Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Loops in this category are the endomorphisms (all of which are actually automorphisms). In abstract algebra, the endomorphisms of an abelian group "X" form a ring. A symmetric tensor is a tensor that is invariant under all these endomorphisms. The Sturmian endomorphisms form a submonoid of the monoid of endomorphisms of "B". The endomorphisms of a nonabelian group generate an algebraic structure known as a near-ring. A ', or a ', which is invariant under surjective endomorphisms. Then, the "inner condition" requires a bijection property from endomorphisms also to arrays. Still more information about endomorphisms of finite-length indecomposables is provided by the Fitting lemma. End("M"), the ring of "R" module endomorphisms of "M". One writes formula 6 for the set of all endomorphisms between a module "M". The composita of the endomorphisms satisfy the relations formula 307 and formula 308. Therefore, one may consider the elements formula 28 as endomorphisms and assign formula 29, accordingly. Over a field of characteristic 2 we consider instead the alternating endomorphisms. The associate endomorphisms are called the homotheties of "G".