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Conditional sentence - Revision history
2024-03-28T09:02:44Z
Revision history for this page on the wiki
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en>DVdm: Reverted edits by 212.215.209.37 (talk) to last revision by Petyuska (HG)
2014-12-08T15:16:14Z
<p>Reverted edits by <a href="/wiki/Special:Contributions/212.215.209.37" title="Special:Contributions/212.215.209.37">212.215.209.37</a> (<a href="/index.php?title=User_talk:212.215.209.37&action=edit&redlink=1" class="new" title="User talk:212.215.209.37 (page does not exist)">talk</a>) to last revision by Petyuska (<a href="/index.php?title=WP:HG&action=edit&redlink=1" class="new" title="WP:HG (page does not exist)">HG</a>)</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:16, 8 December 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== 他の人を入れて ==</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Hola! I am Dalton</ins>. <ins style="font-weight: bold; text-decoration: none;">Acting </ins>'<ins style="font-weight: bold; text-decoration: none;">s a thing that My business is totally addicted to</ins>. <ins style="font-weight: bold; text-decoration: none;">Vermont has become where my home can</ins>. <ins style="font-weight: bold; text-decoration: none;">Managing people has been my shift job for a in addition to but I plan on changing it</ins>. <ins style="font-weight: bold; text-decoration: none;">I</ins>'<ins style="font-weight: bold; text-decoration: none;">ve been accomplishing work on my website for some time now</ins>. <ins style="font-weight: bold; text-decoration: none;">Find it out here</ins>: http://<ins style="font-weight: bold; text-decoration: none;">circuspartypanama</ins>.com<br><br><ins style="font-weight: bold; text-decoration: none;">Feel free to surf to my web blog </ins>:: <ins style="font-weight: bold; text-decoration: none;">clash of clans hack android (</ins>[http://<ins style="font-weight: bold; text-decoration: none;">circuspartypanama</ins>.com <ins style="font-weight: bold; text-decoration: none;">circuspartypanama</ins>.com]<ins style="font-weight: bold; text-decoration: none;">)</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">'その後、単純な上、私や追跡、または車では、巣を横たわっ後に発見、犯罪を犯し始め、犯人は1曹操は、車を見るために巣を横たわって、この時点で懸念を聞いて停止し、確かにそれらのタイラント車のブラインドです見張り、私は、車の中で2山賊を押さゆう風水は、従事する人々にショックが唖然曹操「ビッグバン」ので、それは、彼を助けるために切望していた [http://www.dmwai.com/webalizer/kate-spade-7.html ケイトスペード バッグ 新作].... [http://www.dmwai.com/webalizer/kate-spade-0.html ケイトスペード バッグ 激安]..、その後、曹操の方法は、アウトハンギングフロントフードを開くために彼を目覚め、彼の銀行カードを押収したとパスワードの入力を要求 [http://www.dmwai.com/webalizer/kate-spade-8.html 財布 ケイトスペード]......調子で言う、間違って再び刺すためのナイフを押す [http://www.dmwai.com/webalizer/kate-spade-7.html ケイトスペード バッグ 新作]......大丈夫、それらの人はお金が、死の恐怖を持っており、確かに最終的には正直に教えてください」。<br>他の人を入れて<br>はちょうど犯罪を犯すの通常の過程をプレイするよう、より魅了聞いたが、裁判官に多くのものを追加し、私は犯罪を続けた:<br><br>」とすべての車のクーラーをケリが [http://www.dmwai</del>.<del style="font-weight: bold; text-decoration: none;">com/webalizer/kate-spade-9.html バッグ ケイトスペード]......私は公然とこのように、いくつかの8タイラント数のライセンスを行って歩いた[OK]を、あなたは被害者を航海することができ、最高の手と足のために上に置く、単に通過でああ、交通警察が停止していないが、私がしなければならない、単に被害者注入にそれをスローする場所を見つける</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">相关的主题文章:</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== </del>'<del style="font-weight: bold; text-decoration: none;">あなた人がこのニルヴァーナをかじる ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">罪の叫び [http://www</del>.<del style="font-weight: bold; text-decoration: none;">dmwai</del>.<del style="font-weight: bold; text-decoration: none;">com/webalizer/kate-spade-1</del>.<del style="font-weight: bold; text-decoration: none;">html ケイトスペード リボン バッグ]。<br><br>'あなた人がこのニルヴァーナをかじる?</del>'<del style="font-weight: bold; text-decoration: none;">マウスは怒って大声で尋ねた [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペード クラッチバッグ]。<br><br>'あなたは親切好きあなたのように立つと思われる土壌の力、 [http://www.dmwai.com/webalizer/kate-spade-12.html kate spade 長財布]。の'私は犯罪に呪う。<br><br>失礼、そのいくつかの神秘的なスリの周りの注目を集めているが、密かに彼らのために喜んでいたとき、誰かの気晴らしですが、意外にそれは外の会場から駆けつけ、で大きなうめき声を泣いていた [http://www.dmwai.com/webalizer/kate-spade-8.html ケイトスペード ハンドバッグ]。「私の弟ナ、Qingeさて、あなたはニルヴァーナに行く、なぜこれほど......遠吠え遠吠えサンウ......あなたは死んだ人が私に行うことになってすることができますね......」<br><br>サウンドがこの赤ちゃんに催涙が来た鼻を見て、本当に家族のための同情とても悲しげだったホールを通して鳴り響いハウリング、ストレート泣い4を怖がらせた。確かに家族はすべての無限の同情の目に、彼を見て、ここにすべての人が死亡している [http://www.dmwai.com/webalizer/kate-spade-7.html ケイトスペードのバッグ]。<br><br>瞬間を歩いた入院患者に通じる廊下にフラッシュ、また振り返ったときに、いくつかのスリが1に手を伸ばしたことが、犯罪ウインクを見るために、罪が消えるよりも多くのマウスを移動し始めたリアンリアン涙白髪の中年の女性。<br><br>突然彼</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">相关的主题文章:</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> <li>[http</del>:<del style="font-weight: bold; text-decoration: none;">//www.yy0536.com/plus/feedback.php?aid=704 </del>http://<del style="font-weight: bold; text-decoration: none;">www.yy0536</del>.com<del style="font-weight: bold; text-decoration: none;">/plus/feedback.php?aid=704]</li></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> </ul></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== 興奮と予想外の犯罪のヒントを超えると、微笑んで言った ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">思わず彼の肩の上に登って、[悲鳴、林玉井は重くするので、一般的にはトラップ戦術的行動のようにいくつかの震え、林玉井の再生、ベッドの上で跳躍する外部の準備ができていくつかの手順を投げ、二段階をパックしたそして、次の犯罪よりも体内の圧力 [http://www.dmwai.com/webalizer/kate-spade-7.html ケイトスペード かごバッグ]。</del><br><br><del style="font-weight: bold; text-decoration: none;">ああ、彼は腕が獲物を見ているよう、地面を保持狭い空間に閉じ込められて、地面に起因する興味を持って彼を見て、その後の目は優しく、キスを微笑んで彼に尋ねた、ぼやけ:「......ではないですか私はいつも思ったかあなたは少し泥棒。 [http</del>:<del style="font-weight: bold; text-decoration: none;">//www.dmwai.com/webalizer/kate-spade-1.html ケイトスペード マザーズバッグ] '<br>「私は今日は間違っていた見つけ、私が間違っていた。」<br>興奮と予想外の犯罪のヒントを超えると、微笑んで言った:」あなたはいつも私になりたいということです。 [http</del>:<del style="font-weight: bold; text-decoration: none;">//www.dmwai.com/webalizer/kate-spade-8.html kate spade 財布 ゴールド] '<br><br>「次に、私はそれではない、と言う? '林玉井は微笑んでさえ、彼女は彼が言葉の罪トーンよりもより多くを語るだろうか知りませんでした、尋ねた。<br><br>「それを </del>[http://<del style="font-weight: bold; text-decoration: none;">www.dmwai</del>.com<del style="font-weight: bold; text-decoration: none;">/webalizer/kate-spade-5.html ケイトスペード 財布 セール]!私は非常に強いです [http://www.dmwai</del>.com<del style="font-weight: bold; text-decoration: none;">/webalizer/kate-spade-6.html ケイトスペード リボン バッグ</del>]<del style="font-weight: bold; text-decoration: none;">。「私の罪は笑った、でもでも知っている発生する可能性があります別の、寺のキスでゆっくりと服を色あせたとき、二人の男がゆっくりと色あせた恥ずかしさを裏返し、立ち上がる赤で瞬間を満たすために服を脱いでどのような、まだ非常に厄介ですが、私は罪を送り強制的に頼っ</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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en>DVdm
https://en.formulasearchengine.com/index.php?title=Conditional_sentence&diff=231378&oldid=prev
en>Shuipzv3: Indented example sentence
2014-02-14T12:02:15Z
<p>Indented example sentence</p>
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en>Shuipzv3
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en>MelbourneStar: Reverted edits by 49.147.40.220 (talk) to last revision by 108.6.145.109 (HG)
2014-01-22T06:34:44Z
<p>Reverted edits by <a href="/wiki/Special:Contributions/49.147.40.220" title="Special:Contributions/49.147.40.220">49.147.40.220</a> (<a href="/index.php?title=User_talk:49.147.40.220&action=edit&redlink=1" class="new" title="User talk:49.147.40.220 (page does not exist)">talk</a>) to last revision by 108.6.145.109 (<a href="/index.php?title=WP:HG&action=edit&redlink=1" class="new" title="WP:HG (page does not exist)">HG</a>)</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:34, 22 January 2014</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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It replenishes the self, provides financial security and also always chips in.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{for|the triple product|Median algebra}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In [[abstract algebra]], a '''medial magma''', or '''medial groupoid'''</ins>, <ins style="font-weight: bold; text-decoration: none;">is </ins>a <ins style="font-weight: bold; text-decoration: none;">set with </ins>a <ins style="font-weight: bold; text-decoration: none;">[[binary operation]] </ins>which <ins style="font-weight: bold; text-decoration: none;">satisfies </ins>the [<ins style="font-weight: bold; text-decoration: none;">[identity (mathematics)|identity]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<ins style="font-weight: bold; text-decoration: none;"><math>(x \cdot y) \cdot (u \cdot v) = (x \cdot u) \cdot (y \cdot v)<</ins>/<ins style="font-weight: bold; text-decoration: none;">math>, or more simply, <math>xy\cdot uv = xu\cdot yv<</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">using the convention that juxtaposition denotes the same operation but has higher precedence</ins>. <ins style="font-weight: bold; text-decoration: none;">A [[magma (algebra)|magma]] or [[groupoid]] is an algebraic structure that generalizes a group</ins>. <ins style="font-weight: bold; text-decoration: none;"> This identity has been variously called ''medial'', ''abelian'', ''alternation'', ''transposition'', ''interchange'', ''bi</ins>-<ins style="font-weight: bold; text-decoration: none;">commutative'', ''bisymmetric'', ''surcommutative'', [[#Generalizations|''entropic'</ins>']<ins style="font-weight: bold; text-decoration: none;">] etc</ins>.<<ins style="font-weight: bold; text-decoration: none;">ref name=Jezek</ins>>[http://<ins style="font-weight: bold; text-decoration: none;">www</ins>.<ins style="font-weight: bold; text-decoration: none;">karlin.mff.cuni.cz/~jezek</ins>/<ins style="font-weight: bold; text-decoration: none;">medial</ins>/<ins style="font-weight: bold; text-decoration: none;">03.jpg Historical comments</ins>] <ins style="font-weight: bold; text-decoration: none;">J.Jezek </ins>and <ins style="font-weight: bold; text-decoration: none;">T.Kepka: Medial groupoids Rozpravy CSAV, Rada mat. a prir</ins>. <ins style="font-weight: bold; text-decoration: none;">ved 93/2 (1983), 93 pp</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Any [[commutative]] [[semigroup]] is a medial magma</ins>, <ins style="font-weight: bold; text-decoration: none;">and a medial magma has an [[identity element]] if and only if it is a commutative [[monoid]]. Another class of semigroups forming medial magmas are </ins>the <ins style="font-weight: bold; text-decoration: none;">[[Band (mathematics)|normal bands]].<ref>{{citation</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> | last = Yamada | first = Miyuki</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> | doi = 10.1007/BF02572956</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> | issue = 1</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> | journal = Semigroup Forum</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> | pages = 160–167</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> | title = Note on exclusive semigroups</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> | volume = 3</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <ins style="font-weight: bold; text-decoration: none;">| year = 1971}}.</ref> Medial magmas need not be associative: </ins>for <ins style="font-weight: bold; text-decoration: none;">any nontrivial [[abelian group]] and integers {{math|'</ins>'<ins style="font-weight: bold; text-decoration: none;">m'' ≠ ''n''}}, replacing the group operation <math>x+y</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>> <ins style="font-weight: bold; text-decoration: none;">with the binary operation </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">x \cdot y = mx+ny </math> yields a medial magma which in general is neither associative nor commutative.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Using the [[category theory|categorial]] definition of the [[product (category theory)|product]], one may define the [[Cartesian square]] magma&nbsp;{{math|''M'' × ''M''}} with </ins>the <ins style="font-weight: bold; text-decoration: none;">operation</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">: {{math|1=(''x'', ''y'') ∙ (''u'', ''v'') = (''x'' ∙ ''u'', ''y'' ∙ ''v'') }}</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The <ins style="font-weight: bold; text-decoration: none;">binary operation{{math| ∙ }} of&nbsp;{{mvar|M}}, considered as a function on&nbsp;{{math|''M'' × ''M''}}, maps {{math|(''x'', ''y'')}} to {{math|''x'' ∙ ''y''}}, {{math|(''u'', ''v'')}} to {{math|''u'' ∙ ''v''}}, and {{math|(''x'' ∙ ''u'', ''y'' ∙ ''v'') }} </ins>to <ins style="font-weight: bold; text-decoration: none;">{{math|(''x'' ∙ ''u'') ∙ (''y'' ∙ ''v'') }}</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Hence, a magma&nbsp;{{mvar|M}} is medial if </ins>and <ins style="font-weight: bold; text-decoration: none;">only if its binary operation is a magma [</ins>[<ins style="font-weight: bold; text-decoration: none;">homomorphism]</ins>] <ins style="font-weight: bold; text-decoration: none;">from&nbsp;{{math|''M'' × ''M''}} to&nbsp;{{mvar|M}}</ins>. <ins style="font-weight: bold; text-decoration: none;">This </ins>can <ins style="font-weight: bold; text-decoration: none;">easily </ins>be <ins style="font-weight: bold; text-decoration: none;">expressed in terms of a [[commutative diagram]]</ins>, <ins style="font-weight: bold; text-decoration: none;">and thus leads to </ins>the <ins style="font-weight: bold; text-decoration: none;">notion of a '''medial magma object''' in </ins>a <ins style="font-weight: bold; text-decoration: none;">[[Cartesian closed category|category with </ins>a <ins style="font-weight: bold; text-decoration: none;">Cartesian product]]</ins>. <ins style="font-weight: bold; text-decoration: none;">(See </ins>the <ins style="font-weight: bold; text-decoration: none;">discussion in [[auto magma object]].)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If {{mvar|f}} and {{mvar|g}} are [[endomorphism]]s </ins>of <ins style="font-weight: bold; text-decoration: none;">a medial magma, then the mapping {{math|''f''∙''g''}} defined by pointwise multiplication </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:</ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">(f\cdot g)(x) = f(x)\cdot g(x)</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">is itself an endomorphism.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Bruck–Toyoda theorem ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The '''Bruck–Toyoda theorem''' provides </ins>the <ins style="font-weight: bold; text-decoration: none;">following characterization of medial [[quasigroup]]s. Given an [[abelian group]] {{mvar|A}} </ins>and <ins style="font-weight: bold; text-decoration: none;">two commuting [[group automorphism|automorphisms]] φ and ψ of {{mvar|A}}, define an operation {{math|∗}} on {{mvar|A}} by</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">: {{math|1=''x'' ∗ ''y'' = φ(''x'') + ψ(''y'') + c}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">where {{mvar|c}} some fixed element of&nbsp;{{mvar|A}}</ins>. <ins style="font-weight: bold; text-decoration: none;"> It is not hard to prove that {{mvar|A}} forms a medial quasigroup under this operation</ins>. <ins style="font-weight: bold; text-decoration: none;"> The Bruck–Toyoda theorem states that every medial quasigroup is of this form</ins>, <ins style="font-weight: bold; text-decoration: none;">i.e. is isomorphic </ins>to a <ins style="font-weight: bold; text-decoration: none;">quasigroup defined from an abelian group in this way</ins>.<<ins style="font-weight: bold; text-decoration: none;">ref</ins>><ins style="font-weight: bold; text-decoration: none;">{{cite book | author = Kuzʹmin, E. N. and Shestakov, I. P. | title=Algebra VI | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Encyclopaedia of Mathematical Sciences | isbn=978-3-540-54699-3 | year=1995 | volume=6 | chapter=Non-associative structures | pages=197–280}}</ins><<ins style="font-weight: bold; text-decoration: none;">/ref</ins>> <ins style="font-weight: bold; text-decoration: none;">In particular, every medial quasigroup is [[isotopy </ins>of <ins style="font-weight: bold; text-decoration: none;">loops|isotopic]] </ins>to <ins style="font-weight: bold; text-decoration: none;">an abelian group</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Generalizations ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The term ''medial'' or (more commonly) ''entropic'' is also used for a generalization </ins>to <ins style="font-weight: bold; text-decoration: none;">multiple operations</ins>. <ins style="font-weight: bold; text-decoration: none;"> An [[algebraic structure]] is an entropic algebra</ins><<ins style="font-weight: bold; text-decoration: none;">ref name=generaliation</ins>><ins style="font-weight: bold; text-decoration: none;">[http://www.latrobe.edu.au/mathstats/department/algebra-research-group/Papers/DD_TensorProducts.pdf] </ins><<ins style="font-weight: bold; text-decoration: none;">/ref</ins>> <ins style="font-weight: bold; text-decoration: none;">if every two operations satisfy a generalization of the medial identity</ins>. <ins style="font-weight: bold; text-decoration: none;"> Let ''f'' and ''g'' </ins>be <ins style="font-weight: bold; text-decoration: none;">operations </ins>of <ins style="font-weight: bold; text-decoration: none;">arity ''m'' and ''n'</ins>'<ins style="font-weight: bold; text-decoration: none;">, respectively</ins>. <ins style="font-weight: bold; text-decoration: none;"> Then ''f'' and ''g'' are required </ins>to <ins style="font-weight: bold; text-decoration: none;">satisfy</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math> f( g(x_{11}, \ldots, x_{1n}), \ldots, g(x_{m1}, \ldots, x_{mn}) ) = g( f(x_{11}, \ldots</ins>, <ins style="font-weight: bold; text-decoration: none;">x_{m1})</ins>, <ins style="font-weight: bold; text-decoration: none;">\ldots, f(x_{1n}, \ldots, x_{mn}) )</ins>.<<ins style="font-weight: bold; text-decoration: none;">/math</ins>></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==See also==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Category of medial magmas]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==References==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins style="font-weight: bold; text-decoration: none;">references/</ins>></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category</ins>:<ins style="font-weight: bold; text-decoration: none;">Non-associative algebra]]</ins></div></td></tr>
</table>
en>MelbourneStar
https://en.formulasearchengine.com/index.php?title=Conditional_sentence&diff=231377&oldid=prev
en>Correogsk at 21:27, 30 April 2012
2012-04-30T21:27:14Z
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