Triple system¶
Equip a vector space with a trilinear product returning to that space, creating a generic ternary algebra whose added identities specialize it into Lie, Jordan, and geometry-bearing triple systems.
Core Idea¶
A triple system, or ternar, is a vector space \(V\) over a field \(F\) equipped with an \(F\)-trilinear map \(V\times V\times V\to V\); Lie and Jordan triple systems impose additional distinct polynomial identities on that product. Three vectors enter one multilinear operation whose value remains in the carrier; identities among nested triple products control inner derivations and permit Lie or Jordan structures to be encoded without choosing a binary product.
Its autonomous residual is the vector-space-valued trilinear algebraic carrier, not a set of three objects, a scalar-valued trilinear form, a binary algebra with three generators, or the union of Lie and Jordan subclasses alone.
Scope of Application¶
Triple system applies when the analyst can specify a vector space over a specified field together with a ternary operation on vectors and establish that the carrier is a vector space and one declared ternary product is linear separately in all three arguments and takes every input triple back into that vector space. The entry treats algebraic triple systems on vector spaces. Steiner triple systems, ternary quasigroups, triple categories, and physical three-body systems are separate typed concepts.
Clarity¶
A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because triple system is overloaded across nonassociative algebra and combinatorics, while authors sometimes use it narrowly for Lie or Jordan systems rather than the generic trilinear carrier. The disciplined statement is that the object counts as Triple system exactly when the carrier is a vector space and one declared ternary product is linear separately in all three arguments and takes every input triple back into that vector space
Manages Complexity¶
The abstraction compresses generic ternars, Lie and Jordan triple systems, associative-derived products, hermitian forms of Jordan triples, positive and nondegenerate systems, super and graded variants, and Jordan pairs into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.
Abstract Reasoning¶
- Type the carrier. Establish a vector space over a specified field together with a ternary operation on vectors and reject examples from a different problem. 2. Lock the rule. Express that the carrier is a vector space and one declared ternary product is linear separately in all three arguments and takes every input triple back into that vector space independently of one notation or implementation.
Knowledge Transfer¶
Transfer within nonassociative algebra is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from Every Lie algebra yields a Lie triple system by \([x,y,z]=[[x,y],z]\), whose skew and derivation identities follow from the binary Lie bracket and Jacobi identity. to The tangent space at a point of a symmetric space carries a natural Lie triple product, and its inner derivations assemble with the tangent space into a graded Lie algebra. demonstrates that continuity.
Relationships to Other Abstractions¶
Current abstraction Triple system Domain-specific
Parents (1) — more general patterns this builds on
-
Triple system is a kind of Function (Mapping) Prime
The proposed strict upward parent is
prime:function_mapping.
Hierarchy path (1) — routes to 1 parentless root
- Triple system → Function (Mapping)
Neighborhood in Abstraction Space¶
Triple system sits in a crowded region of the domain-specific corpus (36th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Ambient Structures & Local Geometry (9 abstractions)
Nearest neighbors
- Frobenius normal form — 0.91
- Outer product — 0.90
- Linear map — 0.90
- Subrepresentation — 0.90
- Linear group — 0.90
Computed from structural-signature embeddings · 2026-09-08