Homomorphisms
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the ancient Greek language: (homos) meaning "same" and (morphe) meaning "form" or "shape".
Homomorphisms of vector spaces are also called linear maps, and their study is the object of linear algebra.
The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory.
Being an isomorphism, an automorphism, or an endomorphism is a property of some homomorphisms, which may be defined in a way that may be generalized to any class of morphisms.
A homomorphism is a map between two algebraic structures of the same type (that is of the same name), that preserves the operations of the structures. This means a map between two sets A, B equipped with the same structure such that, if ∗ is an operation of the structure (supposed here, for simplification, to be a binary operation), then
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