What is a homomorphism in abstract algebra?

homomorphism, (from Greek homoios morphe, “similar form”), a special correspondence between the members (elements) of two algebraic systems, such as two groups, two rings, or two fields.

How is homomorphism defined?

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”.

What is morphism in group theory?

In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type.

What is homomorphism in TOC?

A homomorphism is a function from strings to strings that “respects” concatenation: for any x, y ∈ Σ∗, h(xy) = h(x)h(y). (Any such function is a homomorphism.)

What are the types of homomorphism?

There are two main types: group homomorphisms and ring homomorphisms. (Other examples include vector space homomorphisms, which are generally called linear maps, as well as homomorphisms of modules and homomorphisms of algebras.)

What are homomorphic functions?

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn.

What is homomorphic image?

The image of the homomorphism, im(f), is the set of elements of H to which at least one element of G is mapped. im(f) is not required to be the whole of H. The kernel of the homomorphism f is the set of elements of G that are mapped to the identity of H: ker(f) = { u in G : f(u) = 1H }.

What is homomorphism in regular language?

Regular languages are closed under inverse homomorphism, i.e., if L is regular and h is a homomorphism then h−1(L) is regular.

What is homomorphism in finite automata?

DEFINITION:A homomorphism is a mapping h with domain Σ ∗ for some alphabet Σ which preserves concatenation: h(v · w) =h(v) · h(w). Simply speaking in homomorphism we replace each letter in a language with another letter in some other language.