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<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>
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<div class="ChapSects"><a href="chapB_mj.html#X84EFA4C07D4277BB">B <span class="Heading">Filters</span></a>
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<h3>B <span class="Heading">Filters</span></h3>
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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>
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<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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