A homomorphism from an algebraic object to itself: f: A → A that preserves the defining operations of A.

Principle

Endomorphisms capture internal algebraic transformations and always form an algebraic object (for many categories a monoid under composition; in module or vector-space contexts they form a ring or algebra).

Demonstration

A linear endomorphism of a vector space V is a linear map T: V → V; endomorphisms of a group G are homomorphisms G → G and compose to give a monoid End(G).

Misapplication

Calling any endomap an endomorphism without checking structure preservation (for example a function V → V that is not linear), or ignoring composition order when considering algebraic properties.

Consequence

The set of endomorphisms often encodes the internal symmetry and dynamics of the object (e.g., eigenvalues of linear endomorphisms), and algebraic structure on End(A) is central to representation theory and module theory.

Reversal

An automorphism is an endomorphism that is invertible; reversing invertibility produces general endomorphisms which need not be bijective.

Boundary

Must be a map from the object to itself preserving the specified operations; endomorphisms need not be invertible, and additional constraints (continuity, grading, *-structure) are extra hypotheses.

Semantic Tension

Tension with homomorphism: every endomorphism is a homomorphism but the converse does not hold because homomorphisms generally map between distinct objects; also confusion with arbitrary endomaps exists.

Synthesis

An endomorphism is a structure-preserving self-map of an algebraic object whose composition algebra captures internal operations, symmetries, and functional behavior without requiring invertibility.