Definitions

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  • noun Plural form of monoid.

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Examples

  • 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

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