4.13.4 Tensor Additivity with Fixed Slots

Tensor Additivity with Fixed Slots is the extension of the tensor multilinear additivity property to the situation in which some slots of a tensor have already been fixed through partial evaluation, examining how additivity continues to hold in whichever slots remain open once the fixed arguments have been supplied. It clarifies that additivity is not disrupted or altered by the presence of already-fixed slots, and it also examines the complementary question of how the result behaves additively with respect to the fixed arguments themselves.

Additivity in the Open Slots After Fixing

The Remaining Slot Map Inherits Additivity

For a type (p, q) tensor T on a vector space V, suppose a subset S of slots has been fixed by a tensor fixed slot evaluation, producing the tensor remaining slot map T_S on the open slots S̄. For any open slot k ∈ S̄, and any two admissible arguments u and w for that slot,

with every slot in S, already fixed to its assigned value, and every other slot in S̄, held fixed on both sides. This identity follows directly from the general slot additivity of T itself, restricted to slot k, since T_S is defined by substituting the fixed values for the slots in S and leaving the rest of T's behavior untouched.

Fixed Slots Do Not Interfere with Open Slot Additivity

The additivity of T in an open slot is a property of T in that slot alone, independent of what values are supplied to other slots; consequently, fixing some slots to particular values before examining additivity in a remaining slot changes nothing about whether that remaining slot behaves additively, since additivity in slot k was never a joint condition involving the values placed elsewhere.

Additivity in the Fixed Slots Themselves

The Reduced Result Depends Additively on Each Fixed Argument

Separately from additivity in the open slots, the reduced arity result produced by partial evaluation also depends additively on each individual fixed argument, holding the subset S and every other fixed value constant; this is precisely the additive half of tensor partial evaluation linearity, restated here in the specific context of additivity rather than full linearity.

Two Distinct Additivity Statements Coexisting

Additivity in the open slots and additivity in the fixed slots are two logically separate statements about the same underlying tensor T: the first describes how T_S behaves as further arguments are supplied to complete the evaluation, while the second describes how T_S itself, as an object, changes when the already-fixed arguments are varied additively before partial evaluation is even carried out.

Combined Additivity Across Fixed and Open Slots

Simultaneous Variation Is Not Generally a Single Additive Statement

If both a fixed argument and an open argument were varied at the same time, each as a sum of two pieces, the resulting expansion involves four terms rather than two, since additivity must be applied once for the fixed slot's variation and once for the open slot's variation, and these two applications combine multiplicatively in the number of resulting terms rather than simply adding together.

Order of Applying Additivity Does Not Matter

Whether additivity is first applied to expand the fixed argument and then to expand the open argument, or in the opposite order, the same four-term expansion results, since additivity in distinct slots, whether fixed or open, are independent properties of T that can be invoked in any sequence without changing the final outcome.

Practical Role of This Combined Property

Justifying Incremental Construction of Evaluations

Additivity holding both in fixed and in open slots is what justifies building up a full tensor evaluation incrementally: arguments can be supplied to some slots first, decomposed into sums and handled via additivity in those slots, and the remaining slots can subsequently be filled and expanded via additivity in turn, with the two stages composing correctly regardless of the order in which slots are addressed.

Supporting Basis Expansion After Partial Evaluation

When the reduced arity result T_S is itself expanded in a basis for its remaining open slots, this expansion relies on additivity holding in those open slots exactly as it would for the original tensor T, confirming that all of the ordinary tensor machinery, including basis expansion and component evaluation, applies without modification to any tensor obtained via partial evaluation with some slots already fixed.

Diagrammatic Summary

The diagram distinguishes the fixed slots, in which the reduced result behaves additively with respect to the fixed arguments, from the open slots, in which the remaining slot map behaves additively with respect to whatever arguments are eventually supplied to complete the evaluation.