Search results
Results from the WOW.Com Content Network
The most basic example is the inclusion of integers into rational numbers, which is a homomorphism of rings and of multiplicative semigroups. For both structures it is a monomorphism and a non-surjective epimorphism, but not an isomorphism. [5] [7] A wide generalization of this example is the localization of a ring by a multiplicative set ...
Homotopy does lead to a relation on spaces: homotopy equivalence. There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an isotopy between the identity map on X and the homeomorphism from X to Y .
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object.
We define the kernel of h to be the set of elements in G which are mapped to the identity in H ():= {: =}. and the image of h to be ():= {():}. The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism.
An example is the bipartite double cover, formed from a graph by splitting each vertex v into v 0 and v 1 and replacing each edge u,v with edges u 0,v 1 and v 0,u 1. The function mapping v 0 and v 1 in the cover to v in the original graph is a homomorphism and a covering map. Graph homeomorphism is a different notion, not related directly to ...
Diagram of the fundamental theorem on homomorphisms, where is a homomorphism, is a normal subgroup of and is the identity element of .. Given two groups and and a group homomorphism:, let be a normal subgroup in and the natural surjective homomorphism / (where / is the quotient group of by ).
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
In graph theory, two graphs and ′ are homeomorphic if there is a graph isomorphism from some subdivision of to some subdivision of ′.If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in diagrams), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if their diagrams are homeomorphic in the ...