MlB      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Haskell semiringsMITmail@doisinkidney.com experimentalNone/56I  Monoid under  . Analogous to , but uses the  constraint, rather than . Monoid under  . Analogous to , but uses the  constraint, rather than .A  &https://en.wikipedia.org/wiki/SemiringSemiring% is like the the combination of two s. The first is called  ; it has the identity element /, and it is commutative. The second is called  ; it has identity element , and it must distribute over  .LawsNormal  laws (a   b)   c = a   (b   c)    a = a    = a (a   b)   c = a   (b   c)    a = a    = aCommutativity of   a   b = b   aDistribution of   over   a   (b   c) = (a   b)   (a   c) (a   b)   c = (a   c)   (b   c) Annihilation    a = a    = The identity of  .The identity of  . 8An associative binary operation, which distributes over  . -An associative, commutative binary operation.KThe (->)& instance is analogous to the one for .LThe " Lhttps://ncatlab.org/nlab/show/https://ncatlab.org/nlab/show/max-plus+algebraArctic2" or max-plus semiring. It is a semiring where:   =   = " (represented by )   =   (over the inner value)  =  (over the inner value)MThe " /https://ncatlab.org/nlab/show/tropical+semiringTropical2" or min-plus semiring. It is a semiring where:   =   = -" (represented by )   =   (over the inner value)  =  (over the inner value)ZA polynomial in x: can be defined as a list of its coefficients, where the i!th element is the coefficient of x^i6. This is the semiring for such a list. Adapted from  Nhttps://pdfs.semanticscholar.org/702d/348c32133997e992db362a19697d5607ab32.pdfhere.[The   is  for  , and a Cartesian product for  .`  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]     X   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\] 7 6"Some interesting numeric semiringsMITmail@doisinkidney.com experimentalNone5It ;https://en.wikipedia.org/wiki/Semiring#cite_ref-droste_14-0 Wikipedia& has some information on this. Also  Shttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.304.6152&rep=rep1&type=pdfthis3 paper. Apparently used for probabilistic parsing.w ;https://en.wikipedia.org/wiki/Semiring#cite_ref-droste_14-0 Wikipedia& has some information on this. Also  Shttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.304.6152&rep=rep1&type=pdfthis paper.z  is ,   is . Positive numbers only.}  is ,   is Only expects positive numbers!tuvwxyz{|}~ tuvwxyz{|}~ }~z{|wxytuvtuvwxyz{|}~None234IAThe free semiring. Adapted from PureScript's version, available  Zhttps://pursuit.purescript.org/packages/purescript-semirings/3.0.0/docs/Data.Semiring.Freehere9. Only a valid semiring if treated as a multiset, as in: Free [[1],[0]] == Free [[0],[1]]TrueRun a .Run a -, interpreting it in the underlying semiring.%Extremely slow. For testing purposes.  9 (Some QuickCheck properties for SemiringsMITmail@doisinkidney.com experimentalNoneTPlus is associative.Multiplication is associative.Plus is commutative. 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