IIIsom

IIIsom is a term used in abstract algebra and category theory to denote a type of isomorphism that relates three structures simultaneously. The name combines "three" (III) with "isomorphism," reflecting its focus on triadic relationships. While not as widely established as binary isomorphisms, IIIsom appears in discussions of triadic or ternary constructions where a single equivalence must respect three corresponding components at once.

Formal notion: Let C be a category equipped with a ternary operation or a functor ∗: C^3 → D that assigns to a triple (A, B, C) an object D = A ∗ B ∗ C. An IIIsom between two triples (A, B, C) and (A', B', C') consists of a triple of isomorphisms (f: A → A', g: B → B', h: C → C') such that the induced map (f, g, h): A ∗ B ∗ C → A' ∗ B' ∗ C' is an isomorphism in D. In particular cases, such as when the ternary operation is the tensor product or a direct product, IIIsom reduces to a standard isomorphism on the derived construction.

Properties: IIIsom is closed under composition: if (f, g, h) and (f', g', h') are IIIsom between corresponding triples, their componentwise composition is an IIIsom. It is symmetric under simultaneous permutation of components if the underlying ternary operation is symmetric.

Examples: If A, B, C are vector spaces and ∗ is the tensor product, any trio of linear isomorphisms f, g, h induces a linear isomorphism A ⊗ B ⊗ C ≅ A' ⊗ B' ⊗ C'. In a more general setting, IIIsom provides a framework for triadic equivalences in algebraic structures.

See also: isomorphism, tensor product, ternary operation, triadic structures.