ӖTh      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefgNone hijklmnopq hijklmnopq hijklmnopqHaskell semiringsMITmail@doisinkidney.com experimentalNone/2345DIRTThe " .https://ncatlab.org/nlab/show/max-plus+algebraArctic0" or max-plus semiring. It is a semiring where:  = r  = -"  =   = Note that we can't use  from   % because annihilation needs to hold: -"  x = x  -" = -"Taking -" to be s# would break the above law. Using ! to represent it follows the law.The " /https://ncatlab.org/nlab/show/tropical+semiringTropical0" or min-plus semiring. It is a semiring where:  = t  = "  =   = Note that we can't use   from   % because annihilation needs to hold: "  x = x  " = "Taking " to be u" would break the above law. Using " to represent it follows the law.KA suitable definition of a square matrix for certain types which are both v and w%. For instance, given a type like so::{ (data Quad a = Quad a a a a deriving Showinstance Functor Quad where8 fmap f (Quad w x y z) = Quad (f w) (f x) (f y) (f z)instance Applicative Quad where pure x = Quad x x x x+ Quad fw fx fy fz <*> Quad xw xx xy xz =, Quad (fw xw) (fx xx) (fy xy) (fz xz)instance Foldable Quad where6 foldr f b (Quad w x y z) = f w (f x (f y (f z b)))instance Traversable Quad whereD traverse f (Quad w x y z) = Quad <$> f w <*> f x <*> f y <*> f z:})The newtype performs as you would expect:$getMatrix one :: Quad (Quad Integer)@Quad (Quad 1 0 0 0) (Quad 0 1 0 0) (Quad 0 0 1 0) (Quad 0 0 0 1)x1s are another type which works with this newtype::{7let xs = (Matrix . ZipList . map ZipList) [[1,2],[3,4]]7 ys = (Matrix . ZipList . map ZipList) [[5,6],[7,8]]8in (map getZipList . getZipList . getMatrix) (xs <.> ys):}[[19,22],[43,50]]  Monoid under . Analogous to   , but uses the  constraint, rather than y.  Monoid under . Analogous to  , but uses the  constraint, rather than y.A class for semirings with a concept of "negative infinity". It's important that this isn't regarded as the same as "bounded": x   should probably equal .A negative infinite value%Test if a value is negative infinity.xA class for semirings with a concept of "infinity". It's important that this isn't regarded as the same as "bounded": x   should probably equal .A positive infinite value%Test if a value is positive infinity.iUseful for operations where zeroes may need to be discarded: for instance in sparse matrix calculations.z if x is .A  5https://en.wikipedia.org/wiki/Semiring#Star_semirings Star semiring adds one operation,  to a  , such that it follows the law:  x =   x   x =    x  xNFor the semiring of types, this is equivalent to a list. When looking at the v and {; classes as (near-) semirings, this is equivalent to the | operation.Another operation,  , can be defined in relation to :  x = x   xBThis should be recognizable as a non-empty list on types, or the } operation in {.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   = %An ordered semiring follows the laws: x  y => x  z  y  z x  y => x  z  y  z   z  x  y => x  z  y  z  z  x  z  yThe identity of .The identity of .8An associative binary operation, which distributes over .-An associative, commutative binary operation.#Takes the sum of the elements of a . Analogous to  on numbers, or  on s. add [1..5]15add [False, False]Falseadd [False, True]Trueadd [True, undefined]True 'Takes the product of the elements of a . Analogous to  on numbers, or  on s. mul [1..5]120mul [True, True]Truemul [True, False]Falsemul [False, undefined]False!!The product of the contents of a ."The sum of the contents of a .#Transpose the matrix.$Multiply two matrices.This is not a true semiring. In particular, it requires the underlying monoid to be commutative, and even then, it is only a near semiring. It is, however, extremely useful. For instance, this type:  forall a.  ( a)hIs a valid encoding of church numerals, with addition and multiplication being their semiring variants.The (->)& instance is analogous to the one for ~.(getMin . foldMap Min) [1..10]1.0(getMax . foldMap Max) [1..10]10.0A polynomial in x: can be defined as a list of its coefficients, where the i!th element is the coefficient of x^i7. This is the semiring for such a list. Adapted from  Nhttps://pdfs.semanticscholar.org/702d/348c32133997e992db362a19697d5607ab32.pdfhere.  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~=KY%  !"#$% !"  #$    !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~=KY769 NoneIgThe free semiringjRun a g.kRun a g-, interpreting it in the underlying semiring. ghijklmnghijklghilkj ghijklmn9 9 None2345T[thAdds positive and negative infinity to a type. Useful for expressing detectable infinity in types like , etc.x[Adds positive infinity to a type. Useful for expressing detectable infinity in types like , etc.{[Adds negative infinity to a type. Useful for expressing detectable infinity in types like , etc.Not distributive.,Only lawful when used with positive numbers.Doesn't follow  or .,tuvwxyz{|}~tuvwxyz{|}{|}xyztuvw%tuvwxyz{|}~"Some interesting numeric semiringsMITmail@doisinkidney.com experimentalNone%&2345I(Adds a star operation to integral types. () = () () = ()  =   =   0 = 1  _ = *Adds a star operation to fractional types. () = () () = ()  =   =  $ x = if x < 1 then 1 / (1 - x) else  ;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. () = r () = ()  =   =  ;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. () = r x  y = r 0 (x  y  1)  =   = Positive numbers only. () =  () =   =   = $Useful for some constraint problems. () = r () = t  = s  = uOnly expects positive numbers;/(Some functions for generating tests for s.MITmail@doisinkidney.com experimentalNoneTN0Typealias for ternary laws. Can be used like so: -smallCheck 6 (ternaryLaws :: TernaryLaws Int)O/Typealias for binary laws. Can be used like so: +smallCheck 8 (binaryLaws :: BinaryLaws Int)P.Typealias for unary laws. Can be used like so: *smallCheck 10 (unaryLaws :: UnaryLaws Int)QPlus is associative. (x  y)  z = x  (y  z)RMultiplication is associative. (x  y)  z = x  (y  z)SPlus is commutative. x  y = y  xT Multiplication distributes left. x  (y  z) = x  y  x  zU!Multiplication distributes right. (x  y)  z = x  z  y  zVAdditive identity. x   =   x = xWMultiplicative identity. x   =   x = xXRight annihilation of  by .   x = YLeft annihilation of  by . x   = Z$A test for all three unary laws for s (V, W, Y, and X).[#A test for the unary laws for near-s (V, W, and X).\,A test for all of the ternary laws for near-s (Q, R, U).]&A test for all of the binary laws for s (just S).^'A test for all of the ternary laws for s (Q, R, T, U)._The star law for s.  x =   x   x =    x  x`The plus law for s.  x = x   xa The laws for s (_, `).bAddition law for ordered s. x  y => x  z  y  z  z  x  z  ycMultiplication law for ordered s. x  y => x  z  y  z  z  x  z  ydLaws for ordered s (c, b).eLaw for result of  operation. x   =   x =  xfZero is zero law.   = zg The laws for  s (e, f).NOPQRSTUVWXYZ[\]^_`abcdefgNOPQRSTUVWXYZ[\]^_`abcdefgPONVWYXZS]QRTU^[\_`aefgcbdNOPQRSTUVWXYZ[\]^_`abcdefg   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      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