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#W bisets.gd Laurent Bartholdi
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#Y Copyright (C) 2012, Laurent Bartholdi
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## This file defines general bisets.
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# BISET:
# has left/right acting domains, and elementsfamily.
# two representations:
# (1) as homomorphism; then elements are elements in the range
# (2) as machines (maybe twisted, in that the group entries can be from wherever); then elements are pairs (integer,group element) in the "natural" basis
# BASIS: for left-free, stored as (permutation,sequence of group elements).
# OPERATIONS:
# left/right product of elements by acting domains
# tensor product, contragredient
# for a given basis, extract machine, or wreath recursion
# test, and produce, isomorphism of two bisets, in the form of a basis of the second biset that makes them identical.
# multiply biset by homomorphisms on left/right to change coordinates.
# decompose as amalgamated free product, etc.
# congruence of G-G-bisets, by conjugation.
DeclareCategory("IsBiset", IsObject);
DeclareCategory("IsFRBiset",
IsBiset and IsAssociativeROpDProd and IsAssociativeLOpDProd);
DeclareRepresentation("IsFRBisetByFRMachineRep", IsFRBiset and IsAttributeStoringRep, []);
DeclareRepresentation("IsFRBisetByFRSemigroupRep", IsFRBiset and IsAttributeStoringRep, []);
DeclareRepresentation("IsFRBisetByHomomorphismRep", IsFRBiset and IsAttributeStoringRep, []);
DeclareCategory("IsBisetElement",
IsObject);
DeclareRepresentation("IsBisetElementByPair", IsBisetElement and IsComponentObjectRep, []);
DeclareRepresentation("IsBisetElementByElement", IsBisetElement and IsComponentObjectRep, []);
BindGlobal("FRBISET_FAMILY",
NewFamily("FRBisetsFamily",IsObject));
DeclareOperation("BisetByFRMachine", [IsFRMachine]);
DeclareOperation("BisetByFRSemigroup", [IsFRSemigroup]);
DeclareSynonym("BisetByFRMonoid", BisetByFRSemigroup);
DeclareSynonym("BisetByFRGroup", BisetByFRSemigroup);
DeclareAttribute("DualBiset", IsFRBiset);
DeclareOperation("TensorProductOp", [IsList,IsFRBiset]);
DeclareOperation("BisetElement", [IsFRBiset,IsMultiplicativeElement,IsObject]);
DeclareOperation("BisetElement", [IsFRBiset,IsMultiplicativeElement]);
DeclareProperty("IsLeftFree", IsFRBiset);
DeclareProperty("IsRightFree", IsFRBiset);
DeclareProperty("IsLeftTransitive", IsFRBiset);
DeclareProperty("IsRightTransitive", IsFRBiset);
DeclareCategory("IsBisetBasis", IsBasis);
DeclareCategory("IsLeftBisetBasis", IsBisetBasis);
DeclareCategory("IsRightBisetBasis", IsBisetBasis);
DeclareAttribute("Basis", IsFRBiset);
DeclareAttribute("LeftBasis", IsFRBiset);
DeclareOperation("LeftBasis", [IsFRBiset,IsList]);
DeclareOperation("LeftBasis", [IsFRBiset,IsPerm,IsList]);
DeclareOperation("LeftBasis", [IsFRBiset,IsTransformation,IsList]);
DeclareOperation("LeftBasis", [IsFRBiset,IsList,IsList]);
DeclareAttribute("RightBasis", IsFRBiset);
DeclareAttribute("CanonicalBasis", IsFRBiset);
DeclareAttribute("WreathRecursion", IsFRBiset);
DeclareOperation("WreathRecursion", [IsFRBiset,IsLeftBisetBasis]);
DeclareAttribute("FRMachineOfBiset", IsFRBiset);
DeclareOperation("FRMachine", [IsFRBiset]);
DeclareOperation("FRMachine", [IsFRBiset,IsLeftBisetBasis]);
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