PD      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Haskell semiringsMITmail@doisinkidney.com experimentalNone/56I The " 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)The " /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) 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.QThe (->)& instance is analogous to the one for .fA 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.gThe    is  for , and a Cartesian product for .l  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghi     `     !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghi76"Some interesting numeric semiringsMITmail@doisinkidney.com experimentalNone5I ;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. ;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. is ,  is . Positive numbers only. is ,  is Only expects positive numbers! 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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