<h3>4 <span class="Heading">Ideals and left ideals</span></h3>
<p>In this section we describe several functions related to ideals and left ideals of skew braces. References: <a href="chapBib.html#biBMR3647970">[GV17]</a> and <a href="chapBib.html#biBMR3763907">[SV18]</a>.</p>
<p>An left ideal <span class="Math">I</span> of a skew brace <span class="Math">A</span> is a subgroup <span class="Math">I</span> of the additive group of <span class="Math">A</span> such that <span class="Math">\lambda_a(I)\subseteq I</span> for all <span class="Math">a\in A</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list with the left ideals of the skew brace <var class="Arg">obj</var></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StrongLeftIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list with the left ideals of the skew brace <var class="Arg">obj</var> that are normal in the additive group of <span class="Math">A</span></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(24,12);</span>
<skew brace of size 24>
<span class="GAPprompt">gap></span> <span class="GAPinput">strong_left_ideals := StrongLeftIdeals(br);</span>
[ <left ideal in <skew brace of size 24>, (size 24)>,
<left ideal in <skew brace of size 24>, (size 12)>,
<left ideal in <skew brace of size 24>, (size 6)>,
<left ideal in <skew brace of size 24>, (size 4)>,
<left ideal in <skew brace of size 24>, (size 2)>,
<left ideal in <skew brace of size 24>, (size 3)>,
<left ideal in <skew brace of size 24>, (size 1)> ]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftIdeal</code>( <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the subset is a left ideal of <var class="Arg">obj</var></p>
<p>An ideal <span class="Math">I</span> of a skew brace <span class="Math">A</span> is a normal subgroup <span class="Math">I</span> of the additive group of <span class="Math">A</span> such that <span class="Math">\lambda_a(I)\subseteq I</span> and <span class="Math">a\circ I=I\circ a</span> for all <span class="Math">a\in A</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIdeal</code>( <var class="Arg">obj</var>, <var class="Arg">subset</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the <var class="Arg">subset</var> is a left ideal of <var class="Arg">obj</var></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ideals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list with the ideals of the skew brace <var class="Arg">obj</var></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdealGeneratedBy</code>( <var class="Arg">obj</var>, <var class="Arg">subset</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the ideal of <var class="Arg">obj</var> generated by the given <var class="Arg">subset</var></p>
<p>The ideal of a skew brace <span class="Math">A</span> generated by a subset <span class="Math">X</span> is the intersection of all the ideals of <span class="Math">A</span> containing <span class="Math">X</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftSeries</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the left ideals of the left series of <var class="Arg">obj</var></p>
<p>The left series of a skew brace <span class="Math">A</span> is defined recursively as <span class="Math">A^1=A</span> and <span class="Math">A^{n+1}=A*A^n</span> for <span class="Math">n\geq1</span>, where <span class="Math">a*b=\lambda_a(b)-b</span>. Each <span class="Math">A^n</span> is a left ideal.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(8,20);</span>
<skew brace of size 8>
<span class="GAPprompt">gap></span> <span class="GAPinput">LeftSeries(br);</span>
[ <skew brace of size 8>,
<left ideal in <skew brace of size 8>, (size 2)>,
<left ideal in <skew brace of size 8>, (size 1)> ]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightSeries</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the ideals of the right series of <var class="Arg">obj</var></p>
<p>The right series of a skew brace 0<span class="Math">A</span> is defined recursively as <span class="Math">A^{(1)}=A</span> and <span class="Math">A^{(n+1)}=A*A^{(n)}</span> for <span class="Math">n\geq1</span>, where <span class="Math">a*b=\lambda_a(b)-b</span></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallSkewbrace(8,20);</span>
<skew brace of size 8>
<span class="GAPprompt">gap></span> <span class="GAPinput">RightSeries(br);</span>
[ <ideal in <skew brace of size 8>, (size 8)>,
<ideal in <skew brace of size 8>, (size 2)>,
<ideal in <skew brace of size 8>, (size 1)> ]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftNilpotent</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is left nilpotent.</p>
<p>A skew brace <span class="Math">A</span> is said to be left nilpotent if there exists <span class="Math">n\geq1</span> such that <span class="Math">A^n=0</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSimpleSkewbrace</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is simple.</p>
<p>A skew brace <span class="Math">A</span> is said to be simple if <span class="Math">\{0\}</span> and <span class="Math">A</span> are its only ideals.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightNilpotent</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is right nilpotent.</p>
<p>A skew brace <span class="Math">A</span> is said to be right nilpotent if there exists <span class="Math">n\geq1</span> such that <span class="Math">A^{(n)}=0</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftNilpotentIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the list of right or left nilpotent ideals of <var class="Arg">obj</var></p>
<p>An ideal <span class="Math">I</span> of a skew brace <span class="Math">A</span> is said to be left if it is left nilpotent as a skew brace.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightNilpotentIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the list of right or left nilpotent ideals of <var class="Arg">obj</var></p>
<p>An ideal <span class="Math">I</span> of a skew brace <span class="Math">A</span> is said to be right nilpotent if An ideal <span class="Math">I</span> of a skew brace <span class="Math">A</span> is said to be left if it is right nilpotent as a skew brace.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmoktunowiczSeries</code>( <var class="Arg">obj</var>, <var class="Arg">bound</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a list of <var class="Arg">bound</var> left ideals of the Smoktunowicz's series of obj
<p>The Smoktunowicz's series of a skew brace A is defined recursively as A^{[1]}=Aand A^{[n+1]} is the additive subgroup of A generated by A^{[i]}*A^{[n+1-i]} for 1\leq i+j\leq n+1, where a*b=\lambda_a(b)-b.
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">br := SmallBrace(16,145);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmoktunowiczSeries(br,4);</span>
[ <brace of size 16>, <brace of size 8>, <brace of size 4>, <brace of size 2>,
<brace of size 2> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SmoktunowiczSeries(br,5);</span>
[ <brace of size 16>, <brace of size 8>, <brace of size 4>, <brace of size 2>,
<brace of size 2>, <brace of size 1> ]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SocleSeries</code>( <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the socle series of <var class="Arg">obj</var></p>
<p>The socle series of a skew brace <span class="Math">A</span> is defined recursively as <span class="Math">A_1=A</span> and <span class="Math">A_{n+1}=A_n/\mathrm{Soc}(A_n)</span>, see <a href="chapBib.html#biBMR3763907">[SV18]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MultipermutationLevel</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the multipermutation level of the skew brace <var class="Arg">obj</var></p>
<p>The multipermutation level of a skew brace <span class="Math">A</span> is defined as the smallest positive integer <span class="Math">n</span> such that the <span class="Math">n</span>-th term <span class="Math">A_n</span> of the socle series has only one element, see Definition 5.17 of <a href="chapBib.html#biBMR3763907">[SV18]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KernelOfLambda</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the kernel of the map <span class="Math">\lambda</span> as a subset of elements of the skew brace <var class="Arg">obj</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPrimeBrace</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is prime</p>
<p>A skew brace <span class="Math">A</span> is said to be prime if for all non-zero ideals <span class="Math">I</span> and <span class="Math">J</span> one has <span class="Math">I*J\ne 0</span></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPrimeIdeal</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the ideal <var class="Arg">obj</var> is prime</p>
<p>An ideal <span class="Math">I</span> of a skew brace <span class="Math">A</span> is said to be prime if <span class="Math">A/I</span> is a prime skew brace.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimeIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the list of prime ideals of the skew brace <var class="Arg">obj</var></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemiprime</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is semiprime</p>
<p>An ideal <span class="Math">I</span> of a skew brace <span class="Math">A</span> is said to be semiprime if <span class="Math">A/I</span> is a semiprime skew brace.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemiprimeIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the list of semiprime ideals of the skew brace <var class="Arg">obj</var></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBaer</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace <var class="Arg">obj</var> is ia Baer radical skew brace.</p>
<p>A skew brace <span class="Math">A</span> is said to be Baer radical if <span class="Math">A=B(A)</span>, where <span class="Math">B(A)</span> is the Baer radical of <span class="Math">A</span> (i.e., the intersection of all prime ideals of <span class="Math">A</span>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SolvableSeries</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list with the solvable series of the skew brace <var class="Arg">obj</var></p>
<p>The solvable series of a skew brace <span class="Math">A</span> is defined recursively as <span class="Math">A_{1}=A</span> and <span class="Math">A_{n+1}=A_{n}*A_{n}</span> for <span class="Math">n\geq1</span>, where <span class="Math">a*b=\lambda_a(b)-b</span></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMinimalIdeal</code>( <var class="Arg">obj</var>, <var class="Arg">ideal</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if <var class="Arg">ideal</var> is a minimal ideal of <var class="Arg">obj</var> An ideal <span class="Math">I</span> of <span class="Math">A</span> is said to be <em>minimal</em> if does not contain any other ideal of <span class="Math">A</span>. To check if an ideal <span class="Math">I</span> of <span class="Math">A</span> is minimal, one computes the ideals of <span class="Math">I</span> and keep only those that are simple as a skew brace.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalIdeals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of minimal ideals of the skew brace <var class="Arg">obj</var></p>
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