TeXipedia

hom

Represents the set of homomorphisms between two algebraic structures in mathematics.

Overview

Essential in abstract algebra and category theory for denoting morphisms between mathematical objects like groups, rings, or modules.

  • Commonly used in advanced mathematics to describe structure-preserving maps.
  • Appears frequently in commutative diagrams and algebraic proofs.
  • Often combined with subscripts to specify the category or type of morphisms being considered.
  • Particularly important in homological algebra and representation theory.

Examples

Showing the homomorphism between two groups G and H.

hom⁡(G,H)≅K\hom(G,H) \cong K 
\hom(G,H) \cong K

Expressing the dimension of homomorphisms between vector spaces.

dim⁡(hom⁡(V,W))=dim⁡V⋅dim⁡W\dim(\hom(V,W)) = \dim V \cdot \dim W 
\dim(\hom(V,W)) = \dim V \cdot \dim W

Describing the set of ring homomorphisms between rings R and S.

hom⁡R(R,S)={f:R→S∣f is a ring homomorphism}\hom_R(R,S) = \{f:R \to S \mid f \text{ is a ring homomorphism}\} 
\hom_R(R,S) = \{f:R \to S \mid f \text{ is a ring homomorphism}\}