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<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >script type="text/javascript" "https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML " >script >title >GAP (Loops) - Appendix B: Filters</title >meta http-equiv="content-type" content="text/html; charset=UTF-8" />meta name="generator" content="GAPDoc2HTML" />link rel="stylesheet" type="text/css" href="manual.css" />script src="manual.js" type="text/javascript" ></script >script type="text/javascript" >overwriteStyle();</script >head >body class="chapB" onload="jscontent()" >div class="chlinktop" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chap5_mj.html" >5</a> <a href="chap6_mj.html" >6</a> <a href="chap7_mj.html" >7</a> <a href="chap8_mj.html" >8</a> <a href="chap9_mj.html" >9</a> <a href="chapA_mj.html" >A</a> <a href="chapB_mj.html" >B</a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" >Ind</a> </div >div class="chlinkprevnexttop" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chapA_mj.html" >[Previous Chapter]</a> <a href="chapBib_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chapB.html" >[MathJax off]</a></p>"X84EFA4C07D4277BB" name="X84EFA4C07D4277BB" ></a></p>div class="ChapSects" ><a href="chapB_mj.html#X84EFA4C07D4277BB" >B <span class="Heading" >Filters</span ></a>div >span class="Heading" >Filters</span ></h3>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>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 "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 "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 "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" 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>div class="chlinkprevnextbot" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chapA_mj.html" >[Previous Chapter]</a> <a href="chapBib_mj.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a 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