Endomorphism

In the following diagram, the arrows denote implication: Any two endomorphisms of an abelian group, A, can be added together by the rule (f + g)(a) = f(a) + g(a).In any concrete category, especially for vector spaces, endomorphisms are maps from a set into itself, and may be interpreted as unary operators on that set, acting on the elements, and allowing the notion of element orbits to be defined, etc.Depending on the additional structure defined for the category at hand (topology, metric, ...), such operators can have properties like continuity, boundedness, and so on.If S has more than one element, a constant function on S has an image that is a proper subset of its codomain, and thus is not bijective (and hence not invertible).The function associating to each natural number n the floor of n/2 has its image equal to its codomain and is not invertible.
Orthogonal projection onto a line, m , is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism .
Somatotype and constitutional psychologyOrthogonal projectionlinear operatorautomorphismmathematicsmorphismmathematical objectisomorphismvector spacelinear mapgroup homomorphismcategorycategory of setsfunctionscompositionmonoidfull transformation monoidinvertiblesubsetautomorphism group(Homo)morphismEndomorphism ringabelian groupmatricesintegermodulepreadditive categorynear-ringregular moduleconcrete categoryvector spacesunary operatorsactingorbitstopologymetriccontinuityboundednessoperator theorydomaincodomainhomomorphicpermutationsbijectivenatural numberdirected pseudoforestsinvolutionsAdjoint endomorphismEpimorphismFrobenius endomorphismMonomorphismJacobson, NathanEncyclopedia of MathematicsEMS Press