[Up] [Previous] [Index]

# B Filters built into the package

Many implications among properties of loops are built directly into LOOPS. A sizeable portion of these properties are of trivial character or are based on definitions (e.g., alternative loops = left alternative loops + right alternative loops). The remaining implications are theorems.

All filters of LOOPS are summarized below, using the GAP convention that the property on the left is implied by the property (properties) on the right.

    ( IsExtraLoop,  IsAssociative and IsLoop )
    ( IsDiassociative, IsAssociative and IsLoop )
    ( HasInverseProperty, HasRightInverseProperty and IsCommutative )
    ( HasInverseProperty, HasLeftInverseProperty and IsCommutative )
    ( IsMoufangLoop, IsRightBolLoop and IsCommutative )
    ( IsMoufangLoop, IsLeftBolLoop and IsCommutative )
    ( IsMoufangLoop, IsRightBruckLoop and IsCommutative )
    ( IsMoufangLoop, IsLeftBruckLoop and IsCommutative )
    ( IsRightNuclearSquareLoop, IsLeftNuclearSquareLoop and IsCommutative )
    ( IsLeftNuclearSquareLoop, IsRightNuclearSquareLoop and IsCommutative )
    ( HasAutomorphicInverseProperty, HasAntiautomorphicInverseProperty and IsCommutative )
    ( HasAntiautomorphicInverseProperty, HasAutomorphicInverseProperty and IsCommutative )
    ( IsAlternative, IsLeftAlternative and IsCommutative )
    ( IsAlternative, IsRightAlternative and IsCommutative )
    ( HasTwosidedInverses, IsPowerAssociative )
    ( IsPowerAssociative, IsDiassociative )
    ( IsAlternative, IsDiassociative )
    ( IsFlexible, IsDiassociative )
    ( HasLeftInverseProperty, HasInverseProperty )
    ( HasRightInverseProperty, HasInverseProperty )
    ( HasAntiautomoprhicInverseProperty, HasInverseProperty )
    ( HasWeakInverseProperty, HasInverseProperty )
    ( HasInverseProperty, HasLeftInverseProperty and HasRightInverseProperty )
    ( HasInverseProperty, HasLeftInverseProperty and HasWeakInverseProperty )
    ( HasInverseProperty, HasRightInverseProperty and HasWeakInverseProperty )
    ( HasInverseProperty, HasLeftInverseProperty and HasAntiautomorphicInverseProperty )
    ( HasInverseProperty, HasRightInverseProperty and HasAntiautomorphicInverseProperty )
    ( HasInverseProperty, HasWeakInverseProperty and HasAntiautomorphicInverseProperty )
    ( HasTwosidedInverses, HasAntiautomorphicInverseProperty )
    ( IsMoufangLoop, IsExtraLoop )
    ( IsNuclearSquareLoop, IsExtraLoop )
    ( IsCLoop, IsExtraLoop )
    ( IsExtraLoop, IsMoufangLoop and IsLeftNuclearSquareLoop )
    ( IsExtraLoop, IsMoufangLoop and IsMiddleNuclearSquareLoop )
    ( IsExtraLoop, IsMoufangLoop and IsRightNuclearSquareLoop )
    ( IsLeftBolLoop, IsMoufangLoop )
    ( IsRightBolLoop, IsMoufangLoop )
    ( IsFlexible, IsMoufangLoop )
    ( IsDiassociative, IsMoufangLoop )
    ( IsMoufangLoop, IsLeftBolLoop and IsRightBolLoop )
    ( IsLCLoop, IsCLoop )
    ( IsRCLoop, IsCLoop )
    ( IsCLoop, IsLCLoop and IsRCLoop )
    ( IsLeftAlternative, IsLeftBolLoop )
    ( HasTwosidedInverses, IsLeftBolLoop )
    ( IsRightAlternative, IsRightBolLoop )
    ( HasTwosidedInverses, IsRightBolLoop )
    ( IsLeftAlternative, IsLCLoop )
    ( IsLeftNuclearSquareLoop, IsLCLoop )
    ( IsMiddleNuclearSquareLoop, IsLCLoop )
    ( IsPowerAssociative, IsLCLoop )
    ( IsRightAlternative, IsRCLoop )
    ( IsRightNuclearSquareLoop, IsRCLoop )
    ( IsMiddleNuclearSquareLoop, IsRCLoop )
    ( IsPowerAssociative, IsRCLoop )
    ( IsLeftNuclearSquareLoop, IsNuclearSquareLoop )
    ( IsRightNuclearSquareLoop, IsNuclearSquareLoop )
    ( IsMiddleNuclearSquareLoop, IsNuclearSquareLoop )
    ( IsNuclearSquareLoop, IsLeftNuclearSquareLoop and IsRightNuclearSquareLoop
        and IsMiddleNuclearSquareLoop )
    ( IsLeftAlternative, IsAlternative )
    ( IsRightAlternative, IsAlternative )
    ( IsAlternative, IsLeftAlternative and IsRightAlternative )
    ( IsLCCLoop, IsCCLoop )
    ( IsRCCLoop, IsCCLoop )
    ( IsCCLoop, IsLCCLoop and IsRCCLoop )
    ( IsOsbornLoop, IsMoufangLoop )
    ( IsOsbornLoop, IsCCLoop )
    ( HasAutomorphicInverseProperty, IsLeftBruckLoop )
    ( IsLeftBolLoop, IsLeftBruckLoop )
    ( IsLeftBruckLoop, IsLeftBolLoop and HasAutomorphicInverseProperty )
    ( HasAutomorphicInverseProperty, IsRightBruckLoop )
    ( IsRightBolLoop, IsRightBruckLoop )
    ( IsRightBruckLoop, IsRightBolLoop and HasAutomorphicInverseProperty )
    ( IsCommutative, IsSteinerLoop )
    ( HasInverseProperty, IsSteinerLoop )

[Up] [Previous] [Index]