91 lines
3.6 KiB
GDScript3
91 lines
3.6 KiB
GDScript3
## structure.gd RAQ Definitions, generation, and elementary ops and props.
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## GAP Categories and representations
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## Self-distributivity
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# Note these are properties that can and therefore should be defined just at
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# the level of MultiplicativeElements and Magmas, hence although the LOOPS
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# package defines IsLDistributive and IsRDistributive for quasigroups, they
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# would be ambiguous in the case of something like a semiring whose
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# multiplicative structure was a quasigroup
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# (cf. https://arxiv.org/abs/0910.4760). Hence, we implement them in RAQ with
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# 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 collections of elements:
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DeclareProperty("IsLSelfDistributive", IsMultiplicativeElementCollection);
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InstallTrueMethod(IsLSelfDistributive, IsLSelfDistElementCollection);
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# Right self-distributive collections of elements:
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DeclareProperty("IsRSelfDistributive", IsMultiplicativeElementCollection);
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InstallTrueMethod(IsRSelfDistributive, IsRSelfDistElementCollection);
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## Idempotence
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# There is already a property IsIdempotent on elements, but to definw
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# structures which will automatically be quandles we need a corresponding
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# collections category:
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DeclareCategoryCollections("IsIdempotent");
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# Collections in which every element is idempotent
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DeclareProperty("IsElementwiseIdempotent", IsMultiplicativeElementCollection);
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InstallTrueMethod(IsElementwiseIdempotent, IsIdempotentCollection);
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## Left and right racks and quandles
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DeclareSynonym("IsLeftRack", IsLeftQuasigroup and IsLSelfDistributive);
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DeclareSynonym("IsRightRack", IsRightQuasigroup and IsRSelfDistributive);
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DeclareSynonym("IsLeftQuandle", IsLeftRack and IsElementwiseIdempotent);
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DeclareSynonym("IsRightQuandle", IsRightRack and IsElementwiseIdempotent);
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## One-sided quasigroups and racks and quandles by generators
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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("LeftQuasigroupNC");
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DeclareGlobalFunction("RightQuasigroup");
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DeclareGlobalFunction("RightQuasigroupNC");
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DeclareGlobalFunction("LeftRack");
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DeclareGlobalFunction("LeftRackNC");
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DeclareGlobalFunction("RightRack");
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DeclareGlobalFunction("RightRackNC");
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DeclareGlobalFunction("LeftQuandle");
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DeclareGlobalFunction("LeftQuandleNC");
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DeclareGlobalFunction("RightQuandle");
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DeclareGlobalFunction("RightQuandleNC");
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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",
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IsLeftQuasigroup and IsFinite,
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1,
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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",
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IsRightQuasigroup and IsFinite,
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2,
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q -> GeneratorsOfRightQuasigroup(q)
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);
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