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<h3>B <span class="Heading">Filters</span></h3>

<p>Many implications among properties of loops are built directly into <strong class="pkg">LOOPS</strong>. A sizeable portion of these properties are of trivial character or are based on definitions (e.g., alternative loops <span class="SimpleMath">\(=\)</span> left alternative loops <span class="SimpleMath">\(+\)</span> right alternative loops). The remaining implications are theorems.</p>

<p>All filters of <strong class="pkg">LOOPS</strong> are summarized below, using the <strong class="pkg">GAP</strong> convention that the property on the left is implied by the property (properties) on the right. <br /> <br /> <code class="code">( IsExtraLoop, IsAssociative and IsLoop )</code> <br /> <code class="code">( IsExtraLoop, IsCodeLoop )</code> <br /> <code class="code">( IsCCLoop, IsCodeLoop )</code> <br /> <code class="code">( HasTwosidedInverses, IsPowerAssociative and IsLoop )</code> <br /> <code class="code">( IsPowerAlternative, IsDiassociative )</code> <br /> <code class="code">( IsFlexible, IsDiassociative )</code> <br /> <code class="code">( HasAntiautomorphicInverseProperty, HasAutomorphicInverseProperty and IsCommutative )</code> <br /> <code class="code">( HasAutomorphicInverseProperty, HasAntiautomorphicInverseProperty and IsCommutative )</code> <br /> <code class="code">( HasLeftInverseProperty, HasInverseProperty )</code> <br /> <code class="code">( HasRightInverseProperty, HasInverseProperty )</code> <br /> <code class="code">( HasWeakInverseProperty, HasInverseProperty )</code> <br /> <code class="code">( HasAntiautomorphicInverseProperty, HasInverseProperty )</code> <br /> <code class="code">( HasTwosidedInverses, HasAntiautomorphicInverseProperty )</code> <br /> <code class="code">( HasInverseProperty, HasLeftInverseProperty and IsCommutative )</code> <br /> <code class="code">( HasInverseProperty, HasRightInverseProperty and IsCommutative )</code> <br /> <code class="code">( HasInverseProperty, HasLeftInverseProperty and HasRightInverseProperty )</code> <br /> <code class="code">( HasInverseProperty, HasLeftInverseProperty and HasWeakInverseProperty )</code> <br /> <code class="code">( HasInverseProperty, HasRightInverseProperty and HasWeakInverseProperty )</code> <br /> <code class="code">( HasInverseProperty, HasLeftInverseProperty and HasAntiautomorphicInverseProperty )</code> <br /> <code class="code">( HasInverseProperty, HasRightInverseProperty and HasAntiautomorphicInverseProperty )</code> <br /> <code class="code">( HasInverseProperty, HasWeakInverseProperty and HasAntiautomorphicInverseProperty )</code> <br /> <code class="code">( HasTwosidedInverses, HasLeftInverseProperty )</code> <br /> <code class="code">( HasTwosidedInverses, HasRightInverseProperty )</code> <br /> <code class="code">( HasTwosidedInverses, IsFlexible and IsLoop )</code> <br /> <code class="code">( IsMoufangLoop, IsExtraLoop )</code> <br /> <code class="code">( IsCLoop, IsExtraLoop )</code> <br /> <code class="code">( IsExtraLoop, IsMoufangLoop and IsLeftNuclearSquareLoop )</code> <br /> <code class="code">( IsExtraLoop, IsMoufangLoop and IsMiddleNuclearSquareLoop )</code> <br /> <code class="code">( IsExtraLoop, IsMoufangLoop and IsRightNuclearSquareLoop )</code> <br /> <code class="code">( IsLeftBolLoop, IsMoufangLoop )</code> <br /> <code class="code">( IsRightBolLoop, IsMoufangLoop )</code> <br /> <code class="code">( IsDiassociative, IsMoufangLoop )</code> <br /> <code class="code">( IsMoufangLoop, IsLeftBolLoop and IsRightBolLoop )</code> <br /> <code class="code">( IsLCLoop, IsCLoop )</code> <br /> <code class="code">( IsRCLoop, IsCLoop )</code> <br /> <code class="code">( IsDiassociative, IsCLoop and IsFlexible)</code> <br /> <code class="code">( IsCLoop, IsLCLoop and IsRCLoop )</code> <br /> <code class="code">( IsRightBolLoop, IsLeftBolLoop and IsCommutative )</code> <br /> <code class="code">( IsLeftPowerAlternative, IsLeftBolLoop )</code> <br /> <code class="code">( IsLeftBolLoop, IsRightBolLoop and IsCommutative )</code> <br /> <code class="code">( IsRightPowerAlternative, IsRightBolLoop )</code> <br /> <code class="code">( IsLeftPowerAlternative, IsLCLoop )</code> <br /> <code class="code">( IsLeftNuclearSquareLoop, IsLCLoop )</code> <br /> <code class="code">( IsMiddleNuclearSquareLoop, IsLCLoop )</code> <br /> <code class="code">( IsRCLoop, IsLCLoop and IsCommutative )</code> <br /> <code class="code">( IsRightPowerAlternative, IsRCLoop )</code> <br /> <code class="code">( IsRightNuclearSquareLoop, IsRCLoop )</code> <br /> <code class="code">( IsMiddleNuclearSquareLoop, IsRCLoop )</code> <br /> <code class="code">( IsLCLoop, IsRCLoop and IsCommutative )</code> <br /> <code class="code">( IsRightNuclearSquareLoop, IsLeftNuclearSquareLoop and IsCommutative )</code> <br /> <code class="code">( IsLeftNuclearSquareLoop, IsRightNuclearSquareLoop and IsCommutative )</code> <br /> <code class="code">( IsLeftNuclearSquareLoop, IsNuclearSquareLoop )</code> <br /> <code class="code">( IsRightNuclearSquareLoop, IsNuclearSquareLoop )</code> <br /> <code class="code">( IsMiddleNuclearSquareLoop, IsNuclearSquareLoop )</code> <br /> <code class="code">( IsNuclearSquareLoop, IsLeftNuclearSquareLoop and IsRightNuclearSquareLoop</code> <br /> <code class="code"> and IsMiddleNuclearSquareLoop )</code> <br /> <code class="code">( IsFlexible, IsCommutative )</code> <br /> <code class="code">( IsRightAlternative, IsLeftAlternative and IsCommutative )</code> <br /> <code class="code">( IsLeftAlternative, IsRightAlternative and IsCommutative )</code> <br /> <code class="code">( IsLeftAlternative, IsAlternative )</code> <br /> <code class="code">( IsRightAlternative, IsAlternative )</code> <br /> <code class="code">( IsAlternative, IsLeftAlternative and IsRightAlternative )</code> <br /> <code class="code">( IsLeftAlternative, IsLeftPowerAlternative )</code> <br /> <code class="code">( HasLeftInverseProperty, IsLeftPowerAlternative )</code> <br /> <code class="code">( IsPowerAssociative, IsLeftPowerAlternative )</code> <br /> <code class="code">( IsRightAlternative, IsRightPowerAlternative )</code> <br /> <code class="code">( HasRightInverseProperty, IsRightPowerAlternative )</code> <br /> <code class="code">( IsPowerAssociative, IsRightPowerAlternative )</code> <br /> <code class="code">( IsLeftPowerAlternative, IsPowerAlternative )</code> <br /> <code class="code">( IsRightPowerAlternative, IsPowerAlternative )</code> <br /> <code class="code">( IsAssociative, IsLCCLoop and IsCommutative )</code> <br /> <code class="code">( IsExtraLoop, IsLCCLoop and IsMoufangLoop )</code> <br /> <code class="code">( IsAssociative, IsRCCLoop and IsCommutative )</code> <br /> <code class="code">( IsExtraLoop, IsRCCLoop and IsMoufangLoop )</code> <br /> <code class="code">( IsLCCLoop, IsCCLoop )</code> <br /> <code class="code">( IsRCCLoop, IsCCLoop )</code> <br /> <code class="code">( IsCCLoop, IsLCCLoop and IsRCCLoop )</code> <br /> <code class="code">( IsOsbornLoop, IsMoufangLoop )</code> <br /> <code class="code">( IsOsbornLoop, IsCCLoop )</code> <br /> <code class="code">( HasAutomorphicInverseProperty, IsLeftBruckLoop )</code> <br /> <code class="code">( IsLeftBolLoop, IsLeftBruckLoop )</code> <br /> <code class="code">( IsRightBruckLoop, IsLeftBruckLoop and IsCommutative )</code> <br /> <code class="code">( IsLeftBruckLoop, IsLeftBolLoop and HasAutomorphicInverseProperty )</code> <br /> <code class="code">( HasAutomorphicInverseProperty, IsRightBruckLoop )</code> <br /> <code class="code">( IsRightBolLoop, IsRightBruckLoop )</code> <br /> <code class="code">( IsLeftBruckLoop, IsRightBruckLoop and IsCommutative )</code> <br /> <code class="code">( IsRightBruckLoop, IsRightBolLoop and HasAutomorphicInverseProperty )</code> <br /> <code class="code">( IsCommutative, IsSteinerLoop )</code> <br /> <code class="code">( IsCLoop, IsSteinerLoop )</code> <br /> <code class="code">( IsLeftAutomorphicLoop, IsAutomorphicLoop )</code> <br /> <code class="code">( IsRightAutomorphicLoop, IsAutomorphicLoop )</code> <br /> <code class="code">( IsMiddleAutomorphicLoop, IsAutomorphicLoop )</code> <br /> <code class="code">( IsLeftAutomorphicLoop, IsRightAutomorphicLoop and HasAntiautomorphicInverseProperty )</code> <br /> <code class="code">( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and HasAntiautomorphicInverseProperty )</code> <br /> <code class="code">( IsFlexible, IsMiddleAutomorphicLoop )</code> <br /> <code class="code">( HasAntiautomorphicInverseProperty, IsFlexible and IsLeftAutomorphicLoop )</code> <br /> <code class="code">( HasAntiautomorphicInverseProperty, IsFlexible and IsRightAutomorphicLoop )</code> <br /> <code class="code">( IsMoufangLoop, IsAutomorphicLoop and IsLeftAlternative )</code> <br /> <code class="code">( IsMoufangLoop, IsAutomorphicLoop and IsRightAlternative )</code> <br /> <code class="code">( IsMoufangLoop, IsAutomorphicLoop and HasLeftInverseProperty )</code> <br /> <code class="code">( IsMoufangLoop, IsAutomorphicLoop and HasRightInverseProperty )</code> <br /> <code class="code">( IsMoufangLoop, IsAutomorphicLoop and HasWeakInverseProperty )</code> <br /> <code class="code">( IsMiddleAutomorphicLoop, IsCommutative )</code> <br /> <code class="code">( IsLeftAutomorphicLoop, IsLeftBruckLoop )</code> <br /> <code class="code">( IsLeftAutomorphicLoop, IsLCCLoop )</code> <br /> <code class="code">( IsRightAutomorphicLoop, IsRightBruckLoop )</code> <br /> <code class="code">( IsRightAutomorphicLoop, IsRCCLoop )</code> <br /> <code class="code">( IsAutomorphicLoop, IsCommutative and IsMoufangLoop )</code> <br /> <code class="code">( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsMiddleAutomorphicLoop )</code> <br /> <code class="code">( IsAutomorphicLoop, IsRightAutomorphicLoop and IsMiddleAutomorphicLoop )</code> <br /> <code class="code">( IsAutomorphicLoop, IsAssociative )</code></p>


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