## Posts tagged ‘free algebra’

### Monads in mathematics 2 : algebras over monads and operads

A large class of monads is actually derived from operads: basic examples form monads in the category of sets and associate to a set *X* a set *TX* of abstract “terms” made of elements of *X*. For example, the monad of vector spaces I mentioned in the previous post is such a monad, and associates to a set the abstract linear combinations of its elements.

*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 in this operad is , which is a ternary operation.

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