Monadic

Monadic is an adjective used in mathematics and computer science to describe concepts related to monads. In category theory, a monad on a category C is a triple (T, η, μ) consisting of an endofunctor T: C → C and two natural transformations: η: Id_C → T (the unit) and μ: T² → T (the multiplication). These data satisfy associativity and unit laws (μ ∘ Tμ = μ ∘ μT and μ ∘ Tη = μ ∘ ηT = id_T). Monads provide an abstract framework for chaining computations and for modeling effects, contexts, or computations embedded within a category.

In functional programming, a monad is a type constructor M together with operations return (or pure) and bind (often written as >>=) or join, satisfying the monad laws: left identity, right identity, and associativity. This structure allows sequencing of operations that produce values in a context while keeping the programming style pure. Examples include the Maybe (option) monad, the List monad, the IO monad, the State monad, and the Either monad, each encoding a different kind of computational effect or context.

In logic, the term monadic may refer to unary predicates or to monadic theories, where all relation symbols have arity at most one. Monadic logic is a restricted fragment of first-order logic and has particular decidability properties that differ from the full theory.

Outside mathematics, the word is sometimes used more loosely to describe constructs or patterns that follow a single-argument, monad-like structure. In common usage, monadic highlights the presence of a single-step, context-bearing computation or value.