13.5.1 Tensor Multiple Contraction Slot Pair Set

Tensor Multiple Contraction Slot Pair Set structures tensor contractions for efficient algebraic operations in multi-dimensional spaces.

Tensor Multiple Contraction Slot Pair Set is the complete collection of individual contravariant-covariant slot pairings selected for a multiple index contraction operation, gathering every pair chosen through repeated slot pair selection into a single set that together specifies the entire operation.

Composition of the Slot Pair Set

Individual Pairs as Members

Each member of the slot pair set is itself a pairing of one contravariant slot and one covariant slot drawn from the input tensor or product of tensors, matching the structure of a single contraction slot pair but considered here as one element among several within the larger set.

T_{j l}^{i k}

A slot pair set for this tensor might consist of the pairing of slot $i$ with slot $j$ together with the pairing of slot $k$ with slot $l$ .

Cardinality Equal to the Number of Contractions

The number of members in the slot pair set equals the number of independent contractions being performed within the multiple index contraction operation, so the size of the set directly reports how many single index contraction operations have been composed together.

Requirement of Disjointness Among Members

No Slot Shared Between Two Pairs

Every slot appearing in one member of the slot pair set must not appear in any other member, since a slot committed to two different pairings would leave the intended summation for that slot ambiguous and undefined.

{(i, j), (k, l)}

Consequence of Disjointness for Independence

Disjointness among the members of the slot pair set is precisely what guarantees that the several contractions act independently of one another, since no summation index introduced by one pair overlaps with a position governed by another pair.

Requirements Applying to Each Member Individually

Opposite Variance Within Each Pair

Every member of the slot pair set must itself satisfy the requirement that its two constituent slots carry opposite variance, since this requirement applies to each individual contraction regardless of how many other pairs are included in the same set.

Equal Dimension Within Each Pair

Every member of the slot pair set must likewise satisfy the dimension requirement, with the two slots forming that particular pair ranging over spaces of identical dimension, a condition checked separately for each member of the set.

The Slot Pair Set as the Specification of the Operation

Fully Determining the Multiple Contraction

Once the slot pair set is specified, the multiple index contraction operation is completely determined, since the summation set, the result structure, and the overall type effect can all be derived directly from knowing which slots have been grouped into pairs.

Distinguishing Different Multiple Contractions on the Same Tensor

A tensor admitting more than one possible slot pair set, formed by grouping its available slots differently, generally yields different result tensors depending on which set is chosen, so the specific slot pair set selected is what distinguishes one legitimate multiple contraction from another applied to the same input.

T_{i k}^{i k} \neq T_{i l}^{i l}

Relationship to Tensor Operation Notation

The slot pair set is represented in tensor operation notation by the complete collection of distinct symbols, each repeated exactly once as a matched upper and lower index within an expression, with each such repeated symbol corresponding to exactly one member of the set and thereby fully encoding the pairing structure the set describes.