Antihomomorphism¶
A map between operation-bearing structures that preserves a binary product in reversed order: the image of a product is the product of the images in the opposite order.
Core Idea¶
An antihomomorphism is a map \(f:G\to H\) between structures with a binary operation satisfying \(f(xy)=f(y)f(x)\) for all \(x,y\). It transports a product in the opposite order. If \(H^{\mathrm{op}}\) has the same elements as \(H\) but product \(a\star b=ba\), the same function is an ordinary homomorphism \(G\to H^{\mathrm{op}}\). The changed target type is essential; an ordinary homomorphism to unchanged noncommutative \(H\) has the different equation \(f(xy)=f(x)f(y)\).[^ref-6fef9c63907b]
An antiautomorphism is the narrower case of a bijective self-antihomomorphism. An anti-involution additionally squares to identity. The frozen Antiautomorphism Wikipedia redirect is therefore provenance, not a synonym for every antihomomorphism.[ref-6fef9c63907b][ref-d0421d7520e4]
Scope of Application¶
Group inversion illustrates the rule: \((gh)^{-1}=h^{-1}g^{-1}\), so \(g\mapsto g^{-1}\) is an anti-involution of any group. In a square matrix algebra over a commutative field, \((AB)^\mathsf{T}=B^\mathsf{T}A^\mathsf{T}\), so transpose is another anti-involution, also preserving addition and the unit. The two carriers differ while the reversal equation remains the same.[ref-6fef9c63907b][ref-d0421d7520e4]
For rings, multiplicative reversal alone is insufficient to claim a ring anti-map: addition and unit conventions also need to be stated. In a commutative target the forward and reverse product equations coincide, so the distinction has little discriminatory force there.[ref-d0421d7520e4][ref-6fef9c63907b]
Clarity¶
The full equation prevents “preserves products” from hiding a factor-order choice. It also distinguishes a map from the opposite structure used to describe it. Retyping the codomain turns anti-preservation into ordinary preservation without changing the function on elements.[^ref-6fef9c63907b]
Inversion and transpose are memorable but unusually strong examples: each is bijective, a self-map and involutive. None of those properties is required by the general definition.
Manages Complexity¶
One product-reversal equation compares maps across groups, rings and matrix algebras. The opposite-target form lets ordinary homomorphism reasoning apply after the target operation is explicitly changed. Compatible compositions follow a simple parity rule: two anti-maps compose to a homomorphism; one anti-map and one homomorphism compose to an anti-map.[^ref-6fef9c63907b]
Abstract Reasoning¶
For a proposed map, first type source and target operations, then establish \(f(xy)=f(y)f(x)\) for arbitrary inputs. If it holds, one may equivalently work with \(f:G\to H^{\mathrm{op}}\) as an ordinary homomorphism. If only same-order preservation is shown in a noncommutative target, the anti claim has not been established.[^ref-6fef9c63907b]
Knowledge Transfer¶
The rule transfers literally from group inversion to matrix transpose, but each setting adds different algebraic constraints. Live Homomorphism is a related same-order map type; live Transformation is a broader map pattern, not an established strict parent for this draft. The node is staged unparented for independent DAG review. A portable idea of compositional reversal might merit future prime study, but the named antihomomorphism remains algebra-specific.
[^ref-6fef9c63907b]: Mariusz Wodzicki, Introduction to Algebra, University of California, Berkeley lecture notes, §§3.2.5–3.2.11, PDF pp.13–15, and §3.3.2, PDF p.15. [^ref-d0421d7520e4]: Victor Ginzburg, Algebra Notes, University of Chicago lecture notes, Lecture 1 PDF p.3.
Neighborhood in Abstraction Space¶
Antihomomorphism sits in a moderately populated region (56th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Formal Models & Logical Foundations (33 abstractions)
Nearest neighbors
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- Kernel — 0.84
Computed from structural-signature embeddings · 2026-10-08