Semilinear map¶
An additive map between vector spaces whose scalar multiplication is respected after applying a fixed field automorphism.
Core Idea¶
A theta-semilinear T satisfies T(av)=theta(a)T(v); linear maps use the identity automorphism and conjugate-linear maps use complex conjugation, while projective geometry naturally identifies semilinear transformations. The map preserves vector addition directly and transports scalars through one automorphism before applying their action in the codomain. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
The load-bearing residual is not the broad topic of linear algebra. It is the domain-specific identity determined by the source and target vector spaces, base field, fixed field automorphism, additive map and twisted scalar equation are explicit.
Scope of Application¶
Semilinear map belongs to linear algebra and is useful where the analyst can specify the typed linear algebra carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the source and target vector spaces, base field, fixed field automorphism, additive map and twisted scalar equation are explicit. The scope is broad within that domain but bounded by the need for the source and target vector spaces, base field, fixed field automorphism, additive map and twisted scalar equation are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the source and target vector spaces, base field, fixed field automorphism, additive map and twisted scalar equation are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Semilinear map can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Semilinear map. Semilinear map compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed linear algebra carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the source and target vector spaces, base field, fixed field automorphism, additive map and twisted scalar equation are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of linear algebra because they reuse the typed linear algebra carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, The map preserves vector addition directly and transports scalars through one automorphism before applying their action in the codomain., and type the carrier, state every parameter and convention in the definition, test that the source and target vector spaces, base field, fixed field automorphism, additive map and twisted scalar equation are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Semilinear map Domain-specific
Parents (1) — more general patterns this builds on
-
Semilinear map is a kind of Linearity Prime
The proposed strict upward parent is
prime:linearity.
Hierarchy path (1) — routes to 1 parentless root
- Semilinear map → Linearity
Neighborhood in Abstraction Space¶
Semilinear map sits in a crowded region of the domain-specific corpus (9th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Matrix Structure & Linear Maps (48 abstractions)
Nearest neighbors
- Transpose of a linear map — 0.95
- Linear map — 0.95
- Scalar multiplication — 0.93
- Linear complex structure — 0.92
- Defective matrix — 0.92
Computed from structural-signature embeddings · 2026-09-08