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 (x,y,z) to 2x+3y+z, which is a ternary operation.

26 February 2009 at 12:50 am 2 comments