Editing Monad (functional programming) (section)

=== Additive monads ===
An '''additive monad''' is a monad endowed with an additional closed, associative, binary operator '''mplus''' and an [[identity element]] under {{mvar|mplus}}, called '''mzero'''.
The <code>Maybe</code> monad can be considered additive, with <code>Nothing</code> as {{mvar|mzero}} and a variation on the [[logical disjunction|OR]] operator as {{mvar|mplus}}.
<code>List</code> is also an additive monad, with the empty list <code>[]</code> acting as {{mvar|mzero}} and the concatenation operator <code>++</code> as {{mvar|mplus}}.

Intuitively, {{mvar|mzero}} represents a monadic wrapper with no value from an underlying type, but is also considered a "zero" (rather than a "one") since it acts as an [[absorbing element|absorber]] for {{mvar|bind}}, returning {{mvar|mzero}} whenever bound to a monadic function.
This property is two-sided, and {{mvar|bind}} will also return {{mvar|mzero}} when any value is bound to a monadic [[zero function]].

In category-theoretic terms, an additive monad qualifies once as a monoid over monadic functions with {{mvar|bind}} (as all monads do), and again over monadic values via {{mvar|mplus}}.<ref name="RJS2015">{{cite conference |last1=Rivas |first1=Exequiel |last2=Jaskelioff |first2=Mauro |last3=Schrijvers |first3=Tom |date=July 2015 |title=From monoids to near-semirings: the essence of MonadPlus and Alternative |url=https://usuarios.fceia.unr.edu.ar/~mauro/pubs/FromMonoidstoNearsemirings.pdf |conference=17th International ACM Symposium on Principles and Practice of Declarative Programming |location=Siena, Italy |citeseerx=10.1.1.703.342}}</ref>{{efn|Algebraically, the relationship between the two (non-commutative) monoid aspects resembles that of a [[near-semiring]], and some additive monads do qualify as such. However, not all additive monads meet the [[distributive property|distributive]] laws of even a near-semiring.<ref name="RJS2015" />}}