## bytable.gd RAQ Quasigroups, racks, and quandles by multiplication tables.

## Self-distributivity
# Predicates on tables:
DeclareProperty( "IsLeftSelfDistributiveTable", IsMatrix );
DeclareProperty( "IsRightSelfDistributiveTable", IsMatrix );
DeclareProperty( "IsElementwiseIdempotentTable", IsMatrix );

## Attributes (typically used on the families of elements created from a
## multiplication table) to store sections and division tables
DeclareAttribute("LeftPerms", HasMultiplicationTable);
DeclareAttribute("RightPerms", HasMultiplicationTable);
DeclareAttribute("LeftDivisionTable", HasMultiplicationTable);
DeclareAttribute("RightDivisionTable", HasMultiplicationTable);

## Create Quasigroups and racks from multiplication tables
DeclareGlobalFunction("LeftQuasigroupByMultiplicationTable");
DeclareGlobalFunction("LeftQuasigroupByMultiplicationTableNC");
DeclareGlobalFunction("LeftRackByMultiplicationTable");
DeclareGlobalFunction("LeftRackByMultiplicationTableNC");
DeclareGlobalFunction("LeftQuandleByMultiplicationTable");
DeclareGlobalFunction("LeftQuandleByMultiplicationTableNC");
DeclareGlobalFunction("RightQuasigroupByMultiplicationTable");
DeclareGlobalFunction("RightQuasigroupByMultiplicationTableNC");
DeclareGlobalFunction("RightRackByMultiplicationTable");
DeclareGlobalFunction("RightRackByMultiplicationTableNC");
DeclareGlobalFunction("RightQuandleByMultiplicationTable");
DeclareGlobalFunction("RightQuandleByMultiplicationTableNC");

## Property of a collection that its elements know their multiplication table
DeclareProperty("IsBuiltFromMultiplicationTable", IsCollection);

## Create quasigroups etc by permutations
DeclareGlobalFunction("LeftQuasigroupByPerms");
DeclareGlobalFunction("LeftRackByPerms");
DeclareGlobalFunction("LeftRackByPermsNC");
DeclareGlobalFunction("LeftQuandleByPerms");
DeclareGlobalFunction("LeftQuandleByPermsNC");
DeclareGlobalFunction("RightQuasigroupByPerms");
DeclareGlobalFunction("RightRackByPerms");
DeclareGlobalFunction("RightRackByPermsNC");
DeclareGlobalFunction("RightQuandleByPerms");
DeclareGlobalFunction("RightQuandleByPermsNC");