"Endomorphism" Example Sentences
1. An endomorphism is a morphism from a mathematical object to itself.
2. The study of endomorphisms is an important topic in algebra.
3. Every ring has an endomorphism that fixes every element.
4. An endomorphism of a group is a homomorphism from the group to itself.
5. The set of endomorphisms of a vector space is a ring.
6. The identity map is an example of an endomorphism.
7. An automorphism is an endomorphism that is also bijective.
8. If the endomorphism is invertible, it is called an isomorphism.
9. The kernel of an endomorphism is a normal subgroup of the domain.
10. The image of an endomorphism is a substructure of the codomain.
11. Endomorphisms of abelian groups commute with each other.
12. The set of endomorphisms of an object forms a monoid under composition.
13. The identity endomorphism is always a member of the set of endomorphisms.
14. Endomorphisms of a field are always injective.
15. The set of endomorphisms of a module is a ring with identity.
16. An endomorphism of a lattice always preserves order.
17. The group of automorphisms of a group is a subgroup of its endomorphism group.
18. Endomorphisms of posets always preserve meets and joins.
19. The set of endomorphisms of a category forms a monoidal category.
20. An endomorphism ring is a ring whose elements are endomorphisms of a module.
21. Endomorphisms of a Sheaf always preserve the stalks of the Sheaf.
22. The algebra of endomorphisms of a space is called endomorphism algebra.
23. An endomorphism of a finite field is necessarily a linear map.
24. The endomorphism algebra of a vector space consists of only linear maps.
25. A group is called simple if it has no nontrivial endomorphisms.
26. Endomorphisms of algebraic varieties always preserve the Zariski topology.
27. An endomorphism of a Boolean algebra always preserves the complement operation.
28. The set of endomorphisms of the abelian category is a ring.
29. An endomorphism of a scheme always commutes with the structure sheaf.
30. Endomorphisms of manifolds always preserve the differentiable structure.
Common Phases
1. "The
endomorphism maps an element to itself;"
2. "The
endomorphism is a linear transformation of a vector space;"
3. "The
endomorphism preserves the structure of the object it operates on;"
4. "The
endomorphism group of a group is the set of all
endomorphisms of that group;"
5. "The
endomorphism of a ring is a map that sends elements of the ring to other elements of the ring;"
6. "The
endomorphism of a topological space is a continuous map from the space to itself;"
7. "The
endomorphism of a graph is a map that sends vertices to vertices and edges to edges."