monoids

Under this definition, "monoids" would be objects (ie Sets - as before), equipped with (a) a function to the set from the disjoint union of the set with itself, and (b) a function to the set from the empty set - ie the empty function.

Planet Haskell 2009

Just as with monoids, the free ring with identity on two generators is not commutative, the polynomials xgy and ygx being distinct.

Algebra Pratt, Vaughan 2007

The monoids of natural numbers and of even integers are both submonoids of the monoid of integers under addition, but only the latter submonoid is a subgroup, being closed under negation, unlike the natural numbers.

Algebra Pratt, Vaughan 2007

We can now use any of the monoids discussed above to compute properties of our trees.

A Neighborhood of Infinity 2009

This gives rise to the third and last requirement we have of monoids:

A Neighborhood of Infinity 2009

By using named monoids we can write type signatures that express our intentions to people reading our code: for example by using Any instead of Bool we make it clear just how our boolean value is to be used.

A Neighborhood of Infinity 2009

But there is a slightly easier way - we can combine two monoids into one 'product' monoid.

A Neighborhood of Infinity 2009

You probably use one or two monoids implicitly with every line of code you write, whatever the language, but you might not know it yet.

A Neighborhood of Infinity 2009

Although monoids come from mathematics (algebra in particular) they are found everywhere in computing.

A Neighborhood of Infinity 2009

Because we want mempty to do nothing when combined with other elements we also require monoids to obey these two rules

A Neighborhood of Infinity 2009