\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 ) ( IsLeftALoop, IsALoop ) ( IsRightALoop, IsALoop ) ( IsMiddleALoop, IsALoop ) ( IsMiddleALoop, IsCommutative ) ( IsALoop, IsLeftALoop and IsCommutative ) ( IsALoop, IsRightALoop and IsCommutative ) ( IsLeftALoop, IsRightALoop and HasAntiautomorphicInverseProperty ) ( IsRightALoop, IsLeftALoop and HasAntiautomorphicInverseProperty ) ( IsFlexible, IsMiddleALoop ) ( HasAntiautomorphicInverseProperty, IsFlexible and IsLeftALoop ) ( HasAntiautomorphicInverseProperty, IsFlexible and IsRightALoop ) ( IsMoufangLoop, IsALoop and IsLeftAlternative ) ( IsMoufangLoop, IsALoop and IsRightAlternative ) ( IsMoufangLoop, IsALoop and HasLeftInverseProperty ) ( IsMoufangLoop, IsALoop and HasRightInverseProperty ) ( IsMoufangLoop, IsALoop and HasWeakInverseProperty ) ( IsLeftALoop, IsLeftBruckLoop ) ( IsLeftALoop, IsLCCLoop ) ( IsRightALoop, IsRightBruckLoop ) ( IsRightALoop, IsRCCLoop ) ( IsALoop, IsCommutative and IsMoufangLoop ) \endtt