within algebraic structures, guiding their use in mathematical frameworks.

Tensor Algebraic Operation Scope is the set of prerequisite conditions shared across the purely component-level operations of tensor algebra, addition, scalar multiplication, tensor product, contraction, and index raising or lowering, distinguishing these operations, which combine tensors directly through their indices, from evaluation-style operations that instead require external vector or covector arguments.

Foundational Setting

Algebraic Operations as a Distinct Category

Tensor operations broadly split into two kinds: those that combine or transform tensors using only their own indices and components, and those that require supplying external vector or covector arguments to be evaluated against a tensor's slots. Algebraic operation scope concerns the former category exclusively, gathering the specific prerequisite conditions common to this group.

The Shared Underlying Requirement

Every operation within this category ultimately requires the tensors involved to be built over the same underlying vector space, since combining tensors from unrelated spaces would leave no shared notion of basis change against which the combined object's transformation behavior could be verified.

Scope Shared by Addition and Scalar Multiplication

Matching Type as the Binding Constraint

Addition applies only to tensors of identical variance type $(p, q)$ :

R_{j}^{i} = S_{j}^{i} + T_{j}^{i}

Scalar multiplication, involving only one tensor and a number rather than two tensors, falls within scope for any variance type without a matching requirement, since there is no second tensor type to reconcile.

Scope Shared by the Tensor Product and Contraction

The Tensor Product's Looser Requirement

The tensor product, unlike addition, does not require its two factors to share a variance type, only a shared underlying vector space, since its purpose is precisely to combine tensors of differing types:

T^{i j} = u^{i} v^{j}

Contraction's Index-Pairing Requirement

Contraction requires identifying one upper index and one lower index, either on a single tensor or across a tensor product, to be summed together:

\sum_{i} T_{i j}^{i}

Falling outside this scope, such as attempting to contract two indices of the same type, produces no well-defined result within the algebraic system.

Scope of Raising and Lowering Within This Category

An Additional Structural Requirement

Raising and lowering share the general vector-space requirement of the other algebraic operations but layer on an additional condition beyond it: the availability of an invertible metric tensor on that vector space, without which these two operations have no defined scope at all:

v^{i} = \sum_{j} g^{i j} v_{j}

Visual Overview of Shared and Distinct Requirements

Diagram of the Algebraic Operation Family

Contrast with Evaluation-Style Operations

No External Arguments Required

Every operation within algebraic operation scope acts entirely on the components and indices of the tensors supplied, requiring no separate vector or covector arguments from outside the tensor algebra itself, distinguishing this category cleanly from evaluation, which explicitly requires such external arguments matched to a tensor's slots.

Composability Within the Category

Because all operations in this category share compatible component-level definitions, they can be freely composed in sequence, forming a tensor product and then contracting, or contracting and then raising an index, provided each individual operation's own specific scope requirement is satisfied at the point it is applied.

Summary of Key Traits

Defining Characteristics

Algebraic operation scope covers addition, scalar multiplication, tensor product, contraction, and raising or lowering, all acting purely on tensor components.
Every operation in this category requires a shared underlying vector space among its inputs.
Individual operations layer additional requirements atop this shared baseline: matching type for addition, an upper-lower pair for contraction, and an invertible metric for raising and lowering.
This category is distinguished from evaluation-style operations by requiring no external vector or covector arguments, and its operations compose freely when each one's specific scope condition is met.