Splittings

Splittings, or splitting, is the act of dividing an object, set, or phenomenon into parts. In mathematics and science, the concept is used to describe a decomposition that reveals underlying structure or separates observable components.

In mathematics, splitting has several precise meanings. A short exact sequence 0 → A → E → B → 0 is said to split if E is (isomorphic to) a direct sum of A and B, or more generally if there is a section s: B → E with the composition to B the identity. When a polynomial f in a field F[x] has a field extension K over which f factors completely into linear factors, K is called a splitting field of f. Splitting also appears in module theory: a submodule N of M yields a split short exact sequence 0 → N → M → M/N → 0 when M ≅ N ⊕ M/N.

In physics and chemistry, splitting describes the division of degenerate energy levels or spectral lines. Zeeman splitting arises when a magnetic field lifts degeneracy, Stark splitting from an electric field, while fine-structure and hyperfine-structure splitting come from relativistic and nuclear interactions, respectively, producing closely spaced lines in spectra.

Other uses include data splitting in statistics and machine learning, where a data set is partitioned into training, validation, and test subsets; or the practice of splitting a problem into modular components. The term can also appear in biology to describe cellular division in some contexts, though standard terminology uses mitosis, meiosis, or binary fission.