"Endomorphisms" Example Sentences
1. Endomorphisms are important in the study of group theory.
2. The set of endomorphisms of a vector space is a ring.
3. Endomorphisms of a graph are often used to analyze its properties.
4. If f is an endomorphism of a ring R, then f(1) is also an identity element.
5. The endomorphisms of a field K form a group under function composition.
6. Every group has at least two endomorphisms, the identity map and the trivial map.
7. An endomorphism of a unitary ring preserves the property of being unitary.
8. The endomorphisms of a semi-simple Lie algebra are often studied using representation theory.
9. The set of endomorphisms of a module form a ring under function composition and pointwise addition.
10. Endomorphisms of a category are self-maps of the objects and arrows of the category.
11. Depending on the structure being studied, endomorphisms can play different roles.
12. The automorphisms of a group are a subset of its endomorphisms.
13. One way to classify a group is to enumerate its endomorphisms up to isomorphism.
14. The endomorphisms of a module over a division ring form a simple ring.
15. Endomorphisms of a monoid form a semigroup.
16. The endomorphisms of a commutative ring form a commutative ring.
17. For any non-abelian group, there exist non-trivial endomorphisms.
18. Endomorphisms of a regular category preserve the reflexive and transitive relations on its objects.
19. The endomorphisms of a bipartite graph are either all even or all odd functions.
20. An endomorphism of a topological space is a continuous map from the space to itself.
21. The set of endomorphisms of a scheme form a ring with unit.
22. Endomorphisms of a graded ring preserve the grading.
23. The endomorphisms of a Lie algebra form a Lie algebra under the bracket operation.
24. A ring is called simple if the only endomorphisms are either the identity or the zero map.
25. The endomorphisms of a group G over a field K form a K-algebra.
26. Endomorphisms of an algebraic variety correspond to regular functions on the variety.
27. The dimension of the vector space of endomorphisms of a finite-dimensional vector space is equal to the square of its dimension.
28. If a group G has no non-trivial endomorphisms, it is said to be rigid.
29. Endomorphisms of a scheme are also called morphisms from the scheme to itself.
30. Every algebraic structure has at least one endomorphism, the identity map.
Common Phases
1.
Endomorphisms are linear transformations that map a vector space onto itself;
2.
Endomorphisms preserve the algebraic structure of the vector space;
3.
Endomorphisms are a special case of linear transformations that do not require two different vector spaces;
4.
Endomorphisms compose among themselves;
5.
Endomorphisms can be represented by matrices of the same size as the vector space;
6.
Endomorphisms can be used to define important mathematical concepts, such as eigenvectors and eigenvalues.