Homomorphism

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: ὁμός (homos) meaning 'same' and μορφή (morphe) meaning 'form' or 'shape'. However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning 'similar' to ὁμός meaning 'same'.An injective homomorphism is left cancelable: If f ∘ g = f ∘ h , {displaystyle fcirc g=fcirc h,} one has f ( g ( x ) ) = f ( h ( x ) ) {displaystyle f(g(x))=f(h(x))} for every x in C, the common source of g and h. If f is injective, then g(x) = h(x), and thus g = h. This proof works not only for algebraic structures, but also for any category whose objects are sets and arrows are maps between these sets. For example, an injective continuous map is a monomorphism in the category of topological spaces.Let f : A → B {displaystyle fcolon A o B} be a homomorphism. We want to prove that if it is not surjective, it is not right cancelable. 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: ὁμός (homos) meaning 'same' and μορφή (morphe) meaning 'form' or 'shape'. However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning 'similar' to ὁμός meaning 'same'. 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. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of those can 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 f : A → B {displaystyle f:A o B} between two sets A, B equipped with the same structure such that, if ⋅ {displaystyle cdot } is an operation of the structure (supposed here, for simplification, to be a binary operation), then for every pair x, y of elements of A. One says often that f preserves the operation or is compatible with the operation. Formally, a map f : A → B {displaystyle f:A o B} preserves an operation μ of arity k, defined on both A and B if for all elements a1, ..., ak in A. The operations that must be preserved by a homomorphism include 0-ary operations, that is the constants. In particular, when an identity element is required by the type of structure, the identity element of the first structure must be mapped to the corresponding identity element of the second structure.