Copyright	(c) 2020-2021 Reed Mullanix Emily Pillmore
License	BSD-style
Maintainer	Reed Mullanix <reedmullanix@gmail.com>, Emily Pillmore <emilypi@cohomolo.gy>
Stability	stable
Portability	non-portable
Safe Haskell	Safe-Inferred
Language	Haskell2010

Data.Group.Free.Product

Description

This module provides definitions for the FreeProduct of two groups, along with useful combinators.

Synopsis

newtype FreeProduct g h = FreeProduct {
- runFreeProduct :: Seq (Either g h)
}
simplify :: (Eq g, Eq h, Monoid g, Monoid h) => FreeProduct g h -> FreeProduct g h
coproduct :: Monoid m => (a -> m) -> (b -> m) -> FreeProduct a b -> m
injl :: a -> FreeProduct a b
injr :: b -> FreeProduct a b

Documentation

newtype FreeProduct g h Source #

The free product on two alphabets

Note: This does not perform simplification upon multiplication or construction. To do this, one should use simplify.

Constructors

FreeProduct
Fields runFreeProduct :: Seq (Either g h)

Instances

Instances details

Bifunctor FreeProduct Source #
Instance details Defined in Data.Group.Free.Product Methods bimap :: (a -> b) -> (c -> d) -> FreeProduct a c -> FreeProduct b d # first :: (a -> b) -> FreeProduct a c -> FreeProduct b c # second :: (b -> c) -> FreeProduct a b -> FreeProduct a c #
Functor (FreeProduct g) Source #
Instance details Defined in Data.Group.Free.Product Methods fmap :: (a -> b) -> FreeProduct g a -> FreeProduct g b # (<$) :: a -> FreeProduct g b -> FreeProduct g a #
(Eq g, Eq h) => Eq (FreeProduct g h) Source #
Instance details Defined in Data.Group.Free.Product Methods (==) :: FreeProduct g h -> FreeProduct g h -> Bool # (/=) :: FreeProduct g h -> FreeProduct g h -> Bool #
(Ord g, Ord h) => Ord (FreeProduct g h) Source #
Instance details Defined in Data.Group.Free.Product Methods compare :: FreeProduct g h -> FreeProduct g h -> Ordering # (<) :: FreeProduct g h -> FreeProduct g h -> Bool # (<=) :: FreeProduct g h -> FreeProduct g h -> Bool # (>) :: FreeProduct g h -> FreeProduct g h -> Bool # (>=) :: FreeProduct g h -> FreeProduct g h -> Bool # max :: FreeProduct g h -> FreeProduct g h -> FreeProduct g h # min :: FreeProduct g h -> FreeProduct g h -> FreeProduct g h #
(Show g, Show h) => Show (FreeProduct g h) Source #
Instance details Defined in Data.Group.Free.Product Methods showsPrec :: Int -> FreeProduct g h -> ShowS # show :: FreeProduct g h -> String # showList :: [FreeProduct g h] -> ShowS #
Semigroup (FreeProduct g h) Source #
Instance details Defined in Data.Group.Free.Product Methods (<>) :: FreeProduct g h -> FreeProduct g h -> FreeProduct g h # sconcat :: NonEmpty (FreeProduct g h) -> FreeProduct g h # stimes :: Integral b => b -> FreeProduct g h -> FreeProduct g h #
Monoid (FreeProduct g h) Source #
Instance details Defined in Data.Group.Free.Product Methods mempty :: FreeProduct g h # mappend :: FreeProduct g h -> FreeProduct g h -> FreeProduct g h # mconcat :: [FreeProduct g h] -> FreeProduct g h #
(Group g, Group h) => Group (FreeProduct g h) Source #
Instance details Defined in Data.Group.Free.Product Methods invert :: FreeProduct g h -> FreeProduct g h # (~~) :: FreeProduct g h -> FreeProduct g h -> FreeProduct g h # pow :: Integral x => FreeProduct g h -> x -> FreeProduct g h #
(GroupOrder g, GroupOrder h) => GroupOrder (FreeProduct g h) Source #
Instance details Defined in Data.Group.Free.Product Methods order :: FreeProduct g h -> Order Source #

simplify :: (Eq g, Eq h, Monoid g, Monoid h) => FreeProduct g h -> FreeProduct g h Source #

O(n) Simplifies a word in a FreeProduct. This means that we get rid of any identity elements, and perform multiplication of neighboring gs and hs.

coproduct :: Monoid m => (a -> m) -> (b -> m) -> FreeProduct a b -> m Source #

The FreeProduct of two Monoids is a coproduct in the category of monoids (and by extension, the category of groups).

injl :: a -> FreeProduct a b Source #

Left injection of an alphabet a into a free product.

injr :: b -> FreeProduct a b Source #

Right injection of an alphabet b into a free product.