82 lines
3.4 KiB
GDScript3
82 lines
3.4 KiB
GDScript3
## structure.gd RAQ Definitions, representations, and elementary operations.
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## GAP Categories and representations
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## Self-distributivity
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# Note these are properties that should be defined just at the level of
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# MultiplicativeElements and Magmas, hence although the LOOPS package defines
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# IsLDistributive and IsRDistributive for quasigroups, they would be ambiguous
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# in the case of something like a semiring whose multiplicative structure was
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# a quasigroup (cf. https://arxiv.org/abs/0910.4760). Hence, we implement them
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# in RAQ with new, non-conflicting terms.
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# An element that knows that multiplication in its family is left
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# self-distributive:
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DeclareCategory("IsLSelfDistElement", IsMultiplicativeElement);
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DeclareCategoryCollections("IsLSelfDistElement");
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# An element that knows that multiplication in its family is right
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# self-distributive:
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DeclareCategory("IsRSelfDistElement", IsMultiplicativeElement);
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DeclareCategoryCollections("IsRSelfDistElement");
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# Left self-distributive magmas:
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DeclareProperty("IsLSelfDistributive", IsMagma);
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InstallTrueMethod(IsLSelfDistributive,
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IsMagma and IsLSelfDistElementCollection);
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# Right self-distributive magmas:
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DeclareProperty("IsRSelfDistributive", IsMagma);
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InstallTrueMethod(IsRSelfDistributive,
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IsMagma and IsRSelfDistElementCollection);
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# Left and right racks
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DeclareSynonym("IsLeftRack", IsLeftQuasigroup and IsLSelfDistributive);
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DeclareSynonym("IsRightRack", IsRightQuasigroup and IsRSelfDistributive);
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## One-sided quasigroups
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# Returns the closure of <gens> under * and LeftQuotient;
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# the family of elements of M may be specified, and must be if <gens>
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# is empty (in which case M will be empty as well).
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DeclareGlobalFunction("LeftQuasigroup");
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DeclareGlobalFunction("RightQuasigroup");
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DeclareGlobalFunction("LeftRack");
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DeclareGlobalFunction("RightRack");
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# Underlying operation
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DeclareGlobalFunction("CloneOfTypeByGenerators");
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# Attributes for the generators
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# Generates the structure by \* and LeftQuotient. Note that for finite
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# structures, these are the same as the GeneratorsOfMagma but in general more
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# elements might be required to generate the structure just under *
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DeclareAttribute("GeneratorsOfLeftQuasigroup", IsLeftQuasigroup);
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InstallImmediateMethod(GeneratorsOfMagma,
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"finite left quasigroups *-generated by left quasigroup generators",
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IsLeftQuasigroup and IsFinite,
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q -> GeneratorsOfLeftQuasigroup(q)
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);
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# Generates the structure by \* and \/, same considerations as above
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DeclareAttribute("GeneratorsOfRightQuasigroup", IsRightQuasigroup);
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InstallImmediateMethod(GeneratorsOfMagma,
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"finite right quasigroups *-generated by right quasigroup generators",
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IsRightQuasigroup and IsFinite,
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q -> GeneratorsOfRightQuasigroup(q)
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);
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## And now, build them from multiplication tables
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# Need attributes on the families to store the sections and division tables
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DeclareAttribute("LeftSectionList", IsFamily and HasMultiplicationTable);
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DeclareAttribute("RightSectionList", IsFamily and HasMultiplicationTable);
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DeclareAttribute("LeftDivisionTable", IsFamily and HasMultiplicationTable);
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DeclareAttribute("RightDivisionTable", IsFamily and HasMultiplicationTable);
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# And the builders
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DeclareGlobalFunction("LeftQuasigroupByMultiplicationTable");
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DeclareGlobalFunction("LeftRackByMultiplicationTable");
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DeclareGlobalFunction("RightQuasigroupByMultiplicationTable");
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DeclareGlobalFunction("RightRackByMultiplicationTable");
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