Homoelements

Homoelements is a term that appears irregularly in mathematical writing and is not part of the standard vocabulary of most algebra texts. When used, it denotes elements that correspond to one another under a homomorphism between algebraic structures. In this sense, a homoelement of a triple (A, B, φ) consists of an element a in A together with its image φ(a) in B, highlighting the transfer of structure through φ.

Examples of usage are informal. In a group homomorphism φ: G → H, the element g ∈ G and its image φ(g) ∈ H can be regarded as a homoelement pair; similarly, in ring homomorphisms, elements r map to φ(r) in the target ring. The idea is to emphasize the relationship between elements across the mapped structures rather than focusing solely on the elements within a single structure.

Applications are generally about how properties, equations, or relationships carried by elements are preserved or translated under the homomorphism. Studying homoelements can aid in understanding how a problem in one structure reflects in a related structure, or how fibers and preimages of elements behave under the map.

Relation to standard notions is formalized in the language of homomorphisms and images. The term “homoelement” is typically avoided in rigorous texts; writers instead speak of the image of an element, corresponding elements, or elements under a homomorphism.

Limitations include the lack of a fixed, widely accepted definition. Because usage varies, readers should check how a given author defines or employs the term. See also: homomorphism, image, preimage, kernel, isomorphism, element.