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@philzook58

philzook58/intorder.py

Last active May 2, 2020
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python category theory monoids, groups, preorders
class IntOrderCat():
def __init__(self, dom, cod):
assert(dom <= cod)
self.cod = cod
self.dom = dom
self.f = ()
def idd(n):
return IntOrderCat(n,n)
def compose(f,g):
assert( f.dom == g.cod )
return IntOrderCat( g.dom, f.cod )
def __repr__(self):
return f"[{self.dom} <= {self.cod}]"
# our convention for the order of composition feels counterintuitive here.
IntOrderCat(3,5).compose(IntOrderCat(2,3)) # [2 <= 5]
IntOrderCat.idd(3) # [3 <= 3]
class PlusIntMonoid(int):
def mplus(self,b):
return self + b
def mzero():
return 0
class TimesIntMonoid(int):
def mplus(self,b):
return self * b
def mzero():
return 1
class ListMonoid(list):
def mplus(self,b):
return self + b
def mzero():
return []
class UnionMonoid(set):
def mplus(self,b):
return self.union(b)
def mzero():
return set()
ListMonoid([1,2]).mplus(ListMonoid([1,2])) # [1,2,1,2]
UnionMonoid({1,2}).mplus(UnionMonoid({1,4})) # {1,2,4}
TimesIntMonoid(3).mplus(TimesIntMonoid.mzero()) # 3
class PlusIntCat(int):
def compose(self,b):
return self + b
def idd():
return 0
def dom(self):
return () # always return (), the only object
def cod(self):
return ()
class TimesIntCat(int):
def compose(self,b):
return self * b
def idd():
return 1
def dom(self):
return ()
def cod(self):
return ()
class ListCat(int):
def compose(self,b):
return self + b
def idd():
return []
def dom(self):
return ()
def cod(self):
return ()
class UnionSetCat(set):
def compose(self,b):
return self.union(b)
def idd(self,b):
return set()
def dom(self):
return ()
def cod(self):
return ()
PlusIntCat(3).compose(PlusIntCat.idd()) # 3
class SubSetCat():
def __init__(self,dom,cod):
assert( dom.issubset(cod))
self.cod = cod
self.dom = dom
def compose(f,g):
assert(f.dom == g.cod)
return SubSetCat(g.dom, f.cod)
def idd(s):
return SubSetCat(s,s)
def __repr__(self):
return f"[{self.dom} <= {self.cod}]"
SubSetCat( {1,2,3} , {1,2,3,7} ).compose(SubSetCat( {1,2} , {1,2,3} )) # [{1, 2} <= {1, 2, 3, 7}]
from sympy.combinatorics.free_groups import free_group, vfree_group, xfree_group
from sympy.combinatorics.fp_groups import FpGroup, CosetTable, coset_enumeration_r
def fp_group_cat(G, catname):
# A Category generator that turns a finitely presented group into categorical python class
Cat = type(catname, (), vars(G))
def cat_init(self,a):
self.f = a
Cat.__init__ = cat_init
Cat.compose = lambda self,b : G.reduce(self.f * b.f)
Cat.dom = lambda : ()
Cat.cod = lambda : ()
Cat.idd = lambda : Cat(G.identity)
return Cat
F, a, b = free_group("a, b")
G = FpGroup(F, [a**2, b**3, (a*b)**4])
MyCat = fp_group_cat(G, "MyCat")
MyCat(a*a).compose(MyCat.idd())
MyCat.dom()
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