\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 )
( IsExtraLoop, IsCodeLoop )
( IsCCLoop, IsCodeLoop )
( HasTwosidedInverses, IsPowerAssociative )
( IsPowerAlternative, IsDiassociative )
( IsFlexible, IsDiassociative )
( HasAntiautomorphicInverseProperty, HasAutomorphicInverseProperty and IsCommutative )
( HasAutomorphicInverseProperty, HasAntiautomorphicInverseProperty and IsCommutative )
( HasLeftInverseProperty, HasInverseProperty )
( HasRightInverseProperty, HasInverseProperty )
( HasWeakInverseProperty, HasInverseProperty )
( HasAntiautomorphicInverseProperty, HasInverseProperty )
( HasTwosidedInverses, HasAntiautomorphicInverseProperty )
( HasInverseProperty, HasLeftInverseProperty and IsCommutative )
( HasInverseProperty, HasRightInverseProperty and IsCommutative )
( 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, HasLeftInverseProperty )
( HasTwosidedInverses, HasRightInverseProperty )
( HasTwosidedInverses, IsFlexible and IsLoop )
( IsMoufangLoop, IsExtraLoop )
( IsCLoop, IsExtraLoop )
( IsExtraLoop, IsMoufangLoop and IsLeftNuclearSquareLoop )
( IsExtraLoop, IsMoufangLoop and IsMiddleNuclearSquareLoop )
( IsExtraLoop, IsMoufangLoop and IsRightNuclearSquareLoop )
( IsLeftBolLoop, IsMoufangLoop )
( IsRightBolLoop, IsMoufangLoop )
( IsDiassociative, IsMoufangLoop )
( IsMoufangLoop, IsLeftBolLoop and IsRightBolLoop )
( IsLCLoop, IsCLoop )
( IsRCLoop, IsCLoop )
( IsDiassociative, IsCLoop and IsFlexible)
( IsCLoop, IsLCLoop and IsRCLoop )
( IsRightBolLoop, IsLeftBolLoop and IsCommutative )
( IsLeftPowerAlternative, IsLeftBolLoop )
( IsLeftBolLoop, IsRightBolLoop and IsCommutative )
( IsRightPowerAlternative, IsRightBolLoop )
( IsLeftPowerAlternative, IsLCLoop )
( IsLeftNuclearSquareLoop, IsLCLoop )
( IsMiddleNuclearSquareLoop, IsLCLoop )
( IsRCLoop, IsLCLoop and IsCommutative )
( IsRightPowerAlternative, IsRCLoop )
( IsRightNuclearSquareLoop, IsRCLoop )
( IsMiddleNuclearSquareLoop, IsRCLoop )
( IsLCLoop, IsRCLoop and IsCommutative )
( IsRightNuclearSquareLoop, IsLeftNuclearSquareLoop and IsCommutative )
( IsLeftNuclearSquareLoop, IsRightNuclearSquareLoop and IsCommutative )
( IsLeftNuclearSquareLoop, IsNuclearSquareLoop )
( IsRightNuclearSquareLoop, IsNuclearSquareLoop )
( IsMiddleNuclearSquareLoop, IsNuclearSquareLoop )
( IsNuclearSquareLoop, IsLeftNuclearSquareLoop
    and IsRightNuclearSquareLoop and IsMiddleNuclearSquareLoop )
( IsFlexible, IsCommutative )
( IsRightAlternative, IsLeftAlternative and IsCommutative )
( IsLeftAlternative, IsRightAlternative and IsCommutative )
( IsLeftAlternative, IsAlternative )
( IsRightAlternative, IsAlternative )
( IsAlternative, IsLeftAlternative and IsRightAlternative )
( IsLeftAlternative, IsLeftPowerAlternative )
( HasLeftInverseProperty, IsLeftPowerAlternative )
( IsPowerAssociative, IsLeftPowerAlternative )
( IsRightAlternative, IsRightPowerAlternative )
( HasRightInverseProperty, IsRightPowerAlternative )
( IsPowerAssociative, IsRightPowerAlternative )
( IsLeftPowerAlternative, IsPowerAlternative )
( IsRightPowerAlternative, IsPowerAlternative )
( IsAssociative, IsLCCLoop and IsCommutative )
( IsExtraLoop, IsLCCLoop and IsMoufangLoop )
( IsAssociative, IsRCCLoop and IsCommutative )
( IsExtraLoop, IsRCCLoop and IsMoufangLoop )
( IsLCCLoop, IsCCLoop )
( IsRCCLoop, IsCCLoop )
( IsCCLoop, IsLCCLoop and IsRCCLoop )
( IsOsbornLoop, IsMoufangLoop )
( IsOsbornLoop, IsCCLoop )
( HasAutomorphicInverseProperty, IsLeftBruckLoop )
( IsLeftBolLoop, IsLeftBruckLoop )
( IsRightBruckLoop, IsLeftBruckLoop and IsCommutative )
( IsLeftBruckLoop, IsLeftBolLoop and HasAutomorphicInverseProperty )
( HasAutomorphicInverseProperty, IsRightBruckLoop )
( IsRightBolLoop, IsRightBruckLoop )
( IsLeftBruckLoop, IsRightBruckLoop and IsCommutative )
( IsRightBruckLoop, IsRightBolLoop and HasAutomorphicInverseProperty )
( IsCommutative, IsSteinerLoop )
( IsCLoop, IsSteinerLoop )
( IsLeftAutomorphicLoop, IsAutomorphicLoop )
( IsRightAutomorphicLoop, IsAutomorphicLoop )
( IsMiddleAutomorphicLoop, IsAutomorphicLoop )
( IsMiddleAutomorphicLoop, IsCommutative )
( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsCommutative )
( IsAutomorphicLoop, IsRightAutomorphicLoop and IsCommutative )
( IsLeftAutomorphicLoop, IsRightAutomorphicLoop and HasAntiautomorphicInverseProperty )
( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and HasAntiautomorphicInverseProperty )
( IsFlexible, IsMiddleAutomorphicLoop )
( HasAntiautomorphicInverseProperty, IsFlexible and IsLeftAutomorphicLoop )
( HasAntiautomorphicInverseProperty, IsFlexible and IsRightAutomorphicLoop )
( IsMoufangLoop, IsAutomorphicLoop and IsLeftAlternative )
( IsMoufangLoop, IsAutomorphicLoop and IsRightAlternative )
( IsMoufangLoop, IsAutomorphicLoop and HasLeftInverseProperty )
( IsMoufangLoop, IsAutomorphicLoop and HasRightInverseProperty )
( IsMoufangLoop, IsAutomorphicLoop and HasWeakInverseProperty )
( IsLeftAutomorphicLoop, IsLeftBruckLoop )
( IsLeftAutomorphicLoop, IsLCCLoop )
( IsRightAutomorphicLoop, IsRightBruckLoop )
( IsRightAutomorphicLoop, IsRCCLoop )
( IsAutomorphicLoop, IsCommutative and IsMoufangLoop )
\endtt
