Operads are a generic structure giving a more precise definition of these terms. An operad is an abstract set of operations of various arities (an ugly word to precise the number or arguments taken by an operation: a ternary operation is said to have arity three), subject to relations between them or their compositions. For example, the operad of vector spaces consists of two basic operations: sum and product by scalars (which are actually infinitely many operations), which are tied by distributivity, commutativity and associativity among other relations. An example of operation of this monad is (x, y, z) → 2x + 3y + z, which is a ternary operation. An operad T defines a natural associated monad, which associates to a set X the set
Is not an “operad” a particular case of a “signature”/“theory” in mathematical logic? Why invent a new word?