ENDOMORPHISM Meaning and
Definition
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An endomorphism, in the field of mathematics, specifically in the realm of abstract algebra, refers to a function or mapping that operates on a set and maps each element to another element within the same set. More precisely, it is a morphism or map that commences and concludes in the same mathematical structure. Typically, an endomorphism involves objects that are closed under a specific binary operation.
Formally, if we have a set S, an endomorphism is a mapping f: S → S that assigns each element s in S to another element f(s) in S. In other words, the domain and the codomain of an endomorphism are the same set S.
Endomorphisms commonly occur in algebraic structures, such as groups, rings, and vector spaces, where the set S possesses a specific algebraic structure and the function f preserves this structure. For example, in a group, an endomorphism will preserve the group operation, while in a ring, it will preserve both the addition and multiplication operations.
An important aspect of endomorphisms is that they can be composed with one another, resulting in the creation of new endomorphisms. This composition, known as endomorphism composition, produces a new mapping that first applies one endomorphism and then the other. This operation holds associative properties.
In summary, an endomorphism is a self-mapping or self-transformation of a set that remains within the same structure, often found in various mathematical fields, including abstract algebra.
Common Misspellings for ENDOMORPHISM
- wndomorphism
- sndomorphism
- dndomorphism
- rndomorphism
- 4ndomorphism
- 3ndomorphism
- ebdomorphism
- emdomorphism
- ejdomorphism
- ehdomorphism
- ensomorphism
- enxomorphism
- encomorphism
- enfomorphism
- enromorphism
- eneomorphism
- endimorphism
- endkmorphism
- endlmorphism
- endpmorphism
Etymology of ENDOMORPHISM
The word "endomorphism" has its roots in the Greek language. It is formed by combining two Greek words, "endo" (ἔνδον), meaning "within", and "morphē" (μορφή), meaning "form" or "shape". Therefore, the literal translation of "endomorphism" would be "within form" or "within shape". In mathematics, it specifically refers to a linear transformation of a mathematical object onto itself, usually a vector space or an algebraic structure.
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