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< 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 >
2017-10-30 03:54:13 +00:00
< 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" > ( IsRightNuclearSquareLoo
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