\Chapter{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.

\begintt
( 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 )
\endtt
