https://wiki.lojban.io/index.php?action=history&feed=atom&title=socnrpanrnji%27akobi 2026-05-04T14:35:18Z Revision history for this page on the wiki MediaWiki 1.45.3 https://wiki.lojban.io/index.php?title=socnrpanrnji%27akobi&diff=17160&oldid=prev 2026-01-13T16:38:33Z

<p>Import words via API</p> <p><b>New page</b></p><div>==== valsi ====<br /> socnrpanrnji&#039;akobi<br /> ==== type ====<br /> fu&#039;ivla<br /> ==== creator ====<br /> [[personal/krtisfranks|krtisfranks]]<br /> ==== time entered ====<br /> Wed Feb 15 17:13:45 2023<br />  <br /> == English ==<br /> === Definition #73146 - Preferred ===<br />  <br /> ==== definition ====<br /> x&lt;sub&gt;1&lt;/sub&gt; is a binary operator in structure x&lt;sub&gt;2&lt;/sub&gt; which exhibits the Jacobi property with respect to binary operator x&lt;sub&gt;3&lt;/sub&gt; (which also endows x&lt;sub&gt;2&lt;/sub&gt;) and element/object x&lt;sub&gt;4&lt;/sub&gt; (which is an element of the underlying set which form x&lt;sub&gt;2&lt;/sub&gt;).<br /> ==== notes ====<br /> <br /> This word uses a classifier which involves experimental gismu &quot;[[socni|socni]]&quot; as a veljvo. Let &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;&lt;/span&gt; be denoted by &quot;&lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;f&lt;/i&gt;&lt;/span&gt;&quot;, &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/span&gt; be denoted by &quot;X&quot;, &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt;&lt;/span&gt; be denoted by &quot;&lt;span class=&quot;MATH&quot;&gt;+&lt;/span&gt;&quot;, and &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;4&lt;/sub&gt;&lt;/span&gt; be denoted by &quot;e&quot;. Then &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;f&lt;/i&gt;&lt;/span&gt; exhibits the Jacobi property iff, for any &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;, &lt;i&gt;y&lt;/i&gt;, &lt;i&gt;z&lt;/i&gt;&lt;/span&gt; in &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;X&lt;/i&gt;&lt;/span&gt;, the following is true: MATH<br /> $f(x, f(y, z)) + f(z, f(x, y)) + f(y, f(z, x)) = e$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;f&lt;/i&gt; (&lt;i&gt;x&lt;/i&gt;, &lt;i&gt;f&lt;/i&gt; (&lt;i&gt;y&lt;/i&gt;, &lt;i&gt;z&lt;/i&gt;)) + &lt;i&gt;f&lt;/i&gt; (&lt;i&gt;z&lt;/i&gt;, &lt;i&gt;f&lt;/i&gt; (&lt;i&gt;x&lt;/i&gt;, &lt;i&gt;y&lt;/i&gt;)) + &lt;i&gt;f&lt;/i&gt; (&lt;i&gt;y&lt;/i&gt;, &lt;i&gt;f&lt;/i&gt; (&lt;i&gt;z&lt;/i&gt;, &lt;i&gt;x&lt;/i&gt;)) = &lt;i&gt;e&lt;/i&gt;&lt;/span&gt;. Notice that &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;f&lt;/i&gt;&lt;/span&gt; may not be commutative; it may be necessary to further specify that &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;e&lt;/i&gt;&lt;/span&gt; is the identity element in &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;X&lt;/i&gt;&lt;/span&gt; for operstor &#039;&lt;span class=&quot;MATH&quot;&gt;+&lt;/span&gt;&#039;, assuming that such is appropriate.<br /> <br /> ==== gloss words ====<br /> <br /> &lt;ul&gt;&lt;li&gt;[[natlang/en/Jacobi property|Jacobi property ; for binary operators; in a sense, it measures associativity.]]&lt;/li&gt;&lt;/ul&gt;<br /> <br /> ==== created by ====<br /> [[personal/krtisfranks|krtisfranks]]<br /> ==== vote information ====<br /> 1<br /> <br /> ==== time ====<br /> Wed Feb 15 17:15:55 2023<br /> <br /> [[comments.html?valsi=34388;definition=73146|[View<br /> Comments For This Definition]]] <br /> <br /> &lt;br/&gt;&lt;font size=&quot;+1&quot;&gt;Examples&lt;/font&gt;<br /> &lt;hr/&gt;<br /> &lt;dl&gt;<br /> &lt;/dl&gt;<br /> <br /> === Etymology ===</div>

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